BORSUKULAM PROPERTY OF FINITE GROUP ACTIONS ON MANIFOLDS AND
APPLICATIONS
By
YURI A. TURYGIN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2007
02007 Yuri A. Turygin
To my parents
ACKNOWLEDGMENTS
I would like to thanks my advisor Alexander Dranishnikov for many encouraging
conversations over the years on various topics in topology which were of a great influence
on my mathematical education. I also would like to thank Yuli Rudyak for being
alvi, able to find time to discuss topology with me. His influence on my mathematical
education has also been substantial.
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS .................
ABSTRACT .........................
CHAPTER
1 BORSUKULAM THEOREMS ............
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
Introduction .. .............
BorsukUlam Theorem for (Zp)kactions
Calculation of i,_ 1 (l) .. .....
Euler Class of rlc: EG x Ic(G) BG .
Proof of Theorem 1.2.1 .. ........
BorsukUlam Theorem for Z2actions .
Necessary Lemmas .. .........
Computation of Norms .. ........
Computation of Euler Class e(( mp) .
Proof of Theorem 1.6.2 .. ........
2 HUREWICZ THEOREM AND APPROXIMATION OF MAPS ....
2.1 Introduction . . . . . . . .
2.2 Proof of Theorem 2.1.7 ........................
2.3 Proofs of the Approximation Theorems ...............
3 BORSUKULAM THEOREMS FOR MAPS WITH INFINITE FIBERS
Bula's Property ..........
Lipschitz Compactification ...
The Construction .........
APPENDIX
A BORSUKULAM THEOREM FOR Zp
OF SPHERES .............
REFERENCES ................
BIOGRAPHICAL SKETCH .........
x ... x ZpKACTIONS ON PRODUCTS
..................
..................
..................
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
BORSUKULAM PROPERTY OF FINITE GROUP ACTIONS ON MANIFOLDS AND
APPLICATIONS
By
Yuri A. Turygin
May 2007
C('!, : Alexander N. Dranishinikov
Major: Mathematics
This dissertation is devoted to several topics in geometric topology and dimension
theory. In the first chapter we discuss BorsukUlam theorems. We viewed the history of
the subject, stated a few classical results in this area and described a general approach to
proving BorsukUlam type theorems. The results of the author in this area are also stated
and proved in this chapter.
In the second chapter we discuss two closely related questions in dimension theory.
Namely, a fiberwise version of the classical theorem by Hurewicz about 0dimensional
maps of kdimensional compact into kdimensional cube and a conjecture by V.V.
Ui ni1:ii about approximation of kdimensional maps between compact by kdimensional
simplicial maps of polyhedra.
In the third chapter we outline a general geometric construction which shows how
it might be possible to use BorsukUlam type theorems for constructing an example of a
1dimensional map between compact which cannot be approximated by 1dimensional
simplicial maps of polyhedra.
CHAPTER 1
BORSUKULAM THEOREMS
1.1 Introduction
The famous BorsukUlam theorem is well known. It states that every continuous
map of a sphere S" into Euclidean space R' will necessarily collapse at least one pair of
antipodal points. It has been generalized by many authors. Among the first and most
memorable generalizations were by C.T. Yang [29] and D.G. Bourgin [1]:
Theorem 1.1.1. Let T be a fixed point free involution on a sphere S" and let f: S' R'"
be a continuous map into Euclidean space. Then the dimension of the coincidence set
A(f) {x c S f(x) = f(Tx)} is at least n m.
The later theorem stimulated a lot of interest in generalizations of the BorsukUlam
theorem. It became a starting point in the research of many authors on this subject. The
next important generalization belongs to P.E. Conner and E.E. Floyd [2] and it has first
appeared in their famous book.
Theorem 1.1.2. Let T be a differentiable involution on a sphere S" and let f: S' M'"
be a continuous map into a differentiable i,,' ,:'.l ./ M' of dimension m. Assume that
f*: H,(S'; Z2) + Hn(M; Z2) is trivial. Then the dimension of the coincidence set
A(f) {x c S'f(x) = f(Tx)} is at least n m.
The Theorem 1.1.2 became a cornerstone in the development of the BorsukUlam
type theorems. Its proof helped to shape up the general approach to proving generalizations
of the BorsukUlam theorem. Its importance can hardly be overestimated also due to the
fact that it has the famous theorem by J. Milnor [11] as one of its corollaries. The theorem
of J. Milnor asserts that every element of order two in a group which acts freely on a
sphere must be central (see [15] for details). The later theorem p1 i,'. d an important role
in the solution of the so called "spherical space form pin I i which aim was to give a
classification of all finite groups which admit a free action on a sphere.
In their consequent works H. Munkholm [12] and M. Nakaoka [15] showed that the
differentiability condition on the involution T in the formulation of Theorem 1.1.2 can
be dropped provided the target topological manifold M" is assumed to be compact.
Moreover, they generalized the previous theorem to the case of free actions of a cyclic
group Zp on (mod p) homology spheres. Their result reads as follows:
Theorem 1.1.3. Let a ;,/. 1.,' p'.rq' Zp of a prime order act freely on a (mod p) .,,., i'/.
nsphere N", and let f: N" ' Mm be a continuous map into a compact t'. '.y '..: '
i,,,,,.:f1 ,/,/ Mm of dimension m. If p is odd also assume that M is orientable. Suppose
that f,: H,(N; Zp) H H,(M; p) is trivial. Then the dimension of the coincidence set
A(f) = {x N f(x) = f(gx) Vg e Zp} is at least n m(p 1).
1.2 BorsukUlam Theorem for (Zp)actions
The purpose of this section is to ,I . 1 another generalization of the BorsukUlam
theorem which initially appeared in [24]. Further and until the rest of the dissertation p is
alvv assumed to be a prime number.
Theorem 1.2.1. Let M := N'" x ... x N" be a product of (mod p) b,,,.,I,; n ispheres
and let p: (Z,)k 0 M be the product of free actions p: Zp 0 N" (1 < i < k). If p is
odd also assume that all ni's are odd. For a map f: M R"I I ;,L.: a coincidence set
A(f) : {x MIf(x) = f(gx) Vg e (Z)k}. Then
dimA(f) > dimM (p 1)
provided ni > mpi'(p 1) for all i(1 < i < k).
Remark. For p = 2 and m = 1 the theorem above was ii,1',l. /ili proved by A.N.
Dranishnikov in [/3. In the case ni > m(pk 1) for all i(1 < i < k) the theorem above was
proved by V. V. Volovikov in [27]. Moreover, in the Volovikov's theorem the action p can be
assumed an arbitrary free action.
