
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UF00097868/00001
Material Information
 Title:
 Cohomology for normal spaces
 Creator:
 McWaters, Marcus Mott, 1939
 Place of Publication:
 Gainesville, Fla.
 Publisher:
 University of Florida
 Publication Date:
 1966
 Copyright Date:
 1966
 Language:
 English
 Physical Description:
 iv, 44 leaves. : illus. ; 28 cm.
Subjects
 Subjects / Keywords:
 Axioms ( jstor )
Category theory ( jstor ) Graduates ( jstor ) Homomorphisms ( jstor ) Integers ( jstor ) Isomorphism ( jstor ) Mathematical theorems ( jstor ) Mathematics ( jstor ) Topological theorems ( jstor ) Topology ( jstor ) Dissertations, Academic  Physics  UF Generalized spaces ( lcsh ) Group theory ( lcsh ) Homology theory ( lcsh ) Physics thesis Ph. D
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Bibliography:
 Bibliography: leave 43.
 Additional Physical Form:
 Also available on World Wide Web
 General Note:
 Manuscript copy.
 General Note:
 Thesis  University of Florida.
 General Note:
 Vita.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 021716480 ( AlephBibNum )
13289888 ( OCLC ) ACX4040 ( NOTIS )

Downloads 
This item has the following downloads:

Full Text 
COHOMOLOGY FOR NORMAL SPACES
By
MARCUS MOTT McWATERS, JR.
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVIERSITY' OF FLORIDA
April, 1966
To
Pat, Mom, and Dad
ACKNOWLEDGMENTS
The author would like to express his sincere
appreciation to his director, Professor Alexander R.
Bednarek, for his patience and encouragement, as well as for
his professional assistance in the preparation of this
manuscript.
The author recognizes a special debt to Professor
Alexander D. Wallace, for suggesting the topic for this
dissertation, and for providing a general perspective
throughout its development.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS .................................. iii
INTRODUCTION ..................................... 1
Chapter
PRELIMINARY RESULTS AND DEFINITIONS ...... 4
II. THE COHOMOLOGY GROUPS OF A SPACE MODULO
A SUBSET .................................
III. A GROUP ASSIGNMENT FOR A SPECIAL CLASS OF
SPACES ................................... 35
IV. FINAL COMMENTS AND QUESTIONS ............. 40
BIBLIOGRAPHY 43
BIOGRAPHIICAL SKETCH .............................. 44
INTRODUCTION
This paper presents a definition of cohomology
groups of a topological space relative to a subset of the
space. The definition employed was suggested to the author
by A.D. Wdallace, and is a modification of an earlier de
finition, also due to Wallace, which was exploited by
Spanier in [:7 ]. Both definitions, as will be seen below,
involve the notion of pfunctions and hence have their
roots in the works of Alexander and Kolmogoroff. These
definitions agree on compact Hausdorff spaces and, after
a suitable shift in dimension, yield groups isomorphic to
those in [6 J.
If X is a topological space we lot Xp+1 denote the
cartesian product of X with itself p+1 times and define
C (X) = [ rviv: Xp+1 > G), where G is a fixed, though arbi
trary, abelian group. Then C (X) is itself an abolian group,
if addition of two elements in C (X) is defined pointwise;
this group is called the group of pfunctions. If 2( is
an open covering of X, we set a (p+1) = U( u~tJ p+ u 6).
Then for each subset A of X, and each integer p O C, we
may define C (X,A) = (egp 6 C (X) and there exists an open
cover `U of A such that cD = 0 on 1 (P+1) n np+1j
For each integer p O there is a homomorphism, de
fined in Chapter I, F: C (X) > Cp+1(X) having the pro
perties that i;E = O and E[C (X,X)] is contained in Cp+1(X,Xr
Then other subgroups of C (X) may be defined by Z (X,A)=
C (X,A) OE B[Cp+1(XX)]; B (X,A) = E[C (1X,A)] + C (X,X)
(for p = 0, B (X,A) = (0)). The definition used by Spanier
of the pth cohomology group of the space X relative to the
subset A, denoted by H P(X,A), is the quotient group
Z (X,A) / B (X,A).
Our departure from this definition is effected by
redefining C (X,A) as the set of pfunctions, cP, for which
there exists a finite open cover 2/ of A such that c3 = O on
q/((p+1) ,, Ap+1. Similar distinction is found between the
Cech cohomology theory based on finite open coverings and
the Cech theory, advanced by Dowker, based on pairs of in
finite coverings.
Spanier showed that the theory developed in [ 7]
was a cohomology theory, in the sense of Eilenburg and
Steenrod [ 3 ], on the category of compact pairs. This re
sult then, carries over to the development presented in
this paper. In fact, most of the axioms of [ 3 ] will be
verified for general topological pairs (the only excep
tion being the Homotopy Axiom). Each time an axiom is veri
fied the axiom will be identified by a parenthetical in
sertion referring to the axiom e::actly as it is numbered
on page fourteen of [ 3 ].
In Chapter I we review some of Spanier's results
and definitions for use throughout this paper.
Chapter II presents our basic definitions and major
results. The development follows closely that of Wallace's
notes on Algebraic Topology [8 ]. Many of the theorems in
[ 8 ] are proved under the hypothesis that the topological
space in question is fully normal. Virtually all of these
same theorems, including the Reduction and Extension Theo
rems, are proved for normal spaces. We employ the resulting
generality to show that a connected, normal, Tl space with
trivial first cohomology group is unicoherent.
In Chapter III we disprove a conjecture that a
particular group assignment, defined for a special class of
spaces, will assign the trivial group to spaces which are
not locally connected.
We conclude with Chapter IV by discussing related
subjects (e.g. codimension) and open questions.
CHAPTER I
PRELIMINARY RESULTS AND DEFINITIONS
We review some of Spanier's results which will be
needed in the sequel. We assume throughout this paper that
G is a fixed, though arbitrary, abolian group. The term
" mapping" will be used to mean continuous function" and
Xp+1 will denote the cartesian product of the topological
space X' with itself p+1 times.
1.1 Definitionl: Let X be a topological space and let
p > 0 be any integer. Then C~X) = (0 ,:: XPtl > G1. For
each pair ,, p E C (X) define ( byi): Xp+1 > G by
( +4+u)(xO,...,x ) = 3(x'O,...,Jx ) + "(x0,...,x ).
1.2 Definition: 1) For any set P we let the diagonal
D(Pn) of Pn bc U((xjn x 6 Pi.
2) If f: X > Y is a function from a
topological space X to a topological space Y, define
f": C (Y) > C (X) by [f (0~)](xO,...,x )
9[f(x0),...,f(x )].
1.3 Definition: We define ~: C (X) > CP+1(X) by
[( (0)](x0,..., xp+1i / i=0(1)i U(xO, i...x ...,xp+
where (x0" i,,xptl) = (xO..,xi1, xi+1,..'xpel)6 Xptl
1.4 Definition: Let f,g: X > Y. Then define for p > O,
D: C (Y) > Cp1(X) by [D(g)](x0,...,xp1
O~(1) m[g(n).9x0),...,gix ), f(x ~),..fxp1] The pro
perties of C (X), f F, and D, of which we will constantly
make use,are collected in the following:
C (X) is an abelian group
ff and are homomorphisms
DB + FD = fR gR if p > 1
Db = fd g" if p = O
Theorem: 1)
2)
r
5)
CHAPTER II
THE COHOMOLOGY GROUPS OF A SPACE MODULO A SUBSET
We shall adhere to the definitions and notation of
Chapter I, and shall introduce new~ definitions, conventions
and algebraic lemmas as they are needed. We omit the proofs
for these lermmas if they are available in standard texts.
2.1Notation: If ZA is a family of sets U(P =
U(IA A 6 u). If f: X 9 Y is a function from a space X to
a space Y, if A C X and if f(A) c BC Y, then we write
2.2 Definition: If A c X we define C (X,A) = (9180 6 C (X)
and ?; a finite open cover 14 of A 9 9 = O on 7tr(p+1) 0 Ap+1ll
We often write C (X) =Cp(X,A) =C (X,X).
2.3 Lemma: 1) C (X,A) is a subgroup of C (X)
2) If f: (X,A) ) (Y,B), f continuous, then
f"f[C (Y,B)] CC(X,A).
3) If A C X, then B[C (X,A)] c Cpfl(XA).
Proof: 1) Let 3, 9b 6 C (X,A) and let ;?/ and 2/ be
finite open covers of A 9 cp = O on A(pfl) AAt and $ = O
on rV(pfl) n Ap+1, then 2 == (u nvlu 6 ;U and v 61/) is a
finite open cover of A such that ? i = O on ZJ(p+1) n Ap+1
Th~us 9 ; E C (X,A).
2) Let ao 6 C (Y,B), then a finite open
cover 22 of B such that cp = O on 1/(p+1) Ptl. Then
4/ = (fl~u) Iu 6 L( ) is a finite open cover of A such that
ffi"() = O on 1/(p+1) r! Ap+1. Thus ft(C")F C (X,A).
3) Let cP 6 C (X,A), then a finite open
cover CM of A such that rp = 0 on fg(p+1) n Apt1, and one
easily checks that 6rC = O on qj(pb2) n A ,~ hence
dC Cp+(,)
2.4 Definition: Let A be a subset of a space X. Then
1) Z (X,A) = C(X,A) n a [C XA
2) B (X,A) = C(0)O p =
C (X,A) + E[C (1X,A)]: p 1 .
2." Lemma: 1) Z (X,A) is a subgroup of C (X,A)
2) B CA c X > C (X,A) CC (X,B): hence
Z (X,A) c Z (X,B)).
3) B p(X,A) is a subgroup of Z (X,A)
4) If A is a closed subset of X and
~p 6 Z (X,A), then 13 a finite open cover l of X with
(i) Bco = 0 on cU(Ib2) (ii) 7 = O on gU(p+1) n A~t
Proof: 1) and 2) are clear. For 3) we recall that
~g= 0 and use Le~mma 2.3, part throo, to establish the ap
propriate inclusions. For 4) assume m 6 Z (X,A), then
i" 6 C (X,A) and ~ E COpfl(X). Hecnce 'I a finite open cover
of A 9 C3 = O on 9 (ptl) n Ap+1 and a finite open cover
2/ of X 9 Be = O on 7/(p+2). Let 8= UAU(XA) and Th'
(v n o v E rV and o E (7 ), then (1J is a finite open cover
of X satisfying (i) and (ii).
2.6 Definition: If A is a subset of a space X, we define
H (X,A) = Z (X,A) / B (X,A).
2.7 Lemma: If f: (X,A) > (Y,B) is continuous, then
1) fulZ (Y,B)] c Z (X,A)
2) f [B (Y,B)] c Z (X,A).
Proof: 1) If g 6 Z (Y,B), then 6 C (Y,B) and
W 6 Cpfl(Y,B), hence f (g) E C (X,A) and gOfH()] = f"[EQ]
Cp+1(tX). Thus f"(c9) 6 Z (X,A).
2) If 9 F B P(Y,B), then cD = V + 56 where $
Cp(Y) and 6 c Cpl(Y,B). Hence f.(c9) = fF($) + fR(E6)=
f (s) + sOf (9)] 6 Cp(X) + E[Cpl(X,A)] = B (X,A), p 1 1.
The case p = O is clear.
2.7 Induced Homomorphism Theorem: Let P be a group with
subgroup PO, let Q be a group with subgroup OO and let f be
a homomorphism of P into Q such that f(PO) C 00. Then if
a: P  P/PO and 8: Q > Q/OO are the natural homomor
phisms, there exists one and only one homomorphism g such
that ga = Bf.
2.8 Theorem: Let f: (X,A) > (Y,B) be continuous, let
P:Z(X,A) > H (X,A) and 6: Z (Y,B) > H (Y,B) be the
natural homomorphisms and define f :Z (,)>Z(,)b
fO(@) = f (9) for each 9 E Z (Y,B). Then R a unique homomor
phim f: H(Y,B) > H (X,A) such that af =f
Proof: Induced Homomorphism Theorem.
2.9 Theorem: 9 E ZO(X) if and only if c: X > G is con
tinuous in the discrete topology of G, and (Ql~g)g 6 G)
is finite.
Proof: If 9 E ZO(X) then 7 a finite open cover of X
such that 6m = O on Li(2). Let cp(x) = g 6 G, then Tu 6 4
with x 6 u. Now if y 6 u then bc(x,y) = cp(y) W(x) = 0.
Thus V(y) = W(x) = g and u c cp(g). Hence c3 is continuous.
Set `U= Cu.11 < i <_ n) and let g. 6 G be such that
1 1
thus if x 6 el(g) for some g 6 G, then an integer k be
tweenI an n wth x6 9 k).Henc (g)= 9 gk) and
we have [vl~g) g 6 G] c ( (lg )l1 < i < n) U[0). Suppose
now that rp: X > G is continuous in the discrete topology
of G and that (9 (~g)lg 6 G] is finite. Then 7 g 6 G such
that x 6 G such that x E m (g), and if y 6 u (g) we have
bP(x,y) = y(y) m(x) = g g = 0. Hence bcp 6 CO0(X) and
*a 6 ZO X).
2.10 Theorem: If G 4 (0), then X is connected if and
only if H (X,x) = (0) for each (for some) x 6 X.
Proof: Let X be connected, then if W 6 ZO(X,x)
HO(X,x), O 6 CO(X,x) n gl[CO1(X)] c CO(X) A 6l[C O(X)] =
ZO(X). Thus cp is continuous in the discrete topology of G
and is thorofore a constant function. Hence c$(x) = 0 >
n(X) = 0, and we have (0) = ZO(X,x) HO(X,xv). Now assume
X is the union of two disjoint open sets A and B. Fix x in
X and assume x 6 A. Define cC: X > G by cp(A) = O and
cp(B) = g \ 0. Then cp is continuous in the discrete topology
of G and [w1(g) g 6 G) = (A,B) is finite. Thus rp 6 ZO(X),
but x 6 A open > rp 6 CO(X,x), so we have Ov 6 p ZO(X,x)
HO(X,x), a contradiction.
Conventions: If A is a subset of a set B and if
f: A > B is defined by f(x) = x for each x in A, then f
is called the inclusion map of A into B and is denoted by
f: A c B.
If f: G > H is a homomorphism from the group G
into the group H, then [h h 6 H and h = f(g) for some
g 6 Gj, the image of f, will be denoted by I(f) and
(g f(g) = 0), the kernel of f, will be denoted by K(f).
2.11 Theorem: Let X be a space, let B cA C X, let
y: Z (A,B) > H (A,B) and a: Zp+~1(X,A) > Hp+1(X,A) be
the natural homomorphisms and let t: A c X. Then
1) For each h 6 H (A,B) there exists such a
c9 6 C (X) that ytff(g) = h and S EQ Zp+1(X,A).
2) If o, 19 E C (X), if ti"(Q), t (t) 6 Z (A,B) and
if yt (0p) = vtH($r), then ~(rp) 6 Bp+1(~X,A).
rr1 1 isahmmrhs rmIp(A
3) 6 = a~t y sahmmrhimfo AB
into Hp+1(X,A).
Proof: We first note that if g 6 C (X) with t*e(9) =
9 6 C (A) then $ 6 C (X,A). For B a finite cover of A by
sets open in A, with 9 = 0 on ((p+1l), hence if we
write U = (vi nA 1 < i < n) where v. is open in X and
(V=Iv.11 < i < n), we have that 1J is a finite open
cover of A 3 9 = 0 on c (p+1) n A~
1) Let b E H (A,B), then r cp E Z (A,B) with y(3) = b.
Define B: XPt1 > G by the equations '1(xO,...,~x )=
W(xO"...x ) if (xO,..., )p E Atl ~(xO,...x '") =O if
(xO,...,x p) 6 Xp+1 A .l Clearly t (l) = 9, hence
y~t"(a)] = y(a) = b. Now au 6 Cp+1(A) and tff[EI] = 6[tP(S)]=
EP ~ E hec 96C+(X,A).
2) Let cp, $ E C (X); t't(c), tP($) 6 Z (A,B) with
yti(c0) = vt (1), then tR(cp5) 6 B (A,B) and hence t (gf)
= 1 2 (~ ) where el 6 C ~(A) and 62 C (1A,B), p > 1.