Let G be a group and let R be a commutative ring with a unit. Then by IR(G) we
denote the augmentation ideal of the group ring R[G], i.e. the kernel of the augmentation
homomorphism R[G] R. In this paper we assume R" to be a ring where multiplication
structure is given by multiplication of the coordinates.
The key ingredient in the proofs of the most BorsukUlam type theorems for maps
into Euclidean spaces is the following basic observation:
Lemma 1.2.1. Let G 0 M be a free action of a finite p. ',a' G on a '.,'.I/..y..,,/ 1 ,,,f,:.1./l
M. For a continuous map f: M R IR" I/. ; ,' a coincidence set A(f) : {x c Ml f(x)
f(gx) Vg E G}. Then A(f) / 0 if and only if the vector bundle : M XG IR (G) + M/G
does not have a non; ,,.I,.:,': i section.
Proof. First, note that every continuous map f: M R IR gives rise to a continuous
section s(f): M/G + M XG R'[G] of the vector bundle M: M xG RI[G] M/G defined
by a formula:
s(f)(xG)= (x, f(xg1)g)G.
gEG
Observe that E = F E where Em is a trivial mdimensional real vector bundle. Therefore
a projection 7r: M XG R m[G]  M X IRm(G) is well defined. Now define a continuous
section s(f): M/G M XG IRm (G) of by a formula s(f) := 7 o s(f). It is easy to see
that s(f)(xG) = 0 if and only if the orbit of x E M is mapped by f to a point.
Conversely, given a continuous section s of it defines a Gequivariant map s: M
M x R7[G] which is due to its equivariance must be of the form s(x) (x, ZcG f(xg1)g)
for some f : M R R, and the lemma follows. E
Usually, to prove a BorsukUlam type theorem for maps into Euclidean spaces one
shows that the Euler class of the vector bundle : M x Ipm (G)  M/G in a suitable
cohomology theory is nontrivial. Then the dimension restrictions on the coincidence set
A(f) follow (see the proof of Theorem 1.2.1). The theorems from [12, 13] were proved in
this way. Unfortunately, when one uses ordinary cohomology theory, Euler class of very
often turns out to be trivial (see [12]). This, in fact, is the reason why all available results
in the area are restricted to the actions of so few groups. In this setting the results of H.
Munkholm from [13] (also see [14]) are especially interesting. In that paper he proves a
BorsukUlam type theorem for Z, 11i i, p is odd, on odd dimensional spheres using a
KUtheory Euler class. The remaining case of Z2kactions on spheres, k > 1, is considered
in [25] and in this dissertation (see Theorem 1.6.2).
The proof the Theorem 1.2.1 is based on the nontriviality of the (mod p) Euler class
of a corresponding vector bundle. The next two sections will be devoted to the calculation
of Euler classes of relevant vector bundles.
1.3 Calculation of i,'_ _1(])
In this section assume that G = (Z2)k. As usual BG stands for the classifying space
of G and EG stands for the total space of the universal Gbundle. This section is devoted
to the calculation of the (mod 2) Euler class of a vector bundle ]: EG XG IR(G) G BG,
i.e. its StiefelWhitney class i,'_ _1(9). These calculations are then needed in the proof of
Theorem 1.2.1 in case p = 2. Recall that H*(BG; Z2) is a polynomial algebra Z2[x, ..., xk]
on 1dimensional generators.
Lemma 1.3.1. ,' _1(]) = I rq i<...
Proof. Let Z2 act on R by an obvious involution. This involution induces on R a structure
of an R[Z2]module which we will denote by V. Denote by pri: BG  RP" a projection
on the ith coordinate. Then by Ai we denote a 1dimensional real vector bundle obtained
from the following diagram:
E(Ak) S' xz, V
Ai I I
BG p RP"
Here S" stands for the infinite dimensional sphere. From the construction of A it
follows that wi(Ai) = xi.
Let r]i be a vector bundle obtained from the following diagram:
E(Ti) S" xz, R[Z2]
rG I IR
BG P"' RP'
___ x 222
From the isomorphism R[Z2] V V (V I V) i V V2 it follows that 'i Ai E A)
where A =2 Ai A is a trivial 1dimensional bundle. Recall the isomorphism of Rmodules:
R[G] R[Z2 ... Z21] 2 R[Z2] *R ... pR R[2]. From this isomorphism it follows that
T] E1 ]1 01 ... 0 Tlk. Therefore, there exists the following chain of isomorphisms of vector
bundles:
k
S (N0(Ai A)A e (A a ... 0A ).
i1 (al,...,ak)EG
It is a well known that the first StiefelWhitney class of a tensor product of
1dimensional real vector bundles equals to the sum of the first StiefelWhitney classes
of the multiplies. Then by this fact and a formula of Whitney we get the following chain
of qualities:
" 1(T7) I" i(?7TI 1) HJ (aixi + ... + akXk)
(a ,....,ak)70
k
=a na
H H (Xi, '+ +x + i').
q= 1 l
1.4 Euler Class of ]c: EG XG Ic(G)  BG
Through out this section assume that p is a fixed odd prime and that G = (Zp)k
In this section we will calculate the (mod p) Euler class of a complex vector bundle
c: EG xG Ic(G) + BG which equals to its ('!l, i class Cpk_l(rlc). These calculations are
then needed in the proof of Theorem 1.2.1 in case of odd primes. Recall that:
H*(BG; ZP) = zp(Y,, ., k) Zp[xl,..., xk],
where Az,(yl, ...,yk) is an exterior algebra on 1dimensional generators and Zp[x, ..., xk is
a polynomial algebra on 2dimensional generators.
('!., i classes of a regular representation of G, i.e. ('! i i classes of the vector bundle
r]c 1 E': EG XG C[G]  BG, were first computed by B.M. Mann and R.J. Milgram in
[10]. The lemma which is stated after the next definition is essentially borrowed from their
paper.
Definition 1.4.1. Lk = 1i Ha z/p(axll ... i+ ailx1 + xi)
The polynomial defined above is called the kth Dickson's polynomial (see [10] for more
details).
Lemma 1.4.1. e(/c) =( 1)kLp'1
Proof. The action of Zp on C by rotations by 2, induces on C a structure of a C[Zp]module
p
which we will denote by L. Let pri: BG B BZP be a projection on the ith coordinate.
Then let Ai be a 1dimensional complex vector bundle obtained from the following
diagram:
E(A) S" xz L
Ai I
BG p B Zp
It is not very difficult to show that c (Ai) = Xi.
Let rli be a vector bundle obtained from the following diagram:
E(Ti) S" xzz C[Z,]
BlG I
BG ri
BCA
BI
BZP
It follows from the isomorphism C[Zp] 2 L E ... E LP, where L = L _[
that Ti Ai ... A4. Here A = A, C ... c Ai. Also note that A4 is a trivial
complex bundle. Recall the isomorphism of Cmodules: C[G] C[Zp E ... E
C[Zp] 0c ... 0c C[Zp]. From this isomorphism it follows that r~ ED F 1 i
Therefore there exists the following chain of isomorphisms of vector bundles:
k
ii1
1 "'" '" 1 L ,
1dimensional
Zp1
... 0 lk.
(al,...,ak)EG
From a formula by Whitney and the fact that the first ('!., i class of a tensor product
of 1dimensional complex bundles equals to the sum of the first ('!i. 1i classes of the
multiples, it follows that
Cp _I(71c) = Cpk _1(7C 1 ) = (a11 + ... +akk)
(al ,...,amc)#0
k
I J[(P i)!]P ni (aixi + ... + aii1 + x)Pl 
i1 (al,..., i_ 1,1,0,...,0)
[(p )!]kLp1 (l)kLP1
The last equality follows from a theorem of Wilson which states that (p 1)!
(1)(mod p). Thus e(T]c) = Cpk_ (c) = (k1)L1.
1.5 Proof of Theorem 1.2.1
In this section we use the results of the previous sections to finish the proof of
Theorem 1.2.1. Here assume that p is any prime number and that G (Zp)k.
Proof of Theorem 1.2.1. Recall that M = N" x ... x N1 is a product of (mod p) homology
nispheres. We will begin the proof by showing that under assumptions of the theorem the
(mod p) Euler class of M : M x, IRm (G) M/G is nontrivial.
By universality property there exists the following commutative diagram:
M x, IRm(G) EG Xc Im(G)
CMI
M/G A BG.
Case p=2. Let q be the vector bundle from section 1.3. Then from the isomorphism IR,
I e... Elg mlR it follows that ( rql... EDq mrl. Thus e2() i '. _1() = i (I)".
By Lemma 1.3.1 we have
) J J (xi +... + X) i
q1 l_
k1
= xi (x, +... + xj,) + Rk,
q=1
where Rk contains monomials in powers less than 2k1. Therefore
e2(0) =X7 X ... X +Qk, (1)
where Qk does not contain monomials of the form x xm ... 'i m. It is easy to verify
that
H*(M/G; Z) Z2 [X, ] / ( +1),
and p*: H*(BG; Z2)  H*(M/G; Z2) is an epimorphism with
Ker 0* ( 1l+1, ..., Xn 1 ).
Thus from (1) and the assumption ni > m2'1 for all i(1 < i < k) it follows that
eC(M) = p*2(()) / 0.
Case p > 2. Let rTc be a vector bundle from section 1.4. Then from the isomorphism
Ic~ Ic ... Ic mlc it follows that C T Erl ... E rl mry, where C is a
complexification of the vector bundle (. We have the following chain of qualities:
e,(0)2 = e,(C) = C2k1(C) = C2k1(c/)T. (2)
By Lemma 1.4.1 we have
p1
eP(rcC) = ( )kL' ( )kL ' (a1X1 + ... + k1X1 + Xk)
S(1)k xk1 + Rk,
where Rk contains Xk in powers less than pkl(p 1). Thus
ep(C) ( l)(kmr mp 1lp1)(p1) +k,
() k !C 1 + @ k,
where Rk contains Xk in powers less than mpkl(p 1). Then by induction it follows that
ep(C() (l)kmnX (P1) mp(p1) mp. 1(p1)+
... X2k + Qk,
where Qk contains no monomials of the form
bxm(Pl) mp(pl) p 1p1), b O, b Z.
Therefore from the previous and (2) it follows that
m(p1) mp(p1) mpk(p1)
ep() = ax, 2 X2 2 .Xk 2 + Qk, (3)
where a2 (m1)(od p) for some q > 0 and Qk contains no monomials of the form
m(p1) mp(p1) mpk1(p1)
bx1 2 x2 2 X 2 b/ 0, be Zp.
It is not very difficult to see that
nl+l nk+l
H*(M/G;p) = Az,(Y,...,yk) 2[X1,...,Xk/(X 2 ,..,Xk ),
where dimxi = 2, and o*: H*(BG; Zp)  H*(M/G; Zp) is an epimorphism with
nl+l nk+1l
Ker ~* = (x1 ,...,Xxk ).
Then from (3) and the assumption ni > mpil(p 1) for all i(1 < i < k) it follows that
e,(M) = c*(e,()) / 0.
Since A(f) is closed and Ginvariant, the set M \ A(f) is also Ginvariant, and
therefore we can consider the following exact sequence of a pair:
... H(M/G, (M \ A(f))/G) ' H'(M/G) > H'((M \ A(f))/G) ..
By Lemma 1.2.1 the vector bundle (M has a nonvanishing section over M \ A(f). Thus
/3(ep({M)) = 0. Therefore there exists a nontrivial element
pe R' )(M/G, (M \A(f))/G)
such that a(p) = ep,(M). Since we are working over coefficients in a field Zp there exists a
corresponding nontrivial element / E H, pkl)(M/G, (M \ A(f))/G). Then by Alexander
duality we have
HdimM 1)(A(f)/G Z) /,
and thus dimZ A(f)/G > dimM m(p 1) (see [7]). Since the group G is finite it easily
follows that
dim A(f) > dimz A(f ) > dimM m(k 1),
and we are done. D
1.6 BorsukUlam Theorem for Z2kactions
In [14] H. Munkholm and M. Nakaoka proved the following generalization of the
BorsukUlam theorem:
Theorem 1.6.1. Let G be a, I ; 1.. .., ii of odd order pk, where p is a prime, and E
be a homote(.,' 2n + 1 sphere on which a free differentiable Gaction is given. Let M
be a differentiable min,,,,,':;'..1 and let f: E + M be a continuous map. Then the set
A(f) = {x e lf(x) = f(gx) Vg E G} has dimension at least 2n + 1 (pk 1)m [m(k 
1)pk (mk + 2)pk1 + m + 3].
The proof of the theorem above is essentially based on nontriviality of a KUtheoretic
Euler class of a certain complex vector bundle. Let us sketch here the main ideas of the
proof needed to show that A(f) / 0 provided that dimension of E is sufficiently large. For
simplicity, we will omit the tricks used to estimate the dimension of A(f).