Let ep2 6 t (9 r3) c Cpl(X) and define cp1 C1 on A t;
cpl J 2 Q) on Xp+1 p1 hnt(1 1 hence
tP (99 0
[Ct";(92)] = tf [ (CS2)], hence (cp1) cp1 2P) on all
Xp+1. Thus b(oS) i(cpl) 66 2;p) = O, from which E(e') =
6(~ ); but t"(pl) 1 C(A) > (1 6 C XA.Tu
F(cpl 6 E[C (X,A)] and 6 (CP9) = (cpl) 6 B (X,A) The case
p = 0 is trivial.
3) 8 is well defined is immediate from 1) and 2).
2.12 Definition: A sequence of homomorphiisms
b b h h
0 1 n1 n
A > A > .> A > A > .. is
O 1 n n1
oxact if and only if hO is a monomorphism and I(h )=
K(b ~) for i > 0.
Notation: The following notations will hold through~
Theorem 2.16: B cA c X, j: (X,B3) c (X,A), i: (A,B) c (X,B),
and t: (A,0) c (X,0). a: Z (X,A) > H (X,A),
8: Z (X,B) > H (X,B), and y: Z (A,B) > H (A,B) are the
natural homomorphisms.
2.13 Lemma: 1) j O(X,A) > HO(X,B) is a monomorphism.
2 I j = K i for p 2 0.
Proof: 1) Let W 6 ZO(X,A) E HO(X,A) and suppose
j p(x) = 0 for each x in X, then qCj(x)] = rp(x) = 0 for
each x in X. Thus j a(0) = O > Bj (9) = O > j"V = O
> o = 0 > am~ = 0 > j is a monomorphism.
2) We first show that I(j ) c K(i ). If
c9 6 Z (X,A) then B a finite open cover %( of A 9 ri = 0 on
t (p+1) ,A t Let J = (u n Alu 6 ii then 0J is a
finite open cover of A with i'jF'm = 0 on L (Ptl). Hence
i jg F Cp(A) C B (A,B). Since the natural homomorphisms
are onto the proof is complete. To prove K(i ) c I(j ) we
suppose i"b E B(A,B), then i"y = (92) where
'2l 6 Cp(A) and cp2 6 Cp1(A,B), pi i Let 9 6 C (1X,B)
with i''S = 02 and define ;I: Xp+1 0 G by $ = ee1 on Ap+1
and $ = r3 ~(6) on Xp+1 At.l Thnp6C(X,A) and if
(xO,...,Ix ) is in Ap+1 we have )(x0,,x X) = Pl(x0,...,x )
=i"9(xO...,x ) b32(xO,...,x ) = P(xO,...,x )
~i 9(xO...,x = o(xO,...,x ) i p~(xO,...,x )
9(x0"...x ) Bq(x0,...,x ). Thus 09568] = EP 85 9
P 6 Cp+1(X). Hence 3 6 Z (X,A) and P j''# =
EB 6 ECp1(X,B)] c B (X,B). Again the case p = O is trivial
and inclusion follows from the fact that the natural homo
morphisms are onto.
2.14 Lemma: I(i ) = K(8)
Proof: I(i ) c K(6). Let p 6 Z (X,B), then
rDEt i't hence t "cp = b. But cp E 6 CO (~X)], so
9q E Cp+1(X), a subset of Bp+1(X,A). The inclusion is thecn
clear. To show K(65) c I(i ) we suppose t3E Z (A,B),
.I 6 tkl~9) with 6; E Bp+1(XA). Then 89 = 9 ~+ ("32) where
..l 6 Cp+1(X) and r?2 E C (X,A). Define 9 = 02, then
a; = 9l 6 Cp+1 (X) hence 6 E Fl[Cp+1[X)]. Also tV =
9 6 C (A,B) and hence $ 6 C (X,B); therefore 6 E C (X,B)
and we have 9 E Z (X,B3). Now cp i"6 = c3 i"# i 92
0 9 i'2 ic2 6 C(A) c B (A,B).
2.15Lomma: I(d) = K(j )
Proof: I(3) c K(j ). Let 9 6 t 9lc with Po 6 Z (A)B)I
then tR@ = 9 E C (A,B) > $ E C (X,B). Hence if E 6[C (X,B)]
c Bp+1(X,B) and j"EWl = Et. To complete the proof we appeal
to the natural homomorphisms. To shows K(j ) c I(8) we let
r, E Zp+1(X,A) with iurp E Bp+1(X,B). Then :P = j 0p =
Cl 82) where m1 6 Cp+1(tlX) and rD2 E C (X,B). Thus
i"*3; 6 C (A,B) and biu(rp ) = i b(P ) = i'cq i 0 6 Cp5+1 A).
Hecnce i'92 6 Z (A,B3) and a2 6 t i (2), so 6t" '7
S((0 ) 6 Cp+1(X,B) and cp 6 ("2) =1 E C p+1 X) c Bp+1(XA).
2.11 Theorem: IO(A _i IO(X,B)) > HO 8
DI(XA > w"(X,A) > Hn(X,B) i
H~n(A,B) > Hn+1(X,A) i> ... is exact. (Axiom 4 c).
Proof: The three previous Lemmas.
2.17 Corollary: H (X,X) = O, for any space X and any
Pi O.0
Proof: One takes B = A = X in the previous theorem
and easily verifies that i j is the identity function, as
well as the zero function, on H (X,X).
2.18 Corollary: If A is a connected subset of a space X
or~~~ ~~ iA O te :H(A) > H (X,A) is the zero func
tion.
Proof: Recall that HO(A) Z Z(A) and HO(X) ZO()
Each W E ZO(A) is continuous in the discrete topology of
G and hence is a constant function, since A is connected.
The constant function r$: X > G defined by extending cp to
all X is such that i"JI = c, where we have taken B = 0.
Thus i", and consequently i `, is an epimorphism and hence
0
H (A) = I(i ) = K(O) if A is not empty. But A = is clear.
2.19 Theorem: 1) If f: X > Y and if g: Y > Z, then
(gf) = f~g".
2) If f: (X,A) >) (Y,B), if g: YB >
(Z,C), and if f and g are continuous, then (gf) =f g.
(Axiom 2 c).
Proof: 1) [(gf) e](xO,...,x ) = '(gf[x0),...,g4f[x ])
= "v(f[x0],...,f[x ] = [fP(g"q)](xO',.x )=
[f~g']3(xO,..,x ). Thus (gf)"Y = [f"g"] m if cp 6 C (X).
2) Let o: Z (X,A) > H (X,A), o: Z (Y,B) 
H (Y,B) and v: Z (Z,C) > H (Z,C). Then f gJ (v ) = f "oqa
2.20 Theorem: Let f: (X,A,B) > (X',A',B') be continuous.
Define u: (X,A) > (X',A'), v: (X,B) > (X',B') and
w: (A,B) > (A ,B') by u(x) = v(x) = w(x) = f(x). Then the
ladder"
6 > p(X I,A') i> H (X ',B')~ > H (A',B')
u v Iw
6 >H (XA)  >i H (,) > H (AB) 6
is analytic (each rectangle of the ladder is analytic).
(Axiom 3 c).
Proof: It is trivial to verify that jv = uj and
uu u& +St & XX
that iv = wi, hence v j = j u and v i = i w Now let
P.:Z(X',A') > H (X',A'), 5': Z (X',B') 9 H (X',B'),
y'; Z (A',B') > H (A',B'), a, 5, and y as usual, and
t': (A',0) c (:K',0). Then if h 6 H (A',B'), cp E v l~h),
and Q F t' (0~c) with a'89t = 5h. Thus u *h=u28
u" $ = clu 9. But t['ur$ = w"cp implies u''# E t"1
khence u"B 6 t"1 y1 (vw''v) and soGu#=6ywm wh
since w h =vw''Q. Thus Ew = u L;. We now compute thus,
MM~~~~ &e 4 XX XX MM
wI ]=(w~i )j = (i v )j =i (v j ) =i (j u ) = X ]
K + + MM Y
Similarly u ti = pi v and v j 6 = j 6w.
2.21 Corollary: If f: (X,A) > (X',A') is continuous
and if f(X) c A', then /: H I(X',A') > H (~X,A) is the
zero function for each p O .
Proof: Recalling that H (X,X) = (0) and noting that
f: (X,X,A) ) (X',A',A') we use the previous theorem to
assert that if b 6 H P(XI,A') then f (h) =f Xj (h)=
j "f (h) 6 j [HP(X,X)] = (0).
2.22 Theorem: If X is connected then HO(X) G, and if X
is a point space then HO(X) EG and H (X) = (0) for p 0.
(Axiom 7 c).
Proof: If X is connected then each h in ZO(X) is a
constant function, hence we may define f: ZO(X) > G by
f(h) = h(X). Clearly f is a monomorphism. If g F G define
h : y x for each x in X. Then h is in
ZO(X) and f(h ) = hence f is an isomorphism and we have
G ZO(X EO(X). Now assume p O and X = (x). If
a 6 Z (X) and cW(xp+1)= g, then rp(xP+2) = p1(1) g = 0,
i=0 p
rl(x ) = g, then g3(xptl=~j (1) g = (1) g=
Si=0
(1) Wp(xp+1). Thus 5$ = or b(Q) = 9, and we have
? E E[Cpl(X)] cB g(X). Hence Z (X) CB g(X). Therefore
equality holds and H (X) = [0).
2.23 Lemma: If f: (X,A) c (X,A), then f :H XAC
H (X,A) for p O. (Axiom 1 c).
Proof: We have f (am) = af (W) = acp.
2.24 Theorem: If f: (X,A) > (Y,B) is a homeomorphism,
then f H(Y,B) > H (X,A) is an isomorphism.
Proof: Let g: (Y,B) 0 (X,A) be such that gf(x() =
x for each x in X and fg(y) = y for each y in Y. Then
fg: (Y,B) c (Y,B), hencegf (fg) : H (Y,B) C H (Y,B)
and gf: (X,A) (X,A), hence f g : H"XA 'XA.Tu
f is one to one and onto.
2.25 Lemma: Let A c X and u be an open subset of the in
terior of A, thnC(X,A) n C (X,Xu) = PX.
Proof:~~ CeryO(X) c C (X,A) A C (X,Xu). If
9 C(X,A) then there exists a finite open cover Ul o
Such that cP= O on 7 (p+1) Ap+1, an 6C(X,Xu)
implies there exists a finite open cover QA2 of Xu 3
P3 = O on 17 (P+1) ,A .l Then if 14 denotes the collec
tion of open sets obtained by intersecting the members of
fl with the interior of A, and the members of C/2 w1ith
the complement of u closure, we have that 2( is a finite
open cover of X such that 0 = O on cl/(p+1). Thus a 6 COf(X).
2.26 Weak Excision Theorem: If k: (Xu,Au) c (X,A) and
if u is an open set contained in the interior of A, then
kc : H X,A) > H (Xu,Au) is an isomorphism. (Axiom b~ c).
Proof: Let mo 6 Z (Xu,Au) and define ): XPf1l G
by the equations ( = 9 on (Xu~p+1: j = O elsewhere.
NowJ p 6 C (Xu,Au) implies there exists a finite cover
C1X of Au by sets open in Xu such that ci = 0 on
] ((p+1) 0(Au)ptl. Writing 9A = [v. A (Au) 1 < i
where v. is open in X and `1/ = [v.11 < i < n) Ulu), we have
that 4/ is a finite open cover of A and rl = 0 on l(p+1l) n
At hs$6C(X,A) and #~ 6 Cp+(X,A), with 6k 9l = 6c 6
Cp+1(Xu) Hence k" 5 CP+1(Xu) and so Fg 6 C +(X,Xu).
By the previous lemma then, alrE 6Cp+1(lX). Thus ii 6 Z (X,A)
and ki is onto. To see that kX is one to one we let
o 6 Z (X,A) such that k (9) 6 B (Xu,Au). Then k (9)=
01 + (2), for some pl 6 C (Xu) and 02 6 Cpl(Xu,Au).
Define d: XP > G by J12 = 2 on (Xu)P #2 = 0 elsewhere.
Then as before 3r2 6 Cp1(X,A) hence 5(02) 6 C (X,A) and so
s E($2) 6 C (X,A). Now k"[v E($ )] =k 0 ~(k62)
k'g 8(qp) = pl 6 COP(Xu). Thus cp (32) 6 C (X,Xu).
Again by the previous lemma, we know there exists li1 6 C (X)
such that 9 E(92) =1. Hence cp = fl +(2) 6 CO(X) +
b[Cp1(,) = B X,A).
2.27 Theorem: Let f,gJ: (X,A) > (Y,B) and let 9 E C (Y).
Let 2/ be an open cover of Y such that 9r = 0 on 12/(P+2)
and cp = e on 1(pel) p+1. Finally let U be a finite
open cover of X such that u E 14 implies flu) U g(u) U v
fo smev 1 .Then f Pge6Z(X,A) and f 9g'
B (X,A).
Proof: (t is a finite open cover of A and if
(xg ,.x ) 6 up+1 AA ,~ where u 6 U1, then x. 6 u
for 0 < i < p. Hence there exists v 6 if such that
flu) U g(u) c v. Then f(x ), g(x ) 6 v n B for O < i < p,
hence (f[xO]'...,f[x ]), (g[xO],...,g]x ]) vp+1 gpfl
and f p(xO,...,x X), g"W~(xO',..., ) Thus f 9c,
gkc E C (X,A). Also [Ef R(Cp) (xO,..., xp+1 [f6y
(xO,...,xp+1) = BEv(flxO],...,f[xp+1]) = O and (EgP(c)]
(f[xO],..,f[xp+1]) and (g[xO],..,g[xp+1]) are in vp+2
andQ~v+2)= Hence Bgf 0 n E Cp+(X) and con
seqenty fP, ~v Z(X,A). Now is a finite open cover
of X and if (x0,...,xp ) up+1 then there exists v 6 7/
such that f(x ), g(x ) 6 v, and consequently [g(x0),..,g(xi
f(x ),.,f(x )] E vp+2 for O
"p[g(x0),..g(xix ) f(x ),...,f(x )]= O and so
' i=0(1) v[g(x0) ,... ,g(x ), f(x ),..., f(x )] =
D ~(xO, ...,x ) = Hence D~rS = O on gy(p+1) which implies
DOc 6 COP(X), and we have that for p = O (fP g")0 =
DB( ", Cp(X) C B (X,A). Also B[DW] 6 "[Cp1(X,A)] if p 1,l
for if (xO,.., ) 6 uP AA then D;3(xO,..., xp1)
O(1) Q[g(x0)...,g(x ), f(x ),...,f~xp1)]. But f(x ),
g(x i) 6 v r0 B for some v E !/ and O < i < p1, hence
[(xg)9(xO),..gx) f(x ),...f(xpl) E +P1 n BP+1 which
implies '2[g(x0),'...,(x ), f(x ),...,f(xp1)]= O and
therefore Dy(u DA ) = 0. Thus DPg 6 C (X,A) and
f 9 _ g"Wp = ED7 + DbP 6 B (X,A).
The following Corollary is an extension of a
fundamental lemma proved, at Wallace's suggestion, by
Capel in [1].
2.20 Corollary: Let B be closed in the space Y and lot
h 6 H (Y,B). Then there exists such a finite open cover,
a y(h), of Y that; if f,g: (X,A) 0 (Y,B) are maps such
that for each x in X,f(x(), g(x) 6 v(x) for some v(x) 6 ()
then f (h) = g (h).
Proof: Let h 6 H (Y,B), then there exists cp E Z (Y,B)
such that BeP = h. Since B is closed we apply lermma 2.5,
part 4, to yield a finite open cover 1/ of Y with im = O on
C/(p+2) and cp = on Bp+1 y(p+1). Define 1R = (f1~)
gl~v) v F \f ). Then / is a finite open cover of X such
that if u 6 U1, there exists v 6 9/ such that flu) U g(u) c
v. Thus f 0 g"rC E B (X,A), and f (h) = g (h).
2.29g Definition: Let C~and 9 be families of subsets of
a space X and let C c X.
1) r2 efines & ( GL < Ig ) iff A E implies
A cB for some B 6 .
2) St(C, C1 ) = U[AA A 61 and A nC 0 O.
3) St( (() = (St(A, CL ) A 6 C1 ).
4) C1 star refines iff St( 6 ) < .
We will need the following two results which are well known
and will be stated without proof.
2.30 Theorem: A space X is normal iff for each finite
open cover 12 of X there exists a finite open cover 3J' such
that St( (f ) < $1
2. 31 Modification Le~mma: If A is a subset of the space
X, if Elis an open cover of X, if 2! is an open cover of
X such that St( ff) < 1F and if P = St(A, qA ) then there
is a function f: (X,P) ) (X,A) such that
i) f(x) = x for x 6 A U (XP)
ii) If u 6 21( then there is a v E j/ such that
u U flu) c v.
2.32Definition: (X,A) is a normal pair iff X is a
normal space and A is a closed subset of X.
2.33 Notation: If i: (P,0) c (R,S) and if b 6 H (R,S),
then h (P,0) = i (h). If A c X we let Ao denote the interior
of A, and A denote the closure of A.
2.3lC Expansion Lemma: If (X,A) is a normal pair and if
h 6H (X,A), then there exists an open set p oA and
hO C H (~X,PX) such that h0(X,A) = .
Proof: Let b 6 H (X,A), 3 6 Z (X,A) such that
ci(g) = b. Then there exists a finite open cover (1/ of X
such that W(vp+1 D A ") = O and 6h(vp+2) = By theorem
2.30 there exists a finite open cover 14 of X such that
St(9/ ) < \f. By lemma 2.31 there exists f. (X,St(A, 24)) 
(X,A) with f(x) = x for x 6 A U (XSt(A, 1(C )) and such
that u 6 24 implies there exists v 6 1/ such that u U flu) c
v. Let P = PO be such that A cP cP~ c St(A, 'U ), then
f E Z (XP ), for if (xO,...,x ) 6 up+1 n p~p+ then
f(x ) F flu) n A and a v E 'l/such that u U flu) c v. Hence
f q(xO,...,x ) = W(f[xO],..., f[x ]) E 0(vp+1 AA 1)= O
then (flxO],...,ffxp+1]) F v 2 for some v 6 17, hence
6f 3(xO, ...,''xp+) =f be3(xO...,xp+1) = 6PP(f[xO],..,f[xcp+1
6 E(vP+2) = .Tu 6 Ol[C+1X)] and so fDm 6
Z (X,P ). Now if i: (X,A) c (X,A), then we have i,f: (X,A)>
(X,A) and c0, 9A 01/ satisfy the conditions of theorem
2.27, hence is f~ E B (X,A). Consequently if we let
s: Z (X,P ) > H P(X,P ) denote the natural homomorphism,
then taking hO = "[f 9 ehv O(S)=b
2.35 Theorem: Let (X,A) be a normal pair and let A C Mo
M c X. Then, if h 6 H (X,M ) and if h (X,A) = 0, there
exists an open set N such that A cN cNX cM and h (X,N )=
0.
Proof: i: (X,A) C (X,M ). Let h 6 H (X,M ),
9 6 Z (X,M ) with SW = h. Then i (h) = O implies i4q 6
B (X,A), hence 0 = 91 + 6 (P2) where V1 6 C (X), s 26 Cp1(X,A)
for p > Thus there exists q1/ a finite open cover of X
such that v2 = O on 1(P 7 A 1J a finite open cover
of X such that el = on /(p+1), and cll a finite open
cover of X such that cq = 0 on 1 pl ~+.Lt1
(Vl 2 ~ 3 i 6 4i i = 1,2,3, }, then q/ is a finite
open cover of X satisfying the same conditions as 11l'
t /2, and q/ separately. Let 14 be a finite open cover of
X such that st(fA ) I If and let N = No be such that
A cN EN` c [St(A, qX ) n M]. Then there exists
f: (X,St(A, qX )) > (X,A) with f(x) = x for x 6 AU
[XSt(A, 12)] and u 6 ZA implies there exists v 6 q/ such
that u U flu) c v. Now as before i,f: (X,N ) 9 (X,N )
and c9, LA 1 satisfy the conditions of theorem 2.27,
hence is f r; 6 (X,N n). But fe = f 'eplf (2)
f"Q1l app"~). By a proof similar to that of the expan
sion lemma f 0 6 Cpl(XN ), and if (xO,...,x ) F up+1
then there exists v such that u U flu) c y so that
(f~x ],...,f[x ]) E vp+1. Thus flb1(xO,.., X)=
Ol(f[xO],...,f~x]) = O and consequently f~al E COP(X). But
then f'p E B (X,N ) so that 7 f' 6 B (X,N ) implies
P (X,N ). Finally Pm = h and we have b (X,N ) = O.
2.3'6 Notation: Through the Map Excision Theorem let
(X,A) and (Y,B) be normal pairs and let f: (X,A) > (Y,B)
be a closed map such that f takes X A topologically onto
Y B, that is, g: (XA) > (YB) where g(x) = f(x) is a
homeomorphism.
2.37 Lemma: If MI' M2 0 A, then Mi lp fM ) for i = 1,
2, and f(MI M2) (1) 2) = [B U f(M1)
[B U f(M2 '
2) If M is an open set containing A, then B U f(M)
is open.
3) If B3c N = B U ffl(N)
4) If N is an open set containing B then f1 N )
[fl(N))]
2.38 Lemma: Let M be an open set containing A and de
fine fO: (X,MY _)> (Y,B U f(M )) by f0(x() = f(x) for
each x 6 X. Then fO: H (Y,B U f(M )) H (X,M ).
Proof: Choose N = NO such that Ar Nc N cM M.
Then if we let fl: (XN, M N) > (Y [B U f(N)],
[B U f(M )] [B U f(N)]) be defined by fl(x) = f(x), we
have that fl is an onto homeomorphism. For f(XN) =
[B U f(X)] [B U f(N)] = Y [B U f(N)] and X A contains
X N. Similarly f(M N) = [B U f(M )] [B U f(N)]. Now
f(N ) = [f(N )]* implies [f(N)]* c f(N ) c f(M), thus
(B U f(N)] = B U [f(N)]X = B U [f(N)]* CB U f(N ) c
B U f(M) c B U f(M ) and hence B U f(M) open implies
(B U f(N)]~ c (B U f(M )]o. Now let i: (Y [B U f(N)],
[B U f(M )] [B U f(N)]) c (Y,B U f(M )); k: (XN, M N) c
(X,M ). By theorems 2.24 and 2~.26 we have that flJ i an
~~~c *nl~3
k are isomorphisms, hence fO k f i is an isomorphism.
2.39 Map Excision Theorem: f : H (Y,B) H (X,A).
Proof: Let h 6 H (X,A), then there exists M = Mo D A
and ho 6 H (PX,M ) such that i (ho) = h; i: (X,A) c (X,M ).
Let h1 0 (gg*lho) 6 H P(Y,B), j: (Y,B) c (Y,B U f(M ));
then f ~l(ho)i) O (h)= O h i (h ) = .
Thus f is an epimorphism. Let h 6 K(f ) then there exists
N = NO DB and ho 6 H (X,N ) such that 1 (ho) = h, 1: (Y,B) c
(Y,N ). Thus if fl: (X,f1 N)) > (Y,N ) is defined by
fl(x) = f(x), then fl1(h, )6 H (X,f1(N)) is such that
f (ho) I(X,A) = 0. Hence there exists M = MO such that
*C 1 + *
AcMcM c f (N) and k fl (o) 1(~ho) (X,M ) = Let
il: (Y,B U f(M )) c (Y,N ), j: (Y,B) c (Y,B U f(M )), then
*C Y h *t + **1
j i =1 and h =j il(ho) O 1gk~(h ) =j f O (0) = 0.
Thus f is a monomorphism.
2.43 Full Excision Theorem: If X is a normal space and
if X = A U)B where A and B are closed subsets of X and if
f: (A,A n B) c (X,B), then f : H (X,B) ~ H (,
Proof: We have A (A n B) = A B = (A U B) B =
X B, and A = A implies that sets closed in A are closed
in X, so that f is closed. Thus the hypotheses of the Map
Excision theorem are satisfied and the conclusion follows.
2.41 Le~mma: In the diagram of groups and homomorphisms,
if the diagram is analytic and b = gf is an isomorphism,
then f is a monomorphism, g is an epimorphism, O=
I(f) t K(g), and g takes I(f) isomorphically onto R.
P h >
2.42 Lomma: In the diagram of groups and homomorphiisms,
if each triangle is analytic, I(fl) = K(g )' I(f ) = K(g2),
and bl, h2 are isomorphisms; then:
i) f: Pl x P2 Q where f(pl' P2 1 flPl 2 f2(P
ii) gl x "2: Q ~ R1 x R
1 1
iii) If q E Q, then q = f2h 2 92(9 1l 91(
1 c 2
2.43 Corollary: If X = A U B, X is normal and A and B are
closed, then the appropriate inclusion maps induce isomor
ph isms; H (X,A) X H (X,B) H (X, )E BA1B
H (A,A n B).
Proof: Consider the diagram:
H (B,A 0 B) i H (A,A 7 B)
pt~ (,A n B)e
II X, A) i H (X, B
we have I(il) K(j ) and I(j 1) K(i2), analyticity, and
by the Full Excision Theorem i j are isomorphisms. Thus
i2 2 and O, defined by C(hl, h2) 1 (hl) +1(h2) are
isomorphisms by the previous lemma.
2.44 Map Addition Theorem: Let X = X1 U X2 be normal
with Xl and X2 closed and A = X OX If fl, 2, f: (X,A)
> (Y,B) are continuous and if f (x) = f(x) for each
x 6 X f(X ) C B,i = 1,2; then f = fl f 2'
Proof: Consider the following diagram, where the
homomorphisms not induced by the mappings mentioned in the
hypotheses are induced by the corresponding inclusion maps.
H (X1,X1 n X2) H (X ,Xl n X )
1 H (X,X1 n X )
H (X,X ) H H(X,X )
H (Y, B)
2 1
W4e use lemma 2.42 to write f (h) = 11~ 2f(h
31 2jf (h). By analyticity then f (h) =il 2(b) + jlf 1(h)=
2(h) + fl(h) and so f= fl + 2.
2.4 Reduction Theorem: Let (X,XO) be a normal pair and
let A be closed in X. If h 6 H XO)adih(AA O
0, then there exists an open set M DA U XO such that
bl(M Y, XO) = Hence there exists an open set N about A
such that hl(N N n XO) = 0.
Proof: Consider the following diagram:
H (M ,XO)H MM X
f g =il 2
k 2
H (A,A XO H, (A X,, l' X O PX
where all the homomorphisms are induced by the corresponding
inclsio mas. et 6 (XXO)andk () =O, then i (h)= 0
since j is an isomorphism by the Full Excision theorem and
ic & RM
kl U ,J hence h 6 K(iO). Also, we get I(il) =K(i0) from
theorem 2.1, applied to the triple (X,A U XO, XO), and so
ther exstsb 6H (X,A U XO) such that il1(ho) = By
the Expansion Lomma there exists an open set M D (A U XO)
and b1 F H"(PX,M X) such that i2(h ) =ho Applying theorem
2.1t to the triple (X, M XO) we have K(f ) = I(g ).
Thus b (M X ) = g (bo1) = h implies f (h) = Now let
M = N for the second part of the theorem.
2.41 Extension Theorem: Let (X,XO) be a normal pair and
let becloed i X.If 6 (A,A n XO) then there exists
let~ ~ ~ ~ n becoe 4 .I
an open set M D (A U XO) and ho 6 H'MX) uhta
ho (AA n XO) = Thus there is an open set N about A and
ho6H (N C, N n XO) such that b (A,A n XO) = .
Proof: Consider the following diagram:
H (X,XO)k> H (A U XOX) >Hp1XAUXO)> XX
i il
H ( O) Hp+1(X,M )
. Lt h6 H(A U XO, XO). Then by the Expansion Lemma, there
exists M = MO D (A U XO) and hl ; H (lX,M ) such that
bl (X,A U XO) = 6(h), thus h1 (X,XO) = 6(h) (X,XO) = 0 by
exactness. Hence there exists b2 E H (M ,XO) such that
8(h ) = bl. Then I[h i (h l)] = 8(h) Gi (h2)
T(h) i S(h ) =Oimplies i (h ) E K(6) = I(k ). Thus
there exists b3 6 H(X,XO) such that k (h )= i(2
Define ho2 + j (h ) E H (M XO), then h l(AUi XOXO)
i [h2 + j (h )] = i (h2) + i j (h ) = i (h2) + k (h )
i (h2) + h in (h)=b.Nwf A XOX) EH (A,A n XO)
henc we et h6 H(A,A n XO) then there exists M = MO o
(A U XO) and ho 6 H (M ,XO) such that i (ho) = f (lh).
Therefore f i (ho) = h, or equivalently ho (A,A n XO) = b.
For the second part of the theorem let M = N.
2.1;7 Lemma: In the diagram of groups and homomorphisms:
t o
H j H2
G G1
2h I k
if each triangle is analytic, I(ja) = K(i ) for r* = 1,2,
1 1
io o = 0, and kl'k2 are isomorphisms, then lk1 1 h k2 t2
2,41 Theorem: Let X be a space and let B cA C X1 n X2 c
(j ,i ,h) be the homomorphi~sms in the exact sequences for
the triples (X,X ,B), (X,X ,A), (XQ,A,B), and (X,A,B) re
spectively for a = 1,2. If kl: (X2,A) c (X,X ) and k2: (Xl'A
C (X,X,) induce isomorphisms for each p O0, then there
exists an exact sequence:
. P p 1 A B (X,B) > H (Xl ,B) x P(XC., B1)
> H P(A,B)>.
where J = il x i ,, I = il i2, and A j~k2 1F~
?t "1
jlkl 12
Proof: Consider the following diagram:
6 H (A, B) 
H (X2,A) H~ (X ,A)
k ,* H (X,A) kJ
12
H (X,1) H X,X,)
HP(X,~izH (XB
H~~ (XB)) H X B
H (A,B)
That the conditions of lemma 2. 7 are satisfied follows
*1 Ic1
from theorem 2.20, hence jlkl1 2 2k2 6l. The proof
proceeds with seven parts.
1) Let b 6 ZO(X,B) with (i () i () O in
ZO(X1,B) x Z0 X2,B), then i (h) = and i (h) = 0. Hence
if x is in Xl then [i (h)](x) = h[il(x)] = h(x) = O),
similarly x in X2 implies h(x) = O, therefore h(x) = 0
for all x in X and we have h = 0. Thus J is a monomor
phism on HO(X,B).
2) I(G) c K(J ~). If h 6 I(A) then there exists
h' 6 H (A,B) such that b = j 1kl 2(h') = j 2 k2 1l(h ).
**1 'e
Hence il(h) ili l[kl 2(h )] 0, similarly i2 (h) = 0.
Thus J (h) = (il~(), i*2 (h)) = (0,0) and h 6 K(J ).
3) K(J ) c I(A). Suppose J (h) = (0,0), then
*e * *
il(h) = 2(h)= O and b 6 K(il) 1~). Hence there exists
hiEP _* *
hl6H (X,X1) such that jl1(hl) = Now j2kl i2 1 and
j~kl(blj 2 1~j(hl) 2 1~(hl) 2 (h) = Hence kl(h1)
K(j ) I1 r) and h1 6 Hpl(A,B) such that 2l)=lh'
3(1 *1
which implies k 2(hl) = b and so jlk1 2 (hi) 1(h1)
b.
") I(J ) K(I ). If (hl,h2) 6 I(J ) then there
exists h F H P(X,B) such that J (h) = (hl,h2) 1 'i~)
i2(b)). Therefore I (hl,h2)= 1*i(h), i2(h)) =ill)
12 2~(h) = i (h) i (h) = O, and. (hl,h ) 6 K(I ).
') K(I ) c I(J WQe note that Tl= ad e
(hl'h ) 6 K(I ). Then i (h ) (2) 0, hence k2 "2(h2)
F'1 1 bl) = 0 Thus h2 6 K(E ) = I(i 2) and there exists
h26 (X,B) such that i2(h2)= h2. Similarly bl 6 I(il)
and hence there exists hl 6 H P(X,B) such that il(hl) = bl.
Now i (blh2) i (hl) i (h2) 1~,( 1 1 i2(h2)
il(hl) 2(h2) = O, hence there exists ho HP(X,A) such
that j (ho) = hl 2 h. But H XA=Ijl ()im
plies ho = jl 1 2 (m2) where ~1 6H(, n
P w *~*~ *~* *
cp, 6 H (X,X2). Thus j (ho) 1 1,> 2) '2 j;1>
j2 M~2) = b h2,r which implies hl j1 pl)= h2 2 ~(1)
Let b = h1 1 ~1) =h2 2 ji(2) 6 H P(X,B). Then J (h) =
M )t X I I
(iillh), i 2(h)) =(il1[hl 1 (Ccl)] 2h2 2 j 2~,]
** w*'
(hl'hp). Therefore (hl,h2) F I(J ).