First, consider a bundle : E xG MG E/G, where MG = fli M and G acts on
MG by permuting the coordinates. Every continuous map f: E  M induces a section
s(f) of the bundle given by the formula:
s(f) (xG) (x, f (xg)g)G.
gCG
One can easily show that functions {f: E + M} which do not collapse any orbit of G to a
point are in onetoone correspondence with sections {s(f)} such that s(f)(E/G) n (E/G x
AM) 0, where AM is the diagonal in the product MG. See Lemma 1.2.1 for details.
Now consider the normal bundle v of E/G x AM in E XG MG. It was shown in
[14] that v has a structure of a complex vector bundle (Proposition 2 [14]). Mainly it
follows from the fact that all irreducible representations of G are complex. Let 0 be the
KUtheoretic Thom class of v. It turns out that for any continuous map f: E  M the
induced homomorphism
s(f)*: KU (E XG MG, E/G x AM) KU (E/G)
in Ktheory maps 0 to the Euler class of the vector bundle : EG x IR (G) + E/G, where
IRm is the kernel of the augmentation homomorphism IR [G]  IR (see Proposition 3 of
[14]). Here and throughout the paper R" is assumed to be a ring with multiplication given
by the multiplication of the coordinates.
It follows from our constructions that if f: E  M is a map which does not collapse
any orbit of G to a point, then s(f)*(0) = e(o) = 0.
The complex Gmodule IRm (G) is a sum of all nontrivial irreducible complex
Gmodules, which makes ( a sum of onedimensional complex vector bundles. This
decomposition allows to compute the KUtheoretic Euler class of From certain
considerations in elementary algebraic number theory it follows that e() / 0 provided
dim E is sufficiently large, which completes the proof of the theorem.
In case G = Z2c, k > 1, neither v nor have a complex structure, simply, because the
dimension of and v is odd. This fact does not allow to use complex Ktheory in order to
prove that ( does not have a nonvanishing section provided dimension of E is sufficiently
large. Presumably, this is the reason the above result of H. Munkholm and M. Nakaoka
is restricted to the case of odd order groups G = Zpf. Note that the Euler class of in
ordinary cohomology theory is trivial if k > 2 (see [12],[14] for details).
Remark. The ,,,inr,:../,/1 E and M in Theorem 1.6.1 do not need to be assumed differen
tiable. The proof works without wi;1, li~.ir, if one assumes = S2n+1, the action G C E
to be free and the in,,,:. .1./ M to be an mdimensional tej'l''.I1j:./ .l i ,,,,./'1
To the best knowledge of the author a BorsukUlam type theorem for free Z2 actions
on spheres has not been published yet. Theorem 1.6.2 which first appeared in [25] covers
this gap.
Theorem 1.6.2. Let S2"+1 be a (2n + 1)dimensional sphere endowed with a free action
of a ;/. 1.:, /i,. '1 Z2k, where k > 1. Let M be an mdimensional te'l' 1I j.:, l ,,,,,' .:. 1.1 and
let f: S2+1 M be a continuous map. Then the coincidence set A(f) = {x c Ef(x)
f(gx) Vg e 2k has dimension at least (2n + 1) [2(m 1) + m2k1(k 1) + 1].
We will postpone the proof of Theorem 1.6.2 until section 1.10. In the next sections
we will state and prove all the necessary results which are needed for the proof of Theorem
1.6.2.
1.7 Necessary Lemmas
Let G = Z2k and suppose that a free action of G on S2n+1 is given. Then consider a
bundle
S: S2n+1 XG MG S2+1/G,
where M is a topological manifold of dimension m and MG = HJIG M. It is assumed here
that G acts on MG by permutation of coordinates. Let
v: E(v) S2"+1/G x AM
be a normal bundle of S2n+1/G x AM in S2n+1 XG MG. Here AM is the diagonal of MG
invariant under the action of G. Let
: S2n+1 X G R (G) S S2n+l/G
be a vector bundle with IRm(G) Ker(F.' [G] R") as a fiber. Then the following
lemma holds:
Lemma 1.7.1. Let i: S2+1/G S2n+/G x AM be an obvious inclusion. Then i*(v) = .
As it was mentioned in section 1.6, the proof of Theorem 1.6.2 heavily relies on the
geometry of the vector bundle In the next lemma we will give a full description of in
terms of "in i! I vector bundles whose geometry is fairly simple.
Let G act on C by rotations by T. This action gives C a structure of a C[G]module
which in its turn gives rise to a vector bundle A: S2n+1 XG C S2n+1/G. Similarly, the
group G acts on R by involutions which gives rise to a vector bundle p: S2"+1 XGR 
S2n+1/G.
Lemma 1.7.2. ( = m(p A A2 2 ... A2 11)
Proof. The proof of this lemma follows immediately from elementary representation
theory. E
Lemma 1.7.3. mn = m(A A 2 ... A2k1)
Proof. This fact is an immediate consequence of Lemma 1.7.2 and the following obvious
equality:
p i = A .
As it follows from the previous lemma, the vector bundle ( E mp admits a structure
of a complex vector bundle. Therefore, one can consider its Euler class e({ E mp) E
0
KU (S2n+1/G). In the following sections we will prove that e( E mp) is nontrivial,
provided dimension of the sphere S2'+1 is sufficiently large.
Let p*: E(p*)  S2n+l/G be a vector bundle induced from a vector bundle
p: S2n+l XG R S2n+/G via a projection pr: S2n+1/G x AM  S2n+. It is easy
to see that there exists the following commutative diagram:
E( Dmp) E(v rmp*)
( mnp I Iv mp*
S2n+1/G S2n+1G x AM.
The existence of this diagram is equivalent to the statement of the following lemma:
Lemma 1.7.4. ( E mp = i*(v E mp*).
It follows now that a vector bundle v E mp* admits a structure of a complex vector
bundle.
It is easy to see that there exists the following commutative diagram:
S2n+1 XG (MG x R) 2n+l XR m
S2+1 X MG S2n+1/G,
where j: S2n+1 xMG S2n+1 XG (MG x Rm) is the obvious inclusion and r oj = id. It is
not very difficult to show that the normal bundle of S2n+1/G xAM in S2n+ XG(MG xRm)
is isomorphic to
v E mrp*: E(v E mrp*) S2n+1/G x AM.
Now let f: S2,+1 M be a continuous map and let
s(f): S2n+/G S2n+ XG MG
be a section of the bundle : S2n+1 x MG S2n+1/G associated to it. Define a map
s(f): S2n+l/ S2~1 XG (MG x R ) as a composition s(f) j o s(f). Consider the
Thom class of the complex vector bundle v E mp*:
T(v m*) G kUO0 (S21 XG (MG x R), S2 G x AM).
Tt e mpl*) E KU (S2"+1 XG (MG x Rm), S2n+l/G x AM).
Note that by excision there exists the following isomorphism:
0 0
KU (E(v e p*), S2n+1/G x AM) KU (S2n+l XG (MG x R), S2n+1/G x AM).
Define 0 E KU (S2"+1 XG (MG x Rt)) by a formula
0 =i'*T(vED mp*),
where
i'*: (S2n+1 XG (MG x RI), 0) (S2n+l XG (MG x R'), S2n+1/G x AM)
is an obvious inclusion. The following proposition p1 i, an important role in the proof of
Theorem 1.6.2:
Proposition 1.7.1. In the notations above, one has the following ,,;,i;.:;.1;
s(f)*(0) = e(B Dmp).
Proof. The idea of the proof is to compare the map s(f) with the ".. section of 0 o r
on the cohomological level. D
1.8 Computation of Norms
Let F/K be a field extension. Recall that a norm map N: F K is a map defined as
N(u) = ((l1)ao)[F:K(u)], where f(x) = x + ... + ao e K[x] is the minimal polynomial of
u e F. Also recall that N: F K is a multiplicative map, i.e. for any u, v F we have
N(uv) = N(u)N(v). Throughout this section and further in the paper 4T(x) stands for
the nth cyclotomic polynomial.
Lemma 1.8.1. For a prime p, let 7 be a primitive root of ; il;', of order pk and let
Q(7)/Q be the corresponding /. l'/.:,,.: extension. Then
(a) N(,y t) p, 0 < 1 < k,
(b) N(4) (7)) pP1 p,1 0 < < k,
(c) N(4mp (7)) 1, if pm, m > 1.
Proof. (a) First, note that yP is a primitive root of unity of order pkl.
minimal polynomial for (,' 1) is pk (x + 1). Thus ao = p)k_(1)
[Q(7) : Q] p Pk1
[Q(7) Q(,' )] [Q) pk1 k11
Therefore,
N,) F1)m (((a) it f s that1 P.
(b) From (a) it follows that
N(p1 1_ )
N(%p( )) N( P 1)_)
(c) Note that, if p { m, then 7"mp
the same reasoning as in (a), one can
have
Therefore, the
p. Also, note that
p.
1p 11
is a primitive root of unity of order pk Following
show that N(7y 1) = N(yP 1) = pP. Thus, we
N(,mp1 )
NV (4)Tnp) ; t
Hdlmpld<,mp' 4d(7
N(Qy m
N(p 
1.9 Computation of Euler Class e( E mp)
0
In this section we will compute the Euler class e( E mp) E KU (S2"+1/G) and will
prove that it is nontrivial provided the dimension of the sphere S2"+1 is sufficiently large.
Proposition 1.9.1. Let Z, act on C by rotations by 2 and let
A: S2n+1 Xm C S2n+l/Zm
be a vector bundle associated to this action. Then
U (s2n+l/ 2f) Z [x]/((x + 1)2k 1 x1+),
where x = e(A).
Proof. Let rl be the universal onedimensional complex vector bundle over CP". It is a
wellknown fact that the total space of the spherization of the mfold tensor product of rl
is homeomorphic to S2n+1//m. The later fact allows to write down the following Gysin
long exact sequence:
0 Ue(lm) 0 0
... KU (CP) KU (CP") KU (S2n+l/Z) ....
0
The ring KU (CP") is a truncated polynomial algebra Z[x]/(x"+l) where x = e(). From
a formula e(rl 0 T) = x2 + 2x it follows that
e(rm) = (x + 1)m 1.
0
Therefore, KU (S2+1/) [x]/ ((x + 1) 1,xn+1). The equality x = e(A) in
0
KU (S2fn+l/Z) follows from the fact that the homomorphism
(~T) : H2(Cpn) H2(S2n+I/Zm)
map the first ('!. i i class of Tr to the first ('!. i i class of A. D
Proposition 1.9.2. e( mTp) [(x 1)(x2 1) .... (x21 1)_].
Proof. The proposition follows from Lemma 1.7.2 and Proposition 1.9.1. D
Proposition 1.9.3. Let d > 0 and I = ((x 1)"1,x2 1) be an ideal in Z[x] generated
by Fr, 'I; i'.rl (x 1)+1 and x2 1. Suppose that
P(x) = (x 1)d[(x 1)(x2 ) .... (x 1)]' c I,
then d > n 2k(Im 1) m2k2(k 1).
Proof. The polynomial P(x) lies in the ideal I if and only if there exist polynomials h(x)
and g(x) such that
(x )d[(x 1)(x2 _) .... (1 ) h (x)(x ),n1 + g(x)(x2 )
Let 4j(x) be the jt cyclotomic polynomial. Then the equality above can be rewritten in
the following form:
+ 2 I2
(x 1)d+ 2k1 [ I ]
h(x)(x 1)n1 + g(x)(x
k
1)j f 2jX.
j=2
Let cj be defined as follows:
1, if j 2k
0 if j t2k
There exist polynomials g(x) and h(x) such that:
h(x)(x )n+ldn2k + g(x)2k(X).
Let 7 be a primitive root of unity of order 2k and let Q(7)/Q be the corresponding
cyclotomic extension. Then in Q(7) we have:
2k1
,,(7)[ ]m'
h(7)(7 1)n+1dnm2k
N(h(7))N((7
))n+ldmn2k 1
So, it follows that
2k
N(( 1))n+ldmr2k I J N(4y))[2]nm
j=2
By Lemma 1.8.1 we have:
2k12k
N(4 )(7)) ]m =
j=2
k1
 H 2(2J2 1)(m2  11)
j=1
2Z j25 1(m2h 11
k1
N(()2J (7)) 2 j 11
j=1
S2E (2_25 )(m2  11)
2) m2k 2(k1)2 1+1
k (7) kI
N(@y(7)[ mI
2k1
S(X) [ .]mn
Note that according to Lemma 1.8.1, N(7 1) = p. Thus, we have
n + 1 d m2k1 < m2k2(k 1) 2k1 + 1
and
d > n 2kl(m 1) m22(k 1).
Corollary 1.9.1. Let d > 0 and suppose that the class e(dA ( mp) is trivial in
0
KU (S2+1/22k). Then d > n 2k1(m 1) m2k2(k 1).
Proof. From Lemma 1.7.2 it follows that:
e(dA E mpn) = ((x + 1) 1)d[((x + 1) 1)((x + 1)2 1) ... (( + 1)2 1 )]_.
If the class e(dA E ( E mp) is trivial in KU (S2n+1/Z2k), it must belong to the ideal
((x + 1)2k 1), x.+1). Now, the statement of the corollary follows from Proposition
1.9.3. E
1.10 Proof of Theorem 1.6.2
In this section we will use the results of the previous sections to finish the proof of
Theorem 1.6.2. Recall that for a map f: S2"+1 + M we define a coincidence set:
A(f) ={x E S2n+ f(x) f(gx) Vg E 2k}.