;) I(I ) c K(A). If (hl,h ) C IIP(X1,B) x HIP(X,B)
then I (bl,h) 1 2?)ihl an ,i:h h,)]=
*n t )t +1 "
? 1(hl) = k1 1l(hl). Thus jlk1 1i bl) 1, l(hp)] =
*1 *
j k 1 kl 1(h0l jlil(hl) O.
7) K(i) c I(I ). Let b 6 K(A), then jlk 1 82(h) = 0,
"1
hene 1 62(h) K(j ) = I(C ). Thus there exists
P ?c1
bl 6 H (X,B) such that 8 (h )= k1 62(h) whiich implies
that kl 1l(h1) '2(h) and so 62 1h 2(h) or
i2 (hl) h] = Since then il(hl) I h 6c K(62) 2 ~i
thr xst 2EP M &
there exsts h2 6H (X2,B) such that i2(h )= il(hl) h
and h lh) 2h 1ih,h2) Consequently, b 6I(I ).
2.49 Absolute MaverVietoris Secuence: Let X be normal
and let X1,X2 and XO be closed subsets of X and let
X = X1 U X2. Then there exists an exact sequence;
...O > H (1X1 n X2, X1 n X2 n XO) o > H (X,XO)
O > H (X1,X10 XO) x H (X2,X2 n XO> O, >H(Xll X2,X1? X2 n XO)
where letting ta: (Xa,~Xa. n XO) c (:KXXO '
s l: (X1 DX X DX nX ) c (X ,X AX ) for a = 1,2, we
hvJO = tl x t2 and IO = s1 s2'
Proof: Write X =X1 U XO, X2 X2 U XO, A=
X1 X2 = (Xl n X2) U XO and B = XO. Then kl: (X2 U XO'
(Xl n X2) U XO) C (X,X1 U XO) and k2: (X2 U XO, (Xl n 2) U
XO) C (X,X2 U XO) are isomorphisms by the Full Excision
Theorem. Hence we combine theorems 2.20 and 2.48 to write
the analytic ladder" below, with exact upper leg".
C> H (X, X ) '~H (X1 uX,XO) X H (X2 U XO'XO)
>H (X,XO) '\H (XX1 ? XO x HP (X2,X2 n XO)
L H ([Xl n X2] U XO,XO) 
O ~HP(X1 n X2,X1 n X2 n XO)
The homomorphisms w.,i = 1,2,3,4 are understood to be in
1 ~
duced by the appropriate inclusion maps and w = wl x w2.
# W W
By the Full Excision Theorem wl1w2, and wq are isomorphisms,
while w3 is the identity isomorphism. We define JO
* * 1 1
w J wI 4 wI w Ao = 3Ar 4 .Now I JO
w4I 3 w 3 = Oq w hence I(JO) C K(IO)
Conversely if IO(z) = O, then wiCI w ('") = 0 which implies
I w ( ) =O and w (:D) 6 K(I ). Thus w (s~) = J 0a~
hence rn = W p JO 3(9) and 0 6 I(JO). Consequently
K(IO) c I(JO). Similar computations show I(I0) = K(."o) and
I(" ) = K(J ) and that I = sl s Jg = tl x t .
2.50Theorom: If X is connected, normal and TI, and if
H1(X) = (0), then X is unicoherent. (G L Oj)
Proof: Let A,B be closed connected subsets of X with
X = AU B)1. Using the MayorVietoris Sequence with Xl
A,X2 = B,XO = (x) for fixed x E A AB we have the following
diagram:
I~~n(X~~~x) ~> x o> H)A)BX II (X,x) IlX
H (B,xU)
Since HO(X,x) = (0) by theorem 2.10, and HI(X,x) ~ H1(X)=
(0) by assumption and by theorem 2.16 and corollary 2.18,
we have that IO is an isomorphism. Applying theorem 2.10
again yields the desired result.
CHAPTER III
A GROUP ASSIGNMENT FOR A SPECIAL CLASS OF SPACES
We let X be a space with the property that the
intersection of two arbitrary open connected sets is the
union of open connected sets. (e.g. any locally connected
space, or the circle after replacing a proper subarc by
the closure of the sin (x1) curve). Then if G is a fixed
abelian group we will define, for each positive integer p,
an abelian group H P(X) in much the same way as in Chaptor II;
with the notable distinction that now the open covers of
interest will consist of connected sets. Initially it was
hoped that this new group assignment would be such that
spaces which were not locally connected would be assigned
the trivial group. We show by example that this hope is
not realized.
3.1 Definition: If 9A is an open cover of X and if
each element of 9JI is a connected set, then we call an
open connected cover of X. We let C (X) be as before and
define Cp(X) = (919 E C (X) and there exists an open
connected cover q14 of X with c0 = O on 9 (p+1))
Remark: Throughout this chapter all spaces under
36
consideration are assumed to have the property for open
connected sets postulated above for X. It should be clear
that this property is sufficient to guarantee that C p(X) is
a group.
3.2 Definition: Let Z (X) = $ [C l~(X)] and p > O, and
B (X) =C (X) + 6 [Cpl(X)], p 1BOX)=0.
3.3 _Theoem: If f: X > Y is such that the inverse
image of an open connected set is open and connected then:
1) f [C (Y)] c C 0(X)
2) B[CP(X)] c Cptl(X)
3) f '[Z (Y)] c Z (X)
4) f [B (Y)] CB I(X)
Proof: 1) Let 0 6 Cp(Y), then there exists an open
connected cover .L of Y such that cp = O on 1/(p+1). Consider
[fl~v)l v 6 ), by assumption this is an open connected
cover of X and if (xO,...,x ) E [f1 p+]1 then
[f'ql(xO,..., X) = W(f[xO],...,f[x ] 6 e(vp+1) = Thus
fKCm 6 CO(X).
2) Let c? 6 CP(X) and 14( be an open connected
cover of X such that cp = O on CLp+1). Let (xO,...~xpel)
6 u ,2 then [6rp](xO,...,x+1) _CP i=0(1)i g (x'" "
^ ...,xp+1) = 0 since (x0,...,xj.....,xp+1) 6 u p+1implies
W(xO,...,x ,...xp+1) = 0. Thus ~Eq 6 Cp1()
3) Let 9p 6 Z (Y) = b1Cp (Y)], then
6cp 6 Cp+1(Y) hence f Eq 6 f"[Cp+1 Y) cCCp+1(X). Now f 0~ =
Ef"m and so f'Wp 6 1[C+() (~X).
4) Let 9 6 B (Y)= COP(Y) + [~Cp1~~
then f 9 6E f [C (Y)] + ft[CCpl(y)] c C P(X) + E[Cp1(X)]
B X p 1. The case p = O is clear.
3.4 Definition: H (X) = Z (X)/B (X).
3.5 Theorem: Let f: X > Y be such that the inverse
image of an open connected set is open and connected, then
there exists a unique homomorphism f : Hi: H(Y) > H (X)
such that af =f A, where a: ZC (X)  H P(X) and
3: Z (Y) L) H (Y) are the natural homomorphisms.
Proof: Induced Homomorphism Theorem.
3.6 Th eorem : If i: X c X, then i : H (X) C H (X).
Proof Letb 6 (X) and 3 6 Z (X) with a = h,
then ibh = i am ai 0 = aO = b.
3.7 Theorm: If f: X > Y, g: Y > Z are such that
the inverse image of open connected sets are open and con
nected then (gf) = f: H (Z) > H(Z) > H (X) .
Proof: We already know from Chapter II, that
(gf) = fg"g. By part three of Theorem 3.3 letting
a: Z (X) > H (X), A: Z (Y) > H (Y) and y: Z (Z) >
H (Z) we may compute as follows: If b E H p(Z) and
S6Z (Z) such that yp3 = h then f g +(h) = f v'
3.8 Example: Let X = A U BUC 'r D whero A=
C = [ x,0 1
1 1
or y = 2 and 1 < x < nr or x = n. and 2 L y < 0),
(see figure 1). Let R c X XX be defined by R = (A x A) U
D(X ) and let Y = X/R. It is easy to see that the removal
of any pair of points from Y disconnects Y, and hence Y is
topologically the unit circle. Since the definition given
in this chapter is equivalent to that given in Wallace's
notes, for locally connected spaces, we kcnowh that H (Y) G.
(Fig. 1)
Let f: X  X/R be the natural map. Then since X is compact
and f is monotone and continuous we have that the inverse
image of an open connected set is open and connected. De
fine g: Y > X by the equations g~f(x)] = x if x A and
g~f(A)] = (O,0). We note that if u is an open connected
subset of X not containing (0,0) then uA is open and
connected. To see that the inverse image under g of an
open connected set u is open and connected we consider two
cases.
1) If (0,0) ( u, then gl~u) = fluA) is open and
connected since f restricted to XA is a homeomorphism.
2) If (0,0) 6 u, then gl~u) = flu) is connected
since f is continuous and is open if there is an open set
about f(A) contained in g1(u). Now (0,0) 6 u implies there
exists t 6 (1,0) and z 6 (O,n1) such that (t,0) x (0) =
Pcu and ((x,sin x1)(O < x < z) = O c u. Letting v=
P U Q U (0,0)) we have f1f(v) = AU PU Q which is open
in X and hence f(A) c f(v) c f(a) = g (~u) implies gl~u)
is open.
Since fg: Y c Y, and X and Y satisfy the relevant
hypotheses, we use theorems 3.5 and 3.6 to assert
(fgX = g f: H (Y) CH(X) and consequently, if
hl, h2 E H (Y) with f (hl) = f (h2), then h1 g f (hl)
g f (h2)= h2 and f is a monomorphism. Thus H (X) is non
trivial, while X is not locally connected.
CHAPTER IV
FINAL COMMENTS AND QUESTIONS
We first show by example that the cohomology groups
as defined in Wallace's notes are different from those de
fined in Chapter II.
4.1 Example: Let G be an additive abelian group with
the property that for each g in G there exists a positive
integer n, depending on g, such that ng = 0. Suppose also
that G is not nilpotent, i.e. there does not exist a posi
tive integer n such that ng = 0 for all g in G. Letting
H (X) denote the pth cohomology group for the space X,
computed with respect to G, as defined in [8], and re
calling that Ho(X) is isomorphic to the group of functions
mapping X into G which are continuous in the discrete topo
logy of G, we easily show that Ho X) is not isomorphic to
H (X) for X = G. For by theorem 2.9, "3 E Zo(X) e Ho(X) if
and only if cp: X > G is continuous in the discrete
topology of G and (9 (~g) g 6 G) is finite. Let
(rp (g) Ig 6 G) = (9 (lg.) 1 < i
positive integers that pig. = 0, 1
pl x p2 x...x pn. Then pm = 0 6 Zo(X) and hence each
element in Ho(X) has finite order. But it is clear that
i: G 2 G is continuous in the discrete topology and has
order zero, hence Ho(X contains an element which is not
of finite order.
Now Haskel Cohen in [2] gave a definition of co
dimension, based on the groups of [8], for locally compact
spaces, and we have just seen that on locally compact spaces
the groups of [8] differ from those of this thesis. It is
natural to ask then, whether a definition similar to
Cohen's will yield a dimension theory and if so, will this
theory differ from Cohen's. Without reproducing the techni
cal definitions we remark that the dimension of a space was
defined in terms of the cohomological structure of its com
pact subspaces, and since the two cohomology theories in
volved agree on compact spaces it turns out that the afore
mentioned similar" definition is, in fact, identical
with that of Cohen's.
We have shown in Chapter II that we have constructed
a cohomology in the sense of Mac Lane [E] but have not de
monstrated that we have a cohomology theory in the sense
of Eilenburg and Steenrod. Indeed, we have failed to verify
the Homotopy Axiom" of C13. Although Lemmas 2.23 and 2.19,
in the presence of the Reduction Theorem, were shown by
Keesee in [4] to imply the Homotopy Axiom for compact pairs,
there is no such short cut for the general case. We are un
able, however, to produce an appropriate counterexample and
hence must leave the question open. Because the ~Cch theory
based on finite coverings fails to satisfy this axiom we
are led to conjecture that the axion also fails for the
theory of Chapter II;.
BIBLIOGRAPHY
l]3 C.E. Capel, Inverse Limit Spaces, Du~ke Mathemati
cal Journal, vol. 21(1954) PP. 233245".
[2] Haskel Cohen, A Cohomological Definition of Di
mension for Locally Compact Hausdorff Spaces, Duke
Mathematical Journal, vol. 21(1954) pp. 209224.
[3] Samuel Eilenburg and Norman Steenrod, Foundations
of Algebraic Topology, Princeton (1952).
[4] John W. Keesee, On the Homotopy Axion, Annals of
Mathematics, vol. 54(1951) pp. 247249.
]' Saunders Mac Lane, Homology, SpringerVerlag (1963).
[6] T. Rado and P.V. Reicheiderfer, Continuous Trans
formations in Analysis, SpringerVerlag (1955)
C7] Edwin H. Spanier, Cobomology Theory for General
Spaces, Annals of Mathematics, vol. 49(1948)
pp. 407427.
[4] A.D. Wallace, An Outline for Algebraic Topology I,
Tulane University (1949)
BIOGRAPHIICAL SKETCH
Marcus Mott McWaters Jr. was born January 21, 1939,
in Little Rock, Arkansas. He was raised in New Orleans and
Metairie, Louisiana, and was graduated in August, 1956,
from East Jefferson High School.
In September, 1958, he entered Louisiana State
University in New Orleans and was awarded the degree of
Bachelor of Science from this institution in June, 1962.
He then became a graduate student at Louisiana State Uni
veristy for one year, after which time he transferred to
the University of Florida. From September, 1963, when he
entered the University of Florida, he has worked as a
graduate assistant in the Department of Mathematics and
pursued his work toward the degree of Doctor of Philosophy.
Marcus Mott McJaters Jr. is married to the former
Patricia Ann Guice and has one daughter, Sharon Lee.
This dissertation was prepared under the direction
of the chairman of the candidate's supervisory committee
and has been approved by all members of that committee. It
was submitted to the Dean of the College of Arts and Sciences
and to the Graduate Council, and was approved as partial ful
fillment of the requirements for the degree of Doctor of
Philosophy.
April 23, 1966
Dean, Colle4 of'~ Ats and Sciences
Dean, Graduate School
Supervisory Committee:
Chairman

Full Text 
xml version 1.0 encoding UTF8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EC9YJ9VTQ_H4T7R3 INGEST_TIME 20110715T21:19:22Z PACKAGE UF00097868_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
PAGE 1
COHOMOLOGY FOR NORMAL SPACES By MARCUS MOTT McWATERS, JR. A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA April, 1966
PAGE 2
UNIVERSITY OF FLORIDA 3 1262 08552 2752
PAGE 3
To Pat, Mom, and Dad
PAGE 4
ACKNOWLEDGMENTS The author would like to express his sincere appreciation to his director. Professor Alexander R. Bednarek, for his patience and encouragement, as well as for his professional assistance in the preparation of this manuscript. The author recognizes a special debt to Professor Alexander D. Wallace, for suggesting the topic for this dissertation, and for providing a general perspective throughout its development. Ill
PAGE 5
TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii INTRODUCTION 1 Chapter I. PRELIMINARY RESULTS AND DEFINITIONS k II. THE COHOMOLOGY GROUPS OF A SPACE MODULO A SUBSET 6 III. A GROUP ASSIGNMENT FOR A SPECIAL CLASS OF SPACES 35 IV. FINAL COMMENTS AND QUESTIONS 40 BIBLIOGRAPFIY 43 BIOGRAPHICAL SKETCH 44 IV
PAGE 6
INTRODUCTION This paper presents a definition of cohomology groups of a topological space relative to a subset of the space. The definition employed was suggested to the author by A.D. Wallace, and is a modification of an earlier definitionj also due to Wallace, which was exploited by Spanier in [ 7 ]. Both definitions, as will be seen below, involve the notion of pfunctions and hence have their roots in the works of Alexander and Kolmogoroff . These definitions agree on compact Hausdorff spaces and, after a suitable shift in dimension, yield groups isomorphic to those in [ 6 ] . If X is a topological space we let X^'^ denote the cartesian product of X with itself p+1 times and define C^(X) = ( cp(cp: X^ Â— > g], where G is a fixed, though arbitrary, abelian group. Then C^(x) is itself an abelian group, if addition of two elements in C^(x) is defined pointwise; this group is called the group of pfunctions. If W is an open covering of X, we set c<^^ ''=U{ u^^ lu?^). Then for each subset A of X, and each integer p >. 0, we may define C^(X,A) = [cpcp Â€ C^(x) and there exists an open cover V of A such that cp = on U^^'^' n A^*].