Let A: S2n+1 x C  S2n+1/Z2 be a onedimensional complex vector bundle where the
action of Z2k0 on C is given by rotations by 2.
Proof of Theorem 1.6.2. Assume dim A(f) < 2d, then a vector bundle dA E ( E mp has a
nonvanishing section by elementary dimension considerations. Thus e(dA)e(( E mp) = 0.
By Proposition 1.7.1 we have
e(dA)s(f)*(0) = e(dA)e( E mp) = 0.
Therefore, by Proposition 1.9.3, we must have:
d > n 2k(m 1) m222(k 1).
It follows now that:
dim A(f) > (2n + 1) [2k(m 1) + m2l1(k 1) + 1].
[]
CHAPTER 2
HUREWICZ THEOREM AND APPROXIMATION OF MAPS
2.1 Introduction
All spaces are assumed to be separable metrizable. By a map we mean a continuous
function, I = [0; 1]. If KC is a simplicial complex then by IICI we mean the corresponding
polyhedron. By a simplicial map we mean a map f: C I which sends simplices to
simplices and is affine on them. We ;? that a map f: X Y has dimension at most
k (dim f < k) if and only if the dimension of each of its fibers is at most k. We recall
that a space X is a Cspace or has property C if for any sequence {an : n E N} of open
covers of X there exists a sequence {I, : nE N} of disjoint families of open sets such that
each p, refines a, and the union of all systems p,, is a cover of X. Each finitedimensional
paracompact space and each countabledimensional metrizable space has property C. By a
Ccompactum we mean a compact Cspace.
In [26] V.V. Ui. i1:;ii introduced the notion of a map admitting an approximation
by kdimensional simplicial maps. Following him we zi that a map f: X Y admits
approximation by kdimensional simplicial maps if for every pair of open covers Wx of the
space X and wy of the space Y there exists a commutative diagram of the following form
X /C
f I I P
Y L I\1,
where Kx is an wxmap, Ky is an wymap and p is a kdimensional simplicial map
between polyhedra II and I I.
In that paper V.V. Usp, ii1:ii proposed the following question and conjectured that in
the general case the answer to it is "no".
(Q1) Does every kdimensional map f: X Y between compact admit
approximation by kdimensional simplicial maps?
In [6] A.N. Dranishnikov and V.V. Usp, iil:;i proved that light maps admit approximation
by finitetoone simplicial maps. In this paper we give some partial results answering the
question of V.V. Up. il1:ii in affirmative.
Theorem 2.1.1. Let f: X Y be a kdimensional map between Ccompacta. Then
for ,i;1 pair of open covers ux of the space X and uy of the space Y there exists a
commutative diagram of the following form
X /C
f I I p
Y  \ ,
where Kx is an wxmap, K y is an uwymap and p is a kdimensional simplicial map
between compact py1;,,. Ji,i,, I/C and L1. Furthermore, one can il;,. n, assume that
dim
Theorem 2.1.2. kdimensional maps between compact admit approximation by (k + 1)
dimensional simplicial maps.
Theorem 2.1.3. kdimensional maps of Bing compact (i.e. compact with each compo
nent her, .:/.r, :,7;; indecomposable) admit approximation by
kdimensional simplicial maps.
It turned out that the question (Qi) is closely related to the next question proposed
by B.A. P iLkov in [16]. We recall that the diagonal product of two maps f: X Y
and g: X Z is a map f A g: X Y x Z defined by f A g(x) = (f(x),g(x)).
(Q2) Let f: X Y be a kdimensional map between compact. Does there exist a
map g: X Ik such that dim(f A g) < 0?
In this paper we prove the following theorem which states that the questions (Q1)
and (Q2) are equivalent.
Theorem 2.1.4. Let f: X Y be a map between compact. Then f admits approx
imation by kdimensional maps if and only if there exists a map g: X Ik' such that
dim(f A g) < 0.
There are a lot of papers devoted to P .i tkov's question ([16],[17],[18],[21],[20],
[8],[9],[22]). In [16] P i ikov announced the following theorem to which the proof
appeared much later in [17] and [18].
Theorem 2.1.5. Let f: X Y be a kdimensional map between finite dimensional
compact. Then there exists a map g: X I"k such that dim(f A g) < 0.
In [21] Torunczyk proved the following theorem which is closely related to the
theorem proved by P .i kov.
Theorem 2.1.6. Let f: X Y be a kdimensional map between finite dimensional
compact. Then there exists a acompact A C X such that dim A < k 1 and dim f IX\A
0.
One can prove that for any map f: X Y between compact the statement of
theorem 2.1.5 holds (for f) if and only if the statement of theorem 2.1.6 holds (for f) (the
proof can be found in [8]).
We improve the argument used by Torunzyk in [21] to prove the next theorem and
the implication "" of theorem 2.1.4.
Theorem 2.1.7. Let f: X Y be a kdimensional map between Ccompacta. Then
(i) there exists a acompact subset A C X such that
dim A < k 1 and dim f IX\A< 0.
(ii) there exists a map g: X ) Ik such that dim(f A g) < 0.
In the next corollary by e dim(X) we mean the extensional dimension of a compact
space X introduced by A.N. Dranishnikov in [5].
Corollary 2.1.1. Let f: X Y be a kdimensional map between Ccompacta. Then
e dim(X) < e dim(Y x Ik).
Proof. From [6] it follows that the extensional dimension cannot be lowered by 0dimensional
maps so the corollary is an immediate consequence of theorem 2.1.7. D
One can understand the statement of the previous corollary as a generalization of the
classical Hurewicz formula.
2.2 Proof of Theorem 2.1.7
Further in this section we assume that every space X is given with a fixed metric
px which generates the same topology on it. By px(A, B) we mean the distance between
subsets A and B in the space X, namely, px(A,B) = inf{px(a,b) a E A,b E B}. The
closure of a subset A will be denoted by [A].
Lemma 2.2.1. Let f: X Y be a map between compact. Suppose that for ,..i closed
disjoint subsets B and C of X there exists a closed subset T of X such that dimT < k 1
and for ,.; y E Y the set T separates fl(y) between B and C. Then there exists a
acompact subset A c X with dim A < k 1 such that dim f X\A < 0.
Proof. Take a countable open base B {UI 7 E F} on X such that the union of any
finite number of sets from B is again a member of B. Define the set A C F x F by the
requirement: (7, p) E A if and only if [U,] n [U~] = 0. Note that A is countable. By
assumption for every pair (7, p) E A there exists a set T(,,,) of dimension at most k 1
separating every fiber f'(y) between [U,] and [U,]. Now define A = U{T(,,)I (y7, ) E A}.
By definition A is ucompact and, obviously, dim A < k 1. It is also easy to see
that dim f IX\A< 0. Indeed, by the additivity property of the base B for every pair of
disjoint closed subsets G and H of a given fiber fl(y) there exists a pair (7, p) E A such
that G c U, and H C U,. Then T(,,,) C A separates fl(y) between G and H. So,
dim(fl(y) \ A) < 0. D
Let F = NO U U{Nk : k > 1} be the union of all finite sequences of positive integers
plus empty sequence No {*}. For every i cE let us denote by il the length of the
sequence i and by (i,p) the sequence obtained by adding to i a positive integer p.
Lemma 2.2.2. Let f: X Y be a map between compact. Let B and C be closed
disjoint subsets of X. Suppose that for every i CE there are sets U(i), V(i) and F(i) such
that:
(a) F(i) is closed in Y, the sets U(i) and V(i) are open subsets of X and [U(i)] n
[V(i)] 0;
(b) U(*) D B, V(*) D C and F(*) = Y;
(c) U(i,p) D U(i) n f (F(i,p)) and V(i,p) D V(i) n f(F(i,p)) for every p E N;
(d) F(i) C U{F(i,p) : p E N} and diamF(i) < T
(e) the set E(i) = f1(F(i)) \ (U(i) U V(i)) admits an open cover of order k and
diameter 1
(f) in notations of (e) the f.indll; {E(i,p) : p e N} is discrete in X.
Then there exists a closed subset T of X such that dimT < k 1 and for ,:; y e Y
the set T separates fl(y) between B and C.
Proof. We define the set T in the following way:
T,= U{E(i): i= n} and T= n{T: n > 0}.
From property (e) it follows that dimT < k 1. Let us show that for every y e Y the set
T separates f (y) between B and C. For every y e Y there exists a sequence {i : n E N}
such that
{y} F(i) n F(i, i2) n...
Then fl(y) \ T C U(y) U V(y) is a desired partition. Here we denote by U(y) and V(y)
the following sets
U(y) f (y) n Uui(1,..., p) rc N},
V(y) f (y) n UIlV,..., 1 ) .p N}.
F
Lemma 2.2.3. Let f: X Y be a kdimensional map between Ccompacta and c be w;:,
positive number. Let U and V be open subsets of X with [U] n [V] = 0, and F be a closed
subset of Y such that f(U) n f(V) D F. Then there exist families of sets {Up}, {V,}, and
{Fp} for p N such that:
(1) Fp is closed in Y, the sets Up and Vp are open subsets of X and [Up] n [Vp] = 0;
(2) Up DU n f1(Fp), Vp D V n f(Fp);
(3) F C U{Fp: p E N} and diamFp < e;
(4) the set Ep = f(Fp) \ (Up U V,) admits an open cover of order k and diameter e;
(5) in notations of (4) the f.nl,,:;/ {Ep : p e N} is discrete in X.
Proof. Let {W 1 : I N} be a countable sequence of open disjoint sets such that each
of them separates X between [U] and [V]. For every y c F let Pi(y) C W1 be a closed
(k 1)dimensional set separating fl(y) between [U] and [V]. Let Ql(y) C W1 be an
open neighborhood of Pl(y) admitting a finite open cover of size C and order k. As the
map f is closed there exists a neighborhood Gl(y) of y in F such that f'([Gl(y)]) 0 Ql(y)
separates f1([Gl(y)]) between [U] and [V] and diamG(y) < e. For every I e N the family
ai { Gi(y) : y E F} is an open cover of F. As F is a Ccompactum there exists a finite
sequence of finite disjoint open families of sets {p' : 1 < N} such that each family pi refines
the cover ac and p = U{p, : 1 < N} is an open cover of F. Further, for every G e p there
are open subsets U(G) and V(G) of X with disjoint closures such that
U(G) D f(G) n [U], V(G) D fl(G)n [V]
and if G is a member of pi then
f (G) \ (U(G) U V(G)) c Ql(y)
for some y c F. Let {F(G) : G E p} be a closed shrinking of the cover p and let
E(G)= f (F(G)) \(U(G) U V(G))
for G E p. Then for G E p and H E p we have E(G) n E(H) C W, n Wm = 0. So the
family {E(G) : G E p} is discrete in X. Let us enumerate the members of p: GC, G2,
To get the desired sets we set
Fp F(G,), Up = U(G), V = V(Gp).
Lemma 2.2.4. Let f: X Y be a kdimensional map between Ccompacta. Then for
i,,,, closed disjoint subsets B and C of X and for 1,,;1 i C F there exist sets U(i), V(i) and
F(i) .rl'fying (a)(f) of Lemma 2.2.2.
Proof. We will construct the sets U(i), V(i) and F(i) by induction on i First set
F(*) = Y and U(*) = U, V(*) =V for some open subsets U and V of X with
[U] n [V] = 0. Assume the sets U(i), V(i) and F(i) are already constructed and satisfy the
conditions (a)(f) of Lemma 2.2.2. Now to get the sets U(i,p), V(i,p) and F(i,p) for all
p C N apply Lemma 2.2.3 to the sets U U= (i), V = V(i), F = F(i) and to c = .
Lemma 2.2.5. Let f: X Y be a map between compact ,il,,:i.:,:, approximations by
kdimensional maps. Then for i;,, closed disjoint subsets B and C of X there exist sets
U(i), V(i) and F(i) './:/;.,,:,i (a)(f) of Lemma 2.2.2.