PAGE 7
For each integer p >^ there is a homomorphism, defined in Chapter 1, 6: cP(X) > C^'(X) having the properties that 6^ = and 5[cP(X,X)] is contained in C^'''(X,X). Then other subgroups of C^(X) may be defined by Z^(X,A) = CP(X,A) n 6^[c^^(X,X)]; bP(X,A) = 6[cP"^(X,A)] + cP(X,X) (for p = 0, bP(X,A) = [0]). The definition used by Spanier of the pth cohomology group of the space X relative to the subset A, denoted by H^(X,A), is the quotient group ZP(X,A) / bP(X,A). Our departure from this definition is effected by redefining C^(X,A) as the set of pfunctions, cp, for which there exists a finite open cover %l of. A such that cp = on '^(P+1) n aP^*. a similar distinction is found between the Cech cohomology theory based on finite open coverings and the 6ech theory, advanced by Dowker, based on pairs of infinite coverings. Spanier showed that the theory developed in [ 7 ] was a cohomology theory, in the sense of Eilonburg and Steenrod [ 3 ], on the category of compact pairs. This result then, carries over to the development presented in this paper. In fact, most of the axioms of [ 3 ] will be verified for general topological pairs (the only exception being the Homotopy Axiom) . Each time an axiom is verified the axiom will be identified by a parenthetical insertion referring to the axiom e::actly as it is numbered on page fourteen of [ 3 ] Â• In Chapter I we review some of Spanier 's results
PAGE 8
and definitions for use throughout this paper. Chapter II presents our basic definitions and major results. The development follows closely that of Wallace's notes on Algebraic Topology [ 8 ] . Many of the theorems in [ 8 ] are proved under the hypothesis that the topological space in question is fully normal. Virtually all of these same theorems, including the Reduction and Extension Theorems, are proved for normal spaces. We employ the resulting generality to show that a connected, normal, Tj^ space with trivial first cohomology group is unicoherent. In Chapter III we disprove a conjecture that a particular group assignment, defined for a special class of spaces, will assign the trivial group to spaces which are not locally connected. We conclude with Chapter IV by discussing related subjects (e.g. codimension) and open questions.
PAGE 9
CHAPTER I PRELIMINARY RESULTS AND DEFINITIONS We review some of Spanier's results which will be needed in the sequel. We assume throughout this paper that G is a fixed, though arbitrary, abelian group. The term " mapping" will be used to mean " continuous function" and p+ 1 X*^ will denote the cartesian product of the topological space X with itself p+1 times. 1.1 Definition : Let X be a topological space and let p 2 be any integer. Then C^ {X) [a5c:.: X^*""*" Â— > g]. For each pair c?, ij; Â€ C^(X) define (>hii/ ) : X^"*Â— > G by (c(>f^')(xQ, ...,Xp) = cp(xq, .. .,Xp) + v(xq,.. .,Xp). 1.2 Definition : 1) For any set P we let the diagonal D(P'^) of P"" be Ulixj^'lx Â€ P]. 2) If f : X Â— > Y is a function from a topological space X to a topological space Y, define f^ CP(Y) Â— > CP(X) by [f^(cD)](xQ,...,Xp) = cp[f(xQ),...,f(Xp)]. 1.3 Definition : We define 6: C^(X) Â— > C^'^''"(x) by YÂ» pfl ^
PAGE 10
p+1 where (xq, . . ,x^, . . ,x^j^) = (xq,..,x^_^, ^i+1' Â• "^pf l) ^ ^ lA Definition : Let f,g: X Â— > Y. Then define for p > 0, D: CP(Y) Â— > CP"^(X) by [D(cp)](xQ,...,Xp_^) = )^ ^^(1)^ o[g(xQ),...,g(x.), f(x.),...,f(Xp_^)]. The properties of C^(X), f^, S", and D, of which we will constantly make use^ are collected in the following: 1.5 Theorem : 1) C^(x) is an abelian group 2) f ''^ and 5 are homomorphisms 3) f''6 = ^f** 4) ^6 = 5) D6 + ^D = f " g** if p 2 1 6) D6 = f ** g*^ if p =
PAGE 11
CHAPTER II THE COHOMOLOGY GROUPS OF A SPACE MODULO A SUBSET We shall adhere to the definitions and notation of Chapter 1, and shall introduce new definitions, conventions and algebraic lemmas as they are needed. We omit the proofs for these lemmas if they are available in standard texts. 2.1 Notation : If 7^ is a family of sets Xi^"^' U(A^a ^11)' If f: X Â— > Y is a function from a space X to a space Y, if A c: x and if f (a) c b c: y, then we write f: (X,A) Â— > (Y,B). 2.2 Definition ; If A c: x we define C^{X,h) = (cpjcp Â€ cP(X) and ? a finite open cover t( of A 3 cp = on t/^^ '' n A^ }. We often write C^(x) = C^(X,A) = cP(X,X). 2.3 Lemma : 1) C^(X,A) is a subgroup of cP(x) 2) If f: (X,A) Â— > (Y,B), f continuous, then f''[cP(Y,B)] c: cP(X,A). 3) If A c X, then Z[C^{X,A)] <= C^^ {X,K) . Proof: 1) Let cp, ^ Â€ cP(X,A) and let 1/ and V be finite open covers of A 3 cp = on %/S^ ' fl A^ and iji = on y^^^) n A^^, then V= {u n vu Â€ U and v Â€ V] is a
PAGE 12
7 finite open cover of A such that cp 'j; = on 1*/^^ ' D A^ . Thus cp j/ Â€ cP(X,A). 2) Let CD Â€ C^(Y,B)j then "^ a finite open cover ^ of B such that cp = on l/S^^' H B^*. Then ^f = [ f ~ (u) u Â€ L{ ] is a finite open cover of A such that f^(cp) = on V(^^) n A^^. Thus f"(cp)^ cP(X,A). 3) Let cp Â€ cP(X,A), then "^ a finite open cover *U of A such that cp = on (Jy^^ I n A^ , and one easily checks that 6cp = on 'Us^^' n A^^, hence 6cp Â€ cP^'(X,A) . 2.4 Definition ; Let A be a subset of a space X. Then 1) zP(x,A) = cP(x,A) n "^[C^^(X,A)] 2) bP(X,A) =^ r [0] , P = 1^CP(X,A) + ^[cP""^(X,A)]: p > 1. 2.5 Lemma : 1) zP(X,A) is a subgroup of cP(x,A) 2) B c A c X Â— > cP(X,A) c cP(X,B): hence zP(X,A) c zP(X,B). 3) bP(X,A) is a subgroup of zP(X,A) k) If A is a closed subset of X and cp 6 Z^(X,A), then 3 a finite open cover 1^ of X with (i) ^cp = on ^((e^2) ^. .) ,:p = on U^^^) n A^^ Proof : 1) and 2) are clear. For 3) we recall that 6^ = and use Le.Tmia 2.3, part three, to establish the appropriate inclusions. For 4) assume cp Â€ Z^(X,A), then cp Â€ C^(X,A) and ^cp Â€ C^ (x) . Hence 1 a finite open cover ^A^ of A 3 cp = on t^^P^^ n A^ and a finite open cover
PAGE 13
8 If of X 3 6cp = on V^^'^^ . Let O = %i U{XA] and V/ = {v n ov Â€ u and o Â€ CT } , then ZJ is a finite open cover of X satisfying (i) and (ii). 2.6 Definition : If A is a subset of a space X, we define hP(x,a) = zP(x,A) / bP(x,a). 2.7 Lemma : If f: (X,A) Â— > (Y,B) is continuous, then 1) f''[zP(Y,B)] c zP(X,A) 2) f''[BP(Y,B)] c zP(X,A). Proof : 1) If p Â€ zP(Y,B), then cp Â€ cP(Y,B) and 6cp Â€ C^'"(Y,B), hence f'*(cp) Â€ cP(X,A) and ^[^{0)] = f''[5cp] C^^(X). Thus f^(cp) Â€ zP(X,A). 2) If cp Â€ bP(Y,B), then cp = ij/ + 66 where i( cg(Y) and 9 f cP~^(Y,B). Hence f^(cp) = f''() + f^(^e) = f*'(M + 6[f''(e)] Â€ CP(X) + 6[cP^(X,A)] = bP(X,A), p > 1. The case p = is clear. 2.7 Induced Homomorphism Theorem : Let P be a group with subgroup P^, let Q be a group with subgroup Q^ and let f be a homomorphism of P into Q such that f(P^) <= Q^^. Then if a : P Â— > P/Pq and ? : Q Â— > Q/Qq are the natural homomorphisms, there exists one and only one homomorphism g such that ga = 8f. 2.8 Theorem: Let f: (x,a) Â— > (Y,B) be continuous, let a: zP(X,A) Â— > hP(X,A) and B: zP(Y, B) Â— > hP(Y,B) be the natural homomorphisms and define f^: zP(Y,B) Â— > Z^{X,A) by fQ(cp) = f'^("p) for each g Â€ zP(y,B). Then ':^ a unique homomor
PAGE 14
phism f : hP(Y, B) Â— > U^{X,h) such that af^^ = ^f . Proof : Induced Homomor phism Theorem. 2.9 Theorem : "4) Â€ Z (x) if and only if "4^: X Â— > G is continuous in the discrete topology of G, and (cp (g) g Â€ g] is finite. Proof : If cp Â€ Z (x) then ?T a finite open cover of X such that 6cp = on U^ ' . Let cp(x) = g Â€ G, then "^ u Â€ 1/ with X Â€ u. Nov^? if y Â€ u then ^cp(x,y) = cp(y) cp(x) = 0. Thus G is continuous in the discrete topology of G and that [cp~ (g) g Â€ G] is finite. Then '^ g Â€ G such that X Â€ G such that x Â€ cp (g)* and if y Â€ cp (g) we have 6cp(x,y) = cp(y) rp(x) = g g = 0. Hence Jcp Â€ Cp,(X) and p Â€ ZÂ°(X). 0' 2.10 Theorem : If G ^ (O), then X is connected if and only if H^(X,x) = [0) for each (for some) x Â€ X. Proof : Let X be connected, then if cp Â€ Z (X,x) ^ HÂ°(x,x), cp Â€ cÂ°(x,x) n 6^[cJ(x)] c cO(x) n r^[cj(x)] = Z (X) . Thus cp is continuous in the discrete topology of G and is therefore a constant function. Hence cp(x) = Â— > cp(X) = 0, and we have [Oj = ZÂ°(X,x) ^ HÂ°(X,x). Now assume
PAGE 15
10 X is the union of two disjoint open sets A and B. Fix x in X and assume x Â€ A. Define cp: X Â— > G by cp(A) =0 and cp(B) = g ^ 0. Then cp is continuous in the discrete topology of G and (cp~ (g)g Â€ G] = {A,B] is finite. Thus cp Â€ Z^(X), but X 6 A open Â— > cp Â€ C (X,x), so we have ^ cp Â€ Z (X,x) 'H (X,x), a contradiction. Conventions: If A is a subset of a set B and if f : A Â— > B is defined by f(x) = x for each x in A, then f is called the inclusion map of A into B and is denoted by f : A c B. If f : G Â— > H is a homomorphism from the group G into the group H, then [hh Â€ H and h = f(g) for some g Â€ G], the image of t, will be denoted by l(f) and {gf(g) =0}, the kernel of f, will be denoted by K(f). 2.11 Theorem ; Let X be a space, let B c A c x, let Y: zP(A,B) Â— > hP(A,B) and a: Z^'^ {yi,K) Â— > H^'(X,A) be the natural homomorphisms and let t: A c x. Then 1) For each h Â€ H^(A,B) there exists such a cp Â€ cP(X) that Yt*("45) = h and 5rp Â€ Z^'(X,A). 2) If cp, 'Ji Â€ cP(x), if t^(cp), t^() Â€ zP(A,B) and if Yt'*(cp) = Yt**(0^ then 6(cp*) Â€ B^'(X,A). 3) 6 = a6t** Y is a homomorphism from H^(A,B) into H^'(X,A). Proof: We first note that if ijf 6 C^(X) with t*(J;) = cp Â€ cP(A) then 'j/ Â€ cP(X,A). For ? a finite cover of A by sets open in A, t( , with cp = on V^^ > , hence if we
PAGE 16
11 write U= [v. nAl<^i<_n) where v. is open in X and \f = [ V. l <^ i <^ n] , we have that 2/ is a finite open cover of A 3 ^ = on ^^'^^) n A^""". 1) Let h Â€ hP(A,B), then '^ cp 6 zP(A,B) with y(^) = h. P+1 Define i/ : X*"^ Â— > G by the equations ij( (x^, . . . ,x ) = D+ 1 cp(xq, . . . ,Xp) if (xQ,...,Xp) Â€ A^ ; J; (xq, . . . ,Xp) = if (xÂ„, ...,x ) Â€ X^"A^*. Clearly t'^('0 = ^, hence Y[t^(i/)] = y(^) = h. Now 6cp Â€ C^^(A) and t^[^H = ^[t^(0) ] = ^cp hence h^ Â€ C^''"(X,A). 2) Let cp, i) Â€ cP(X)j t**(cp), t''(*) 6 zP(A,B) with Yt'^(cp) = Yt^(^), then t*^(cp^) Â€ bP(A,B) and hence t^(cpi() = cp^ + ^(cp^) where cp^ Â€ cP(a) and cp^ Â€ C^*(A,B), p 2 1Let cp' Â€ t ~ (cpp) Â•= cP~^(x) and define cp' = cp on A^*; cp^ = cp _ ij( _ ^(cp^) on X^^*A^". Then t**'(:;) = cp^, hence t'^(cp^) _ cp^ = t*(cpij/) t**(cp^) = t^[(;o^) cp] = l{^^ = 6[t^(cP2)] = t^[6(cp2)], hence (cp>f ) cp = ^(cp^) on all X^ . Thus S^(cpilf) ^(cp) = !^F(cp') = 0, from which ^(cp'ji) = 6(cpj); but t'^(cp) = cp^ Â€ cg(A) Â— > rp^ Â€ cP(X,A). Thus ^(cpj) Â€ 6[cP(X,A)] and 6 (cpi; ) = 6(cp) Â€ bP(X,A) The case p = is trivial. 3) 6 is well defined is immediate from 1) and 2). 2.12 Definition : A sequence of homoraorphisms h. h, ^ _i ^ A^ ^ Â— > At Â— > ... " ^ Â— > A " Â— > A , Â— > ... is 1 n n1 exact if and only if h^. is a monomorphism and l(h.) = K(h^^j^) for i >_ 0Notation : The following notations will hold through
PAGE 17
12 Theorem 2.16: B c a c x, j: (X,B) c= (X,A), i: (A,B) c (X,B), and t: (A,a) c (X,a). a: zP(X,A) Â— > hP(X,A), B: zP(x,B) Â— > hP(X,B), and y: Z^(A,B) Â— > hP(A,B) are the natural homomorphisras. 2.13 Lemma : 1) j : H (X,A) Â— > H (X,B) is a monomorphism. 2) I(j*) K(i*) for p 2 0Proof : 1) Let cp Â€ Z*^(X,A) s hÂ°{X,A) and suppose j^cp(x) = for each x in X, then "45[j(x)] = cp(x) = for each X in X. Thus j a(c?) = Â— > Bj^(cp) = Â— > j ^^cp = Â— > cp = Â— > ap = Â— > j is a monomorphism. 2) We first show that l(j ) c K(i ). If cp Â€ Z^(X,A) then "E a finite open cover U of A 3 cp = on ^LX ^^Â•'Â•^ n A^". Let 'u = [u f] au e U ], then 'U is a finite open cover of A with i^j^^cp = on V ^ ' . Hence i^j'^cp Â€ C?(a) c: b"(A,b). Since the natural homomorphi sms are onto the proof is complete. To prove K(i ) c: i(j ) we suppose i'''"? Â€ B^(A^B), then i'^cp = cp. + 6(cpÂ„) where cpj_ Â€ C^(a) and cp^ Â€ C^'(A,B), p >. 1^^t 9 Â€ cP""'"(X,B) with i'*9 = cp and define 'l; : X^"^ Â— > G by ij; = cp^ on A^^ and (jf = cp 6(e) on X^"A^''. Then '^ 6 C^(X,A) and if (xq, ...,x ) is in A^ we have iji (x^, . . . ,x ) = cPj^(xq, . . . ,x ) = i^cp(xQ,...,Xp) 6:p2(xq, ...,Xp) = cp(xq, . . . ,Xp) 6i^e(xQ, . . . ,Xp) = cp(xq, . . .,Xp) i'*6p(xQ, . . .,x ) = cp(Xq,...,x ) 3'cp(xÂ„, . . . ,x ). Thus o[cp6 9] = ^cp 6^9 = 6cp Â€ C^'(X). Hence ) Â€ zP(X,A) and cp j'Â«v = ^e Â€ 6[C^'(X,B)] e bP(X,B). Again the case p = is trivial
PAGE 18
13 and inclusion follows from the fact that the natural homomorphisms are onto, 2.14 Lemma ; l(i*) = K(6) Proof: I(i ) c= K(6). Let cp Â€ zP(x,B), then c? Â€ t'^''i^'cp hence 6t''"'i^cp = 6cp. But cp Â€ "'[C^"''(X) ] , so 6cp Â€ C^ (X), a subset of B^ (X,A) . The inclusion is then clear. To show K(6) c i(i ) we suppose cp Â€ Z^(A,B), if Â€ t*~"''(cp) with 6'J; Â€ B^"''(X,A). Then 6if = cp^ + ^(cp^) where ^1 ^ ^o*'"'^^'^^ ^^'^ "2 ^ cP(X,A). Define 9 = il/ cp^, then 66 = cp^ Â€ C^'(X) hence 9 Â€ ^"'"[C^'[X) ] . Also t^V = cp Â€ cP(A,B) and hence ^ Â€ C^(X,B); therefore 9 Â€ C^(X,B) and we have 9 Â€ zP(X,B). Now cp 1*^6 = p i**^ i'^cp^ = cp cp i'fcp^ = i'^cp^ Â€ cP(A) c bP(A,B). 2.15 Lemma : l(5) = K(j ) Proof : 1(c) c K(j*). Let ij; Â€ t^'^^'cp with rp Â€ zP(A,B), then t"'^ = cp Â€ cP(A,B) Â— > t ^ cP(X,B). Hence 6') Â€ 6[cP(X,B)] <= B^ {X,B) and j**^']; = 6tj; . To complete the proof we appeal it to the natural homomorphisms. To show K(j ) c: 1(6) we let cp Â€ Z^"^(X,A) with i**cp Â€ B^'(X,B). Then "P = j >":? = cpj^ + 6(P2) where cp^ Â€ C^'(X) and cp^ Â€ cP(X,B). Thus i"'cp2 ^ cP(A,B) and ^i'^(t?2) = i^'^C^g) = ^*" " ^"''l ^ ^r"^^^)Hence i'^cD^ Â€ zP(A,B) and cp^ Â€ t^''^i (^2)' ^Â° 6t''"'i'*('P2) = 6(cp2) e C^^(X,B) and cp 6(cp2) = cp^ Â€ cP^^(x) c bP^^(X,A). 2.16 Theorem : HÂ°(X,A) Â— ^ > H^(X,B) ^^Â— > n^(A,B) > * * . * h^(x,a) Â— i Â— > ... Â— ^> h'''(x,a) Â— JÂ— > h"(x,b) ^^Â— >
PAGE 19
14 6 Â•* h'^(A,B) > H^"*" (X,A) ^ > ... is exact. (Axiom 4 c) . Proof : The three previous Lemmas. 2.17 Corollary : H^(X,X) = 0, for any space X and any p _> 0. Proof : One takes B = A = X in the previous theorem and easily verifies that i j is the identity function, as well as the zero function, on H^(X,X) . 2.18 Corollary : If A is a connected subset of a space X or if A = D, then 6: H (a) > H (X,A) is the zero function. Proof : Recall that HÂ°(A) s^ ZÂ°(A) and HÂ°(X) s ZÂ°(X). Each cp Â€ Z (a) is continuous in the discrete topology of G and hence is a constant function, since A is connected. The constant function jf : X Â— > G defined by extending cp to all X is such that i" il; = cp, where we have taken B = D. tt * Thus i , and consequently i , is an epimorphism and hence * H (a) = I(i ) = K(6) if a is not empty. But A = D is clear. 2.19 Theorem : 1) If f : X Â— > Y and if g: Y Â— > Z, then (gf)^= f^g^ 2) If f: (X,A) Â— > (Y,B), if g: (Y,B) Â— > (Z,C), and if f and g are continuous, then (gf) = f g . (Axiom 2c). Proof: 1) [(gf)'*cp](xQ,. ..,Xp) = cp(gf [x^] , . . . , gf [x^] ) = gÂ«cp(f[xQ],...,f[Xp]) = [f'"(g''cp)](xQ, ...,Xp) = [f''g'*]:?(xQ,...,x ). Thus (gf)"":? = [fg""] cp if cp Â€ C^ {X) .
PAGE 20
15 2) Let o.: zP(X,A) Â— > u'^{X,A), Â°: zP(Y,B) > * * ' r^ r= H^(y,B) and v: Z^(Z,C) Â— > H^(Z,C). Then f g (v) = f Bg af"g'>0 = a(gf)"'^ = (gf)*(v). 2.20 Theorem : Let f: (X,A,B) Â— > (X',A',B') be continuous, Define u: (X^A) Â— > (X',A'), v: (X,B) Â— > (X ' , B ' ) and w: (A,B) Â— > (a',B') by u(x) = v(x) = w(x) = f(x). Then the " ladder" 6 > hP(x',a') Â— ^ Â— > hP(x',b') Â— Â— > hP(a',b') u 6> H (X'A) iP V w > H^(X,B) > h^(a,b) 6 : 1 is analytic (each rectangle of the ladder is analytic) . (Axiom 3 c) . Proof : It is trivial to verify that jv = uj and that iv = wi, hence v j = j u and v i = i w . Now let a' Y' t' zP(X',A') > hP(X',A'), ^': zP(X',B') Â— > H^(X',B'), zP(A',B') Â— > hP(A',B'), Â» Similarly u 6i = ^i v and v j 6 = j 6w . 2.21 Corollary: If f: (X,A) Â— > (X',A') is continuous and if f(X) c: a ' , then f : hP(X',A') Â— > hP(X,A) is the
PAGE 21
16 zero function for each p >^ 0. Proof : Recalling that H^(X,X) = [0] and noting that f: (X,X,A) Â— > (X'jA'^A') we use the previous theorem to assert that if h Â€ E^{X',h' j*f*(h) Â€ j*[hP(X,X)] = (0} assert that if h Â€ E^{X',h') then f (h) = f j (h) = 2.22 Theorem : If X is connected then H (X) ^ G, and if X is a point space then h'^(X) = G and U^{x) = {0] for p 4 0(Axiom 7 c) . Proof: If X is connected then each h in Z (x) is a constant function, hence we may define f: Z (X) Â— > G by f(h) = h(x) . Clearly f is a monomorphism. If g ? G define h : X Â— > G by h (x) = g for each x in X. Then h is in g ^ g^ ^ ^ g Z (x) and f (h ) = g, hence f is an isomorphism and we have G s z'^(X) = HÂ°(X). Now assume p 4 and X = [x]. If p+1 cp Â€ zP(x) and cp(xP^^) = g, then 6cp(x^2) = V (l)S = 0, ^ p . '' i=0 hence ^ (1)^9 = (1)^9Define ij; : [x^] Â— > G by t(xP) = g~ then 6^ (x^^^) = 7 ^ (l)^g = (l)Pg = p+1 "^ ^'^ (l)^cp(x^ ). Thus 6i/ = tp or 6(i];) = rp, and we have cp Â€ 6[cP~''(X)] c B^(X). Hence Z^{X) a B^ {X) , Therefore equality holds and hP(x) = [0]. 2.23 Lemma ; If f: (X,A) e (X,A), then f : U^{X,a) c: hP(X,A) for p >^ 0. (Axiom 1 c) . Proof : We have f (acp) = af **(:?) = acp. 2.24 Theorem : If f: (X,a) Â— > (Y,B) is a homeomorphism, then f : H^(Y,B) Â— > H^(X,A) is an isomorphism.
PAGE 22
17 Proof : Let g: (Y,B) Â— > (X,A) be such that gf(x) = X for each x in X and fg(y) = Y for each y in Y. Then fg: (Y,B) c (Y,B). hence g*f* = (fg)*: hP(Y,B) c hP(Y,B) and gf: (X,A) c: (x,A), hence fV: hP(X,A) c: hP(X,A). Thus f is one to one and onto. 2.25 Lenroa : Let A c X and u be an open subset of the interior of A, then cP(X,A) n cP(X,Xu) = cP(X) . Proof : Clearly cg(X) c cP(X,A) cP(X,Xu) . If rp Â€ cP(X,A) then there exists a finite open cover ^U^ of A such that cp = on ^^^^^ n A^\ and cp Â€ cP(X,Xu) implies there exists a finite open cover XA^ of Xu 3 cp = on V ^^"""^ n A^^. Then if lA denotes the collection of open sets obtained by intersecting the members of ^ with the interior of A, and the members of U^ with the complement of u closure, we have that 1/ is a finite open cover of X such that c? = on U^^^^ Thus cp Â€ cgCx) . 2.26 Weak Excision Theorem : If k: (Xu,Au) c (x,A) and if u is an open set contained in the interior of A, then k*: hP(X,A) Â— > hP(Xu,Au) is an isomorphism. (Axiom 6 c). P+1 s n Proof: Let cp Â€ zP(Xu,Au) and define ): X > G by the equations <1; = cp on (Xu)^^^: '^ = elsewhere. Now cp Â€ cP(Xu,Au) implies there exists a finite cover XX of Au by sets open in Xu such that cp = on 1>((e>^1) n (Au)^^ writing ^ = [v. (Au)l <. i 1 n] where v. is open in X and V = [v. jl 1 i < n] U[u], we have that V is a finite open cover of A and * =0 on 'V H
PAGE 23
18 A^*. Thus ^ Â€ cP(X,A) and Z^ Â€ C^''(X,A), with 6k*'^ = 5cp Â€ cP^'(Xu). Hence k*^ 6 ^ Â€ cP^'(Xu) and so 6i^ Â€ C^^(X,Xu). By the previous lemma then, 6if Â€ C^ (x) . Thus iji Â€ zP(X,A) # Â•stand k is onto. To see that k is one to one we let c? Â€ zP(X,A) such that k^("p) Â€ bP(Xu,Au). Then k^(cp) = cpj^ + ^(cp^), for some cp^ Â€ C^(Xu) and cp^ Â€ C^" (Xu,Au) . Define ij/^: X^ Â— > G by i{f ^ = cp^ on (Xu)^; i; ^ = elsewhere. Then as before ilf^ ^ C^ (X,A) hence ^(il/p) ^ C^(X,A) and so cp 6(^2) ^ CP(X,A). Now k*^[rp ^(ilr^)] = k'^cp ^(k^^^) = k'^cp Z{cf)^) = rp^ Â€ cP(Xu). Thus cp 6 ('If 2) ^ cP(X,Xu). Again by the previous lemma, we know there exists 'K Â€ C^(X) such that cp Z {^ ^) = ^ ^. Hence ^ = ^^ + ^(*2^ ^ *^0^^^ "^ 6[cP^(X,A)] = bP(X,A). 2.27 Theorem : Let f,g: (X,A) Â— > (Y,B) and let cp Â€ cP(y) . Let y be an open cover of Y such that S^cp = on 1/\^'^) and cp = . On U^^'^' D B^"*. Finally let U be a finite open cover of X such that u 6 ?^ implies f(u) U g(u) U v for some v Â€ lA . Then f'^cp, g^cp Â€ Z^(X,A) and f*^p g^cp Â€ bP(x,a). Proof : ?X is a finite open cover of A and if (xq, ...,x ) Â€ u^ A^ , where u ^ %L , then x. Â€ u for <_ i <^ p. Hence there exists v Â€ 1/ such that f(u) U g(u) <= v. Then f(x^), g(x.) Â€ v PI B for <_ i _< p, hence (f [xq] , . . . , f [x^] ) , (gCx^] , . . . , g[Xp] ) Â€ v^^ n B^^ and f cp(xq, . . .,x ), g^cp(xQ, . . . ,x ) = 0. Thus f^cp, g*^cp 6 cP(X,A). Also [6f*(cp)](xQ,...,x^j_) = [f^(6cp)]
PAGE 24
19 (xq, ...,x^^) = ^cp(f[xQ], . ..,f[x^^]) = and [^g^(cp)] \ c P+2 ) t u , since (Xq, ...,x j^) = for (xq,...,x ^) Â€ u^ , since (f[xQ], . . . ,f [Xp^j^]) and (g[xQ] , . . . , g[x^^ ] ) are in v^ and 6cp(v^^) = 0. Hence lq% and Sf'^co Â€ C^''"(X) and consequently f cp, g^cp Â€ Z^(X,A). Now 2^ is a finite open cover of X and if (x^,...,x ) Â€ u then there exists v Â€ 7/ ^0 P such that f(x.), g(x.) Â€ v, and consequently [ g(X),..., g(x. ), f (x. ), . . .,f (x )] Â€ v^^^ for 1 i <. p. Thus ^cp[g(xQ), . . . ,g(x^), f (x^), . . .,f (Xp)] = and so I Lo^^^^^^f^^^^O^^'^^^^i)^ f(x.),...,f(x )] = D^cp(x^, . . . ,x ) = 0. Hence D6cp = on ti^^ ' which implies D6cp Â€ C^(X), and we have that for p = (f*^ g^)cp = D^cp Â€ cg(X) c bP(X,A). Also ^[EKp] Â€ l[C^~^{X,h)] if p 2 1^ for if (xÂ„j . . . ,x ) Â€ u^ n A^ then Dcp(x , . . . i^rji) ~ I i=o(^)^^f'?(^o)'"^(''i)' f(^i).f(Vl^^But f(x.), g(x.) Â€ V n B for some v Â€ l^ and <_ i <_ p1, hence [g(xQ),...,g(x.), f(x.),...,f(Xp_^)] 6 vF^^ B^^ which implies fp[g(xQ) , . . . ,g(x^) , f (x^) , . . . , f (x j^) ] = a nd therefore Dcp(uP n A^) = 0. Thus Dcp Â€ C^''(X,A) and f^cp g^cp = 6Dcp + D^cp Â€ bP(X,A). The following Corollary is an extension of a fundamental lemma proved, at Wallace's suggestion, by Capel in [1] . 2.28 Corollar y: Let B be closed in the space Y and let h 6 H^(Y,B). Then there exists such a finite open cover, ^^\/{'h) , of Y that; if f,g: (X,A) Â— > (Y,B) are maps such
PAGE 25
20 that for each x in X,f(x), g(x) Â€ v(x) for some v(x) Â€ 1/{^), then f*(h) = g*(h) . Proof : Let h Â€ H^(YjB), then there exists cp Â€ zP(Y, B) such that Sep = h. Since B is closed we apply leinma 2.^, part 4, to yield a finite open cover U of Y with S'D = on V^^^^ and cp = on B^^^ n V^"^^) . Define ?^ = {f^v) g (v) v (z \f ] . Then 2^ is a finite open cover of X such that if u Â€ ti , there exists v f If such that f(u) U g(u) c: V. Thus f cp g^cp Â€ bP(X,A), and f (h) = g (h) . 2.29 Definition : Let L/. and (o be families of subsets of a space X and let C cr x. 1) (2 refines ^ ( (^ < (g ) if f A Â€ (^ implies A c B for some B Â€ (8 . 2) St(C,^ ) = U[aa Â€ 0. and A C 4 D]3) st( a ) = (st(A, a )A Â€ a }. 4) (2 star refines (g iff St( d ) < (3 We will need the following two results which are well known and will be stated without proof. 2.30 Theorem : A space X is normal iff for each finite open cover t// of X there exists a finite open cover TJ such that St{ 1/ ) < lA . 2.31 Modification Lemma : If A is a subset of the space X, if 6/ is an open cover of Xj if '?/ is an open cover of X such that St( %i) < If and if P = St(A, %( ) then there is a function f: (X,P) Â— > (X,A) such that
PAGE 26
i) f(x) = X for X Â€ A U (XP) ii) If u Â€ 7/ then there is a v Â€ 1/ such that u U f (u) c V. 2.32 Definition : (X,A) is a normal pair iff X is a normal space and A is a closed subset of X. 2.33 Notation : If i: (P,Q) ^ (R^S) and if h Â€ hP(R,S), then h (P,0) = i (h) . If A c x we let AÂ° denote the interior of A, and A denote the closure of A. 2.3^ Expansion Lemma : If (X,A) is a normal pair and if h ? H^(X,A), then there exists an open set p 3 A and hQ Â€ ir(X,P ) such that h (X,A) = h. Proof: Let h Â€ hP(X,A), cp Â€ 7p{X,h) such that a(cp) = h. Then there exists a finite open cover t/ of X such that cp(v^'' D A^"*) = and 6h(v^^) = 0. By theorem 2.30 there exists a finite open cover 2X of X such that St(^ ) < V' By lemma 2.31 there exists f: (X,St(A, t( )) Â— > (X,A) with f(x) = x for x Â€ A U (XSt(A, H )) and such that u Â€ t/ implies there exists v Â€ 1/ such that u U f(u) c V. Let P = PÂ° be such that AcpcP c=st(A,t^), then f"cp Â€ zP(X,P*), for if (x^, ...,x ) Â€ u^^ p*I^^ then f(x.) Â€ f(u) n A and 3 vÂ€ T/such that u U f(u) c v. Hence f'^cp(xQ, ...,Xp) = cp(f[xQ],...,f[Xp]) Â€ cp(v^^ n A^^) = and so f^'~? 6 C^(X,P ). Similarly if ('^q' ' ' ' Â•Â• '^txl^ ^ ^ * then (f [xÂ„] , . . . , f [x , ] ) <^ v^ for some v f. If , hence ^f ^(Xq, . . .,Xp^^) = f Zc9{xq,. . .,x^^) = ^'Â•p(f[xQ],. .,f[x^j^])
PAGE 27
22 Â€ 6cp(v^^) = 0. Thus f^^cp Â€ ^"'[cP^'(X) ] and so f^^cp Â€ zP(X,P ). Now if i; (X,A) <= (X,A), then we have i,f: (X,A) Â— > (X,A) and co, tY , 1/ satisfy the conditions of theorem 2.2.1, hence i'*:p f'cp Â€ B^(X,A). Consequently if we let q: Z^(X,P ) Â— > H^(X,P ) denote the natural horaomorphism, then taking h^ = e[f'^cp] we have hQ(X,A) = h. 2.35 Theorem: Let (X,A) be a normal pair and let A c= M = M c X. Then, if h Â€ hP(X,M ) and if h(X,A) = 0, there exists an open set N such that A c N c N c M and h (X,N ) = 0. Proof : i: (X,A) c (x,M ). Let h Â€ hP{X,M ), cp Â€ Z^(X,M ) with S'p = h. Then i (h) = implies i'^cp Â€ bP(X,A), hence cp = cp^^ + l{^^) where ^^ Â€ C^ {x) , "Jg^ ^^" (^'^) for p 2 1. Thus there exists %/>> a finite open cover of X such that t:?^ = on U^^^^ H A^, ^^ a finite open cover of X such that c; = on XC^^ ' , and 1/^ a finite open cover of X such that cp = on t/lvP+1) ^ m*^""". Let V = [vj_ n Vp n v,v. Â€ Xf . , i = 1,2,3,}^ then ^ is a finite open cover of X satisfying the same conditions as XJ^, '^Xy^ni and l/^ separately. Let t^ be a finite open cover of X such that St( t/< ) c V and let N = NÂ° be such that A c N c: N c [St(A, t< ) fl M] . Then there exists f: (X,St(A, IX )) Â— > (X,A) with f(x) = x for x Â€ A U [XSt(A, V( )] and u Â€ t/ implies there exists v 6 7/ such that u U f(u) c V. Now as before i,f: (X,N ) Â— > (X,N ) and cp, lA , '%f , satisfy the conditions of theorem 2.27,
PAGE 28
23 hence i'*Â© f'^cp Â€ bP(X,N ). But f% = f^'cp^ + f''^Z{:r, ) = f'cp^ + ^(f cpp) . By a proof similar to that of the expansion lenuna f'cpp Â€ C^ (X,N ), and if (xÂ„, ...,x ) Â€ u^ then there exists v such that u U f(u) c v so that (f[xQ], . . .,f[Xp]) Â€ v^ . Thus f ^cp^(xq, . . .,Xp) = cp, (f [xÂ„], . . . , f [x ]) = and consequently f^cp. Â€ Cq(X) . But then f^'cp e B^{X,N ) so that c? f^cp Â€ B^(X,N ) implies cp Â€ B^(X,N ). Finally Bcp = h and we have h(X,N ) = 0. 2.36 Notation : Through the Map Excision Theorem let (X,A) and (Y,B) be normal pairs and let f: (X,A) Â— > (Y,B) be a closed map such that f takes X A topologically onto Y B, that is, g; (XA) Â— > (YB) where g(x) = f(x) is a homeomorphism. 2.37 Lemma : If M^, M^ => A, then M^ = f" f(M^) for i = 1, 2, and f(Mj^ M^) = f(M^) f(M2) = [B U f(M^)] [B U f{M^)]. 2) If M is an open set containing A, then B U f(M) is open. 3) If B c N = B U ff"'(N) Â— 1 * 4) If N is an open set containing B then f (N ) = 2.38 Lemma : Let M be an open set containing A and define f^: (X,M ) Â— > (Y, B U f(M )) by fQ('<) = f(x) for each X Â€ X. Then f^: H (Y,B U f(M )) ^ H (X,M ). Proof: Choose N = NÂ° such that ACNCN CM=MÂ°.
PAGE 29
24 Then if we let fj_: (XN, M N) Â— > (Y[B U f(N)]^ [B U f(M*)] [B U f(N)]) be defined by f^(x) = f(x), we have that f, is an onto homeomorphism. For f(XN) = [B U f(X)] [B U f(N)] = Y[B U f(N)] and X A contains X N. Similarly f(M N) = [B U f(M )] [B IJ f(N)]. Now f(N*) = [f(N*)]* implies [f(N)] c f(N*) <= f(M), thus [B U f(N)] = B U [f(N)] = B U [f(N)] c B U f(N ) c B U f (M) c: B U f(M ) and hence B U f(M) open implies [B U f(N)]* c: [B U f (M*) ]Â°. Now let i: (Y[B U f(N)], [B U f(M )] [B U f(N)]) c (Y,B U f(M )); k : (XN, M N) c {X,H ). By theorems 2.24 and 2.26 we have that f,, i , and k are isomorphisms, hence f ^^ = k f i is an isomorphism. 2.39 Map Excision Theorem : f : H^(Y,B) s hP(x,a) . Proof ; Let h Â€ hP(X,A), then there exists M = MÂ° =5 A and h^ Â€ hP(X,M ) such that i (h ) = h; i: (X,A) c (x,M ). Let h^ = j*fQ"^(h^) Â€ hP(Y,B), j: (Y,B) c (y,B U f(M*)); then f (h^) = f j fo (h^)=i fofo (^o^ = ^ (^o^ = ^Thus f is an epimorphism. Let h Â€ K(f ) then there exists N = NÂ° 3 B and h Â€ hP(X,N ) such that 1 (h ) = h, 1: (Y,B) c (Y,N ). Thus if f,: (X,f""'"(N)) Â— > (Y,N ) is defined by f^{x) = f(x)> then f ^(h^) Â€ hP(X, f" (N) ) is such that f, (h ) (X,A) = 0. Hence there exists M = M such that A c M c M* c f~"''(N) and k*f*(h^) = f*(h^)  (X,M ) = 0. Let i^: (Y,B U f(M*)) c (Y,N*), j: (Y,B) ci (y, B U f (M )), then j i =1 and h = j i^(h^) = j fQ he f3_(h^) = j fQ (0) = 0. XThus f is a monomorphism.
PAGE 30
25 2.40 Full Excision Tlieorem : If X is a normal space and if X = A U B where A and B are closed subsets of X and if f; (A, A n B) c (X,B), then f*: hP(X,B) ^n^{h,h '^ B) . Proof : We have A (A B) = A B = (A U B) B = X B, and A = A implies that sets closed in A are closed in X, so that f is closed. Thus the hypotheses of the Map Excision theorem are satisfied and the conclusion follows. 2AI Lemma : In the diagram of groups and homomorphisms, if the diagram is analytic and h = gf is an isomorphism, then f is a monomorphisra, g is an epimorphism, Q = 1(f) f K(g), and g takes l(f) isomorphically onto R. y' X > R 2.42 Lemma : In the diagram of groups and homomorphisms, if each triangle is analytic, iCf^) = K(g^), l(f2) = ^(gg)^ and h, , hp are isomorphisms; then: i) f: P^ X P^ 2Q where f(Pj^, P2) = fi(Pi) + f2^P2^ ii) g^ ^ 92' Q " ^1 ^ ^2 iii) If q Â€ Q, then q = f2h2'^g2(q) + f^^ <3i{^) R 2.43 Corollary : If X = A U B, X is normal and A and B are
PAGE 31
26 closed, then the appropriate inclusion maps induce isomorphisms; H^(X,A) X hP(X,B) s U^{X,A B) a E^{B,h H B) K hP(a,a n B) . Proof: Consider the diagram: H^(X,A) H^(X,B) we have l(ii) = K(jp) and l(j.) = K(ip), analyticity, and by the Full Excision Theorem i , j are isomorphisms. Thus ip X jp and cp, defined by cp(h,, hp) = ii (^i) + ji (hp) are isomorphisms by the previous lemma. 2.4^ Map Addition Theorem : Let X = X, U XÂ„ be normal with X, and Xp closed and A = X, Xp. If f^^^, fp, f : (X,A) Â— > (Y,B) are continuous and if f Â• (x) = f(x) for each X Â€ X^, f(X^) c B,i = 1,2; then f = f^ + f** Proof : Consider the following diagram, where the homomorphisms not induced by the mappings mentioned in the hypotheses are induced by the corresponding inclusion maps. hP(Xj^,Xj^ n Xg) hP(x,Xp) n^(x,x^ n x^) H^(Y,B)^^ hP(X2,X^ Xp) hP(x,x^) We use lemma 2.42 to write f (h) = i,i ~ ipf (h) +
PAGE 32
27 * * # jij ~ jpf (h) . By analyticity then f (h) = i,fp(h) + j,f,(h)= f2(h) + fjL(h) and so f = f^ + Â£5* 2.43 Reduction Theorem : Let (X,XÂ„) be a normal pair and let A be closed in X. If h Â€ hP(X,Xq) and if h(A,A H Xq) = 0, then there exists an open set M 3 A U XÂ„ such that h I (M , X^) = 0. Hence there exists an open set N about A such that h(N , N n Xq) = 0. Proof: Consider the following diagram: hP(m*, xÂ„) H^(X,X^)Â« H^(A, A n X^)* 0' H^(M , M n X ) g = 1. "Lip hP(x,a U X^) 0' 11^ (A U XÂ„, X_) 0H^(X, M ) where all the homomorphisms are induced by the corresponding inclusion maps. Let h Â€ H^(X,Xp^) and k (h) = 0, then i (h)= since j is an isomorphism by the Full Excision theorem and k, = j Iqj hence h Â€ K(i ). Also, we get l(i,) = ^(^q) f^o'" theorem 2.1( applied to the triple (X,A U X^^, ^0^' ^"*^ ^Â° there exists h^ Â€ hP(X,A U Xq) such that i^CI^q) = hBy the Expansion Lemma there exists an open set M 3 (A U Xq) and h Â€ Ir (X,M ) such that ip(h ) = h . Applying theorem 2. 16' to the triple (X, M , X^) we have K(f ) = I(g ). Thus h(M , X ) = g (h^) = h implies f (h) = 0. Now let M = N for the second part of the theorem.
PAGE 33
28 2.46 Extension Theorem ; Let (X,X^) be a normal pair and let A be closed in X. If h Â€ H (A,A D X^^) then there exists 0' n open set M r5 (A U XÂ„) and h Â€ R^{M , X ) such that 0' I h I (A, A XÂ„) = h. Thus there is an open set N about A and h hP(A U X^,XÂ„)Â— > H^^(X,A U X^) > H^^(X,X^) hP(m , Xq) > h^^(x,m') 0' Let h Â€ H^(A U XÂ„, ^r)) Â• Then by the Expansion Lemma, there exists M = MÂ° 3 (A U XÂ„) and h, Â€ hP^'(X,M*) such that hj^Kx.A U Xq) = 6(h), thus h^(X,XQ) = 6(h) (X,Xq) = by exactness. Hence there exists hp Â€ H^(M ,Xp^) such that 5(h2) = hj_. Then 6[h i*(h2)] = 6(h) 6i*(h2) = 6(h) i 6(hp) = implies i*(h2) Â€ K(6) = l(k*). Thus there exists h^ Â€ H^(X,X,) such that k (h,) = h i (hp) . Define h^ = h2 + ^* ij^^) ^ hP(M*, X^), then h^  (A U Xq,Xq) = i*[h2 + 3*(h3)] = i*(h2) + i*j*(h^) = ^* {"n^) + k*(h^) = i (hp) + h i (hp) = h. Now f : H^(A U X ,Xq) ^ hP(A,A n Xq) hence we let h Â€ H^(A,A fl X ) then there exists M = MÂ° o 0' (A U Xq) and h^ Â€ H^(M ,Xq) such that i (h^) = f (h) Therefore f i (h ) = h, or equivalently h  (A, A ( For the second part of the theorem let M = N. 0'
PAGE 34
29 2.47 Lemma: In the diagram of groups and homomorphisms if each triangle is analytic, 1(3^,) = K(i ) for a = 1,2, a' i i =0, and k,.k^ are isomorphisms, then hk^ t, = h^k^ t^ o'o 12 111 22? 2.48 Theorem : Let X be a space and let B c A ^^ Xj^ n X'= * * _* _*Â• _ X, U XÂ„ = X. Let (j ,i ,6 ), (T ,i ,5 ), (j ,i ,6 ), and (j ,i ,6) be the homomorphisms in the exact sequences for the triples (X,X^,B), (X,X^,A), (X^,A,B), and (X,A,B) respectively for a = 1,2, If kj^: (XÂ„,a) c= {x,X^) and k^: (Xj^,A) c: (X,Xp) induce isomorphisms for each p >_ 0, then there exists an exact sequence: > hP~^(a,b) ^ Â— > hP(x,b) ^^> hP(x^,b) xhP(X2,b) > hP(a,b) _* _* .* *li where J =i, xii =i^i, and A = jpk^ ^1 ~ .* *lr "3 1^1 ^'2Proof : Consider the following diagram;
PAGE 35
30 hP(X2,A) H^(X,X^) rP/v H^(X^,B) hP1(a,b) ^ hP(x,a) H^(X,B) hP(a,b) hP(x^,a) hP(X,X2) H^(X2.B) That the conditions of lemma 2.47 are satisfied follows from theorem 2.20, hence jik," Â£Â„ = jpkp 6 . The proof proceeds with seven parts. 1) Let h Â€ ZÂ°(X,B) with (ij(h), i^C^^)) = in ZÂ°(Xj_,B) X ZÂ°(X2,B), then i^(h) = and igl^) = 0. Hence if X is in X^ then [i^(h)](x) = h[i^(x)] = h(x) = 0, similarly x in Xp implies h(x) = 0, therefore h(x) = for all X in X and we have h = 0. Thus J is a monomorphism on H (X,B) . 2) 1(A) c K(J ). If h Â€ 1(a) then there exists _* *l. .*. *It h' e H^(A,B) such that h = j ^^^^~^6 ^{h' ) = "Jg^g ^1^^ )* Hence i^(h) = iiJit^i" ^p(^ ) ] = 0^ similarly ip(h) = 0. Thu s J (h) = (iL(h), i2(h)) = (0,0) and h Â€ K(j ). 3) K(J ) e 1(A). Suppose J*(h) = (0,0), then i^(h) = ipi^) ~ and h Â€ K(i,) = l(j,). Hence there exists _* x* * h^ e H^(X,X ) such that J3^(h^) = h. Now Jp^l ^ ^2^1 and
PAGE 36
31 _* * . . * , , * * K{j*) = 1(^2) ^"^ ^1 ^ hP~'(A,B) such that ^^2(^1) = ^i(^i)* which implies k, 62(h,) = h and so jik^ ^P^''^!^ ~ ^l^'^l) ~ h. 4) I(J*) ^ K(I*). If (hj^,h2) Â€ I(J ) then there exists h f H^(X,B) such that J (h) = (h^,h2) = (ij^(h), i2(h)). Therefore I (h^,h2) = I (i^lh), i2(l^)) = iiii(h) I2i2(h) = i*(h) ^* {^) = 0, and (h^,h2) Â€ K(l*). 5) K(I*) c I(jf. We note that 5^1 = k2S2 ^^'^ ^^^ (h^,h2) Â€ K(I*). Then i*(hj^) 12(^2^ " ^' hence ^2^2(^2^ " __)* * 6,i,(h,) = 0. Thus hp Â€ K(6 ) = l(io) and there exists I hp Â€ H^(X,B) such that i2(h2) = hp" Similarly h^ 6 l(ij_) and honce there exists h, Â€ H^(X,B) such that inl^ii) = ^i Â• Nqw i (113^^2) = i (h^) i (ho) = iLi3_(h^) i2i2(h2) = i,(h,) Ip(h2) = 0, hence there exists h (z H^(X,a) such that j*(h^) = hj h' But H^~'^(X,A) = lil^) ' ^(^p) ^^~ plies h^ = llil"^!) + 3'2(''''2^ where cp^ Â€ H^(X,Xj^) and CP2 Â€ hP(X,X2). Thus j (h^) = j ^(^i) + J 3'2(""2) = ^l^'^^l^ + ^2^'"^2^ = h^ h2, which implies h^^ J^^C'^^^) = h2 + J2('^2^* Let h = h^ j*(^o^) = ^2 + J2('^2^ ^ H^(X,B). Tlien J*(h) = (i*(h), i2(h)) = (ii[h{ j*(co^)], i2[h2 + J2(^P2^^^ " y I .^ JL ^> I Jtf. y y I Â«. I (il(hi) ij^jj^Ccp^), i2(h2) i2J2('P2^^ " (ilC^i)' i2^^2^) " (hj^,h2). Therefore (hj^,h2) ^ l(J )Â• 6) 1(1*) c K(A). If (hj^^h^,) f U^{X^,B) X hP(X2,B) * , . * then I (hj^,h2) = ii(hL) i2(h2) and ^2tii(hjL) " i2(^''2^ ^ " ^2^1(^1) = k*6^(h^). Thus j*k*^^^[I*(h^) I*(h,)] = j*k*"^k^^(h^) = j*6^(h^) = 0.
PAGE 37
32 7) K(a) c I(I*). Let h Â€ K(a), then j *k*"'62(h) = 0, At I ^ hence k, 6^{h) c K(j,) = l(5,). Thus there exists h, Â€ H^(X,B) such that 6, (h,) = k, 5p(h) which implies Â•X5t that k^6j^(hj^) = ^^(h) and so ^g^l^^l^ " ^2^^^ Â°^ 6 [ij^(h ) h] =0. Since then IL(hj_) h Â€ K(62) = l(io) there exists h Â€ H^(Xp,B) such that ip(h^) = ii (^i) h and h = i,(h,) Ip(hp) = I (h^^h^). Consequently^ h 6 l(l ), 2.^9 /absolute MayerVietoris Sequence : Let X be normal and let X, ,XÂ„ and XÂ„ be closed subsets of X and let X = X, U Xp. Then there exists an exact sequence; . . . Â— Â— > hP"'(Xj^ n x^, x^ Xg n x^) ^> h^{x,x^) ^> HP(Xj_,x^n Xq) X E^{x^,x^ n Xq)^> HP(Xj_n x^^x^^n X2nxQ) A 2 > where letting t : (X ,X n X^) cz (x,XÂ„), s : (X^ n X^, X, n XÂ„ ri X^) e (x ,X n X^) for a = 1,2, we a "^ 1 2 1 2 0' ^ a a 0'' have J = t, X tÂ„ and IÂ„ = s, Sp. Proof : Write X^^ = X^ U X^, X = X U X^, A = X^ n Xp = (X^ n X ) U X and B = X . Then k, : (X U X , (X^ n X^) U Xq) c (x,X^ U Xq) and k^: (x^ u x^, (x^ n x^) u XÂ„) c: (X,Xp U X^) are isomorphisms by the Full Excision Theorem. Hence we combine theorems 2.20 and 2.48 to write the analytic " ladder" below, with exact upper " leg".
PAGE 38
33 * > hP(X,Xq) ^ ^^hP(X^ U Xq,Xq) X hP(X2 U Xq,Xq) l"^Â„ J"" ,M H 2 2> hP(x,Xq) Â— ^^^Â— >hP(x^,x^ n Xq) X hP(X2,X2 n x,) T* ^Â— >hP([x^ n x^] u Xq,Xq) 1 Wi * Â— ^>hP(Xj_ n x^^x^ n x^ n x^) The homomorphisms w.,i = 1,2,3^^ are understood to be in* * Mduced by the appropriate inclusion maps and w = w, x w By the Full Excision Tlieorera w, ,Wp, and w^ are isomorphisms, * while w^ is the identity isomorphism. We define JÂ„ = w J w^ , I = w^,! w , A = w^AWh . Now I ^ J^ = WhI w w J Wo = W.I J w~ = 0, hence I(Jq) <= K(I ). Conversely if Iq(^) = 0, then Wi,I w ~ (co) = which implies I*w*~'(^;) = and w*""^(:p) Â€ K(l*). Thus w*"''(r;) = J*(9), hence cp = w J (fl) = JqWo(&) and cp Â€ I(Jq)Consequently K(Iq) c I(j ). Similar computations show l(l,^) = K(A ) and I(A^) = K(Jq) and that I^ = s* s*, Jq " ^1 ^ ^2' 2.50 Theorem : If X is connected, normal and T, , and if H (X) = (0], then X is unicoherent. (g 4 [0]) Proof: Let A,B be closed connected subsets of X with X = A U B. Using the MayerVietoris Sequence with X, = A,Xp = ^^^o ~ ^^^ ^^^ fixed x Â€ A fl B we have the following diagram: HÂ°(X,x) Â— 2Â— > X ^> HÂ°(A n B,x) Â— 2_> H^(x,x) s= H^(x) HÂ°(B,x)
PAGE 39
34 Since HÂ°(X,x) = [0] by theorem 2.10, and H'(X,x) ^s H^(x) = (0] by assumption and by theorem 2.l6 and corollary 2.18, we have that 1^ is an isomorphism. Applying theorem 2.10 again yields the desired result.
PAGE 40
CHAPTER III A GROUP ASSIGNMENT FOR A SPECIAL CLASS OF SPACES We let X be a space with the property that the intersection of two arbitrary open connected sots is the union of open connected sets. (e.g. any locally connected space, or the circle after replacing a proper subarc by the closure of the sin (x~ ) curve). Then if G is a fixed abelian group we will define, for each positive integer p, an abelian group H^(X) in much the same way as in Chapter II; with the notable distinction that now the open covers of interest will consist of connected sets. Initially it was hoped that this new group assignment would be such that spaces which were not locally connected would be assigned the trivial group. We show by example that this hope is not realized. 31 Definition : If lA is an open cover of X and if each element of l/( is a connected set, then we call U. an open connected cover of X. We let C^(x) be as before and define C^(X) = [cpcp Â€ C^(x) and there exists an open connected cover IX of X with cp = on 'H*^ ' ] . Remark : Throughout this chapter all spaces under
PAGE 41
36 consideration are assumed to have the property for open connected sets postulated above for X. It should be clear that this property is sufficient to guarantee that Cq(X) is a group. 3.2 Definition : Let zP(X) = ^"''"[C^'(x) ] and p >. 0, and bP(X) = cg(X) + 6[CP^(X)], p > 1; BÂ°(X) = [0]. 3.3 Theorem : If f : X > Y is such that the inverse image of an open connected set is open and connected then: 1) f^[cP(Y)] c cP(X) 2) 6[CP(X)] e C^l(X) 3) f''[zP(Y)] c zP(x) 4) f'[BP(Y)] c bP(X) Proof : 1) Let cp Â€ C?(Y), then there exists an open connected cover '//' of Y such that cp = on 1/^^ '' . Consider ( f ~ (v) v Â€ !/ ], by assumption this is an open connected Â— 1 D+l cover of X and if (xÂ„,...,x ) Â€ [f (v) ]^^ then [f'^cp^XQ^..^Xp) = cp(f[xQ],...,f[Xp] Â€ cp(v^^) = 0. Thus f ^cp Â€ C^(X) . 2) Let p Â€ C^(X) and t< be an open connected cover of X such that cp = on %aJ.P^ z . Let (^q^ Â• Â• Â• > ^r^i ) Â€ uF^^, then [ 6cp] (xq, . . . , x^^^) =V ._ (l)^c? (Xq,..., x^,...,x j^) = since (xq, . . . , x^, . . . , x ^) Â€ u^ implies cp(Xq, . . .,x^,. ..,x^j_) = 0. Thus 6cp Â€ C^'(x). 3) Let cp Â€ zP(Y) = 6~^[cP+^(Y)], then 6cp Â€ cP^'(Y) hence f^Scp Â€ f '^[C^'(y) ] c C^^(x). Now f'Scp = Ff'cp and so f^cp Â€ 5~^[C^^(x)] = zP(X) .
PAGE 42
37 4) Let cp Â€ bP(Y) = CP(Y) + ^[cP^Y)], then f'^cp Â€ f^[cP(Y)] + f'^[cP^(Y)] c cP(x) + ^[cP^(X) ] = bP{x), p >^ 1. The case p = is clear. 3.4 Definition : n^{X) = zP(x)/bP(X) . 3.5 Theorem : Let f : X > Y be such that the inverse image of an open connected set is open and connected, then there exists a unique homomorphisra f : H {Y) > hP(x) such that af^= f*S, where a: zP(X) > hP(X) and 3: zP(Y) > hP(y) are the natural homomorphisms. Proof: Induced Homomorphism Theorem. 3.6 Theorem : If i: X rz X, then i : H^(x) c h^(x) . Proof : Let h Â€ hP(x) and p 6 Z^ {X) with ac? = h. then i h = i acD = ai**":; = acp = h. 3.7 Theorem : If f : X > Y, g: Y > Z are such that the inverse image of open connected sets are open and connected then (gf)* = g*f*: hP(z) > hP(Z) > n^{x). Proof ; We already know from Chapter II, that (gf)'^= f'^g". By part three of Theorem 3.3 letting a: ZP(X) > hP(x), q: zP(y) > hP(y) and y: zP(Z) > hP(Z) we may compute as follows: If h Â€ hP(Z) and cp Â€ zP(Z) such that yj? = h then f g (h) = f g (v") = f*P(gÂ«cp) = af''(g^cp) = a(gf)^cp = (gf) y^ = (gf) (h) . 3.8 Example : Let X=AU BU CM D where A = UOiV) 11 1 y 1 1)> B =t(x, sin(x"')) 0 < X < n"^].
PAGE 43
38 C = [(x,0) 11 1 X <. 0} and D = [(x,y) x = 1 and 2 <. y <. 0, or y = 2 and 1 <_ x <_ tt~ , or x = tt and 2 <_ y <. 0], (see figure 1). Let R c: x x x be defined by R = (A x a) U D(X ) and let Y = x/R. It is easy to see that the removal of any pair of points from Y disconnects Y, and hence Y is topologically the unit circle. Since the definition given in this chapter is equivalent to that given in Wallace's notes, for locally connected spaces, we know that H (Y) G. (Fig. 1) Let f: X X/R be the natural map. Then since X is compact and f is monotone and continuous we have that the inverse image of an open connected set is open and connected. Define g: Y > X by the equations g[f(x)] = x if x ^ A and g[f(A)] = (0,0). We note that if u is an open connected subset of X not containing (0,0) then uA is open and connected. To see that the inverse image under g of an open connected set u is open and connected we consider two cases. 1) If (0,0) ^ u, then g~'''(u) = f(uA) is open and connected since f restricted to XA is a homeomorphism. 2) If (0,0) Â€ u, then g~ (u) = f(u) is connected since f is continuous and is open if there is an open set about f(A) contained in g~ (u) . Now (0,0) Â€ u implies there
PAGE 44
39 exists t Â€ (1,0) and z Â€ (0,n~ ) such that (t,0) x [o] = P c u and ((x,sin x~)0
PAGE 45
CHAPTER IV FINAL COMMENTS AND QUESTIONS We first show by example that the cohomology groups as defined in Wallace's notes are different from those defined in Chapter II. 4.1 Example : Let G be an additive abelian group with the property that for each g in G there exists a positive integer n, depending on g, such that ng = 0. Suppose also that G is not nilpotent, i.e. there does not exist a positive integer n such that ng = for all g in G. Letting H^(X) denote the pth cohomology group for the space X, computed with respect to G, as defined in [8], and recalling that H (X) is isomorphic to the group of functions mapping X into G which are continuous in the discrete topology of G, we easily show that H (x) is not isomorphic to HÂ°(X) for X = G. For by theorem 2.9, cp Â€ ZÂ°(X) == HÂ°(X) if and only if cp: X > G is continuous in the discrete topology of G and {cp (g) g Â€ G] is finite. Let {cp(g) g Â€ G} = {tp~ (g.) l <. i <. n) and let p. be such positive integers that p.g. =0, 1 <^ i <_ n, and define p = p. X p X...X p .' Then pep = Â€ Z (X) and hence each 40
PAGE 46
element in H (X) has finite order. But it is clear that i: G '^ G is continuous in the discrete topology and has order zero, hence H (X) contains an element which is not of finite order. Now Haskel Cohen in [2] gave a definition of codimension, based on the groups of [8], for locally compact spaces, and we have just seen that on locally compact spaces the groups of [8] differ from those of this thesis. It is natural to ask then, whether a definition similar to Cohen's will yield a dimension theory and if so, will this theory differ from Cohen's. Without reproducing the technical definitions we remark that the dimension of a space was defined in terms of the cohomological structure of its compact subspaces, and since the two cohomology theories involved agree on compact spaces it turns out that the aforementioned " similar" definition is, in fact, identical with that of Cohen's. We have shown in Chapter II that we have constructed a cohomology in the sense of Mac Lane [<:] but have not demonstrated that we have a cohomology theory in the sense of Eilenburg and Steenrod. Indeed, we have failed to verify the " Homotopy Axiom" of [3]. Although Lemmas 2.23 and 2.19, in the presence of the Reduction Theorem, were shown by Keesee in [4] to imply the Homotopy Axiom for compact pairs, there is no such short cut for the general case. We are unable, however, to produce an appropriate counterexample and hence must leave the question open. Because the Cech theory
PAGE 47
k2 based on finite coverings fails to satisfy this axiom we are led to conjecture that the axion also fails for the theory of Chapter II.
PAGE 48
BIBLIOGRAPHY [1] C.E. Capel, Inverse Limit Spaces, Duke Mathematical Journal, vol. 21(1954) pp. 233245. [2] Haskel Cohen, A Cohomological Definition of Dimension for Locally Compact Hausdorff Spaces, Duke Mathematical Journal, vol. 21(1954) pp. 209224. [3] Samuel Eilenburg and Norman Steenrod, Foundations of Algebraic Topology, Princeton (I952). [4] John W. Keesee, On the Homotopy Axion, Annals of Mathematics, vol. 54(1951) pp. 247249. [5] Saunders Mac Lane, Homology, SpringerVerlag (I963) [6] T. Rado and P.V. Reicheiderfer, Continuous Transformations in Analysis, SpringerVerlag (1955). [7] Edwin H. Spanier, Cohomology Theory for General Spaces, Annals of Mathematics, vol. 49(1948) pp. 407427. [8] A.D. Wallace, An Outline for Algebraic Topology I, Tulane University (I949) 43
PAGE 49
BIOGRAPHICAL SKETCH Marcus Mott McWaters Jr. was born January 21, 1939, in Little Rock, Arkansas. He was raised in New Orleans and Metairie, Louisiana, and was graduated in August, I956, from East Jefferson High School. In September, I958, he entered Louisiana State University in New Orleans and was awarded the degree of Bachelor of Science from this institution in June, I962. He then became a graduate student at Louisiana State Univeristy for one year, after which time he transferred to the University of Florida. From September, I963, when he entered the University of Florida, he has worked as a graduate assistant in the Department of Mathematics and pursued his work toward the degree of Doctor of Philosophy. Marcus Mott McWaters Jr. is married to the former Patricia Ann Guice and has one daughter, Sharon Lee. 44
PAGE 50
This dissertation was prepared under the direction of the chairman of the candidate's supervisory cominittee and has been approved by all members of that committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Councilj and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. April 23, 1966 Dean, Collegoy of^Atts and Sciences Dean, Graduate School Supervisory Committee Chairman . 0. ^,
PAGE 52
4660