Proof. The sets U(i), V(i) and F(i) will be constructed by induction on i First set
F(*) = Y and U(*) = U, V(*) = V for some open subsets U and V of X with
[U] n [V] = 0. Assume the sets U(i), V(i) and F(i) are already constructed and satisfy the
conditions (a)(f) of Lemma 2.2.2. Take e = min{ (U(iV()), i}. By assumption, there
exists a commutative diagram of the following form:
X /C
f I I[P
Y Y L 1,
where KX and Ky are maps with csmall fibers. Let G = x([U(i)]), H = x([V(i)])
and F Ky(F(i)). Note that G n H 0. Let U and V be open subsets of /C with
U D C, V D H and [U] n [V] = 0. Let A, be a Lebesgue number of some open covering on
IICI whose preimage under the map KX is an csmall covering on X. Let A2 be a number
defined similarly for IL and Ky. Let A = min{Ai, A2}. Apply Lemma 2.2.3 to the sets U,
V, F and to A to produce the sets Up, Vp and Fp for all p C N satisfying conditions (1)(5)
of Lemma 2.2.3. Now set U(i,p) = K(Up), V(i,p) = Kix(V) and F(i,p) = Ky(Fp). Since
taking a preimage preserves intersections, unions and subtractions, the sets U(i,p), V(i,p)
and F(i,p) satisfy the conditions (a)(f). E
Proof of theorem 2.1.7. The statements (i) and (ii) are equivalent ([8]), so, it is sufficient
to prove only (i). But (i) immediately follows from Lemmas 2.2.1, 2.2.2 and 2.2.4. E
2.3 Proofs of the Approximation Theorems
Let 7 be an open cover on X. Then by N1 we mean the nerve of the cover 7. Let
{a, : ca A} be some partition of unity on X subordinated to the locally finite cover
7. Then the canonical map defined by the partition of unity {a, : a E A} is a map
K: X N1V defined by the following formula
ux) = aa ax) a a.
aEA
If r is some triangulation of the polyhedron P, then by St(a, r) we mean the star of the
vertex aE r with respect to triangulation r, i.e. the union of all open simplices having a
as a vertex.
Proof of theorem 2.1.4. Let f: X Y be a map between compact admitting
approximation by kdimensional simplicial maps. By [8] to show that there exists a
map g: X Ik with dim(f A g) < 0 it is sufficient to find a acompact subset A in X of
dimension at most k 1 such that dim f IX\A 0. The existence of such subset A follows
immediately from Lemmas 2.2.1, 2.2.2 and 2.2.5.
Now suppose there exists a map g: X I I" such that dim(f A g) < 0. For every
(y, t) E Y x Ik there exists a finite disjoint family of open sets V(y,t) {= {V : 7 E F(y,t)} such
that (f A g)l(y,t) C U"',,,., and v(y,t) > Wx. Let O(y,t) be an open neighborhood of (y,t)
in Y x Ik such that (fA g)1O(,t) C U ,,. Let = {Ua : a A} and L = {Is : 6 e D} be
finite open covers of the spaces Y and Ik such that:
(ai) ; > Wy;
(bi) the order of < does not exceed dimY + 1;
(cl) (, x L) {U x h : (a, 6) e A x D}> {O(,t): (y,t) e Y x Ik.
The partition of unity {ua : a E A} on Y subordinated to the cover gives rise to the
canonical map p: Y IH. Then the map p x i: Y x I" I x I is an (, x t)map.
By r":  x Ik  we denote the projection. Let T and 0 be such triangulations on
polyhedra A x Ik and A respectively such that the following conditions are satisfied:
(1') : A x Ik S A is a simplicial map relative to the triangulations 'T and 0;
(2') {(p x id)1(St(a, )) : a E } c x t;
(3') {p1(St(b, 0)) : b 0} > c.
Let us define {St(a, r) : a E r} and ( {St(b,0) : b E 0}. Define the partition
of unity {I,', : a E r} on 1A x Ik subordinated to the cover ( by letting it, (z) to be the
barycentric coordinate of z E  x Ik with respect to the vertex a E c Analogously,
define the partition of unity {,,. : b 0} on 4 subordinated to the cover (. Note that
the projection r:  x Ik sends the stars of the vertices of the triangulation 'T
to the stars of the vertices of the triangulation 0. That is why there is a simplicial map
=: ) NH between the nerves of the covers and (. Moreover, the following diagram
commutes.
I41 x I
XK .
i~X^ I N
Here b and 0 are canonical maps defined by the partitions of unity {"', : a E T}
and {,,. : b E 0} respectively. Let us remark that dim < k. By Wa we will denote the
set (p x id)'(St(a, ,)), by A the cover {W, : a E }c and Tr is p1((). Further, we set
w* = ,,', o (p x id) for each aE T and = ,. o p for each b E 0. The partitions of
unity {w* : a E r} on Y x Ik and {v* : bE 0} on Y are subordinated to the covers A
and TI respectively. We set IC' = A and L = Ab. The simplicial complexes /C' and L are
isomorphic to A/ and A/e that is why the simplicial map q : IC' LI is defined and the
following diagram commutes.
YxIk /IC '
pr { q
Y L1.
Here o and Ky are canonical maps defined by the partitions of unity {w* : a E } and
{, : b e 0} respectively. Obviously, dimq < k.
Recall that p is an {O(y,t)}map. For every a E T pick a point (y,t)a such that
Wa C O(y,t).. Let w* w* o (f A g) and Ba F(y,t). Then
supp (w**) C (f A g)(Wa) C (f A g)(Ot))
Consequently, Uv(v,t), D supp(w**). As the family v(y,t), is disjoint, there exists
a family of nonnegative functions {b3 : 3 Ba} such that w** = EpB, b and
supp(b3) C Vp. Let B U{B : a E 7} and V = {Vp n (f A g)1(Wa) : a E r,3 P B}.
The family {b3 : P3 B} is a partition of unity on X subordinated to the cover V.
Let C be the nerve of the cover V and Kx: X I C the canonical map defined by
the partition of unity {b3 : 3 B}. We define a simplicial map p': IC + IC' by
requiring that the vertex 3 E Ba goes to a. Clearly, the map p' is finitetoone. Indeed,
no two vertices in Ba are connected by an edge. That is why the restriction of p' to any
simplex is a homeomorphism. Finally we define the desired kdimensional simplicial map
p: IC  IL as the composition p = q o p'. Moreover by (b1) we have dim I < dimY
and dim IIC < dim Y + k since p is kdimensional.
The following results of M.Levin [8] and Y.Sternfeld [20] are needed to prove theorems
2.1.2 and 2.1.3:
Theorem 2.3.1. Let f: X Y be a kdimensional map between compact. Then there
exists a map g: X Ijk+1 such that dim(f A g) < 0.
Theorem 2.3.2. Let f: X Y be a kdimensional map of Bing compact. Then there
exists a map g: X Ik such that dim(f A g) < 0.
Proof of theorem 2.1.2. The theorem is an immediate consequence of theorems 2.3.1 and
2.1.4. O
Proof of theorem 2.1.3. The theorem is an immediate consequence of theorems 2.3.2 and
2.1.4. O
CHAPTER 3
BORSUKULAM THEOREMS FOR MAPS WITH INFINITE FIBERS
3.1 Bula's Property
We v that a surjective map f: X + Y satisfies Bula's property if and only if there
exist closed disjoint subsets A and B, A, B C X, such that f(A) = f(B) = Y.
A question about existence of open maps between compact with infinite fibers which
does not satisfy Bula's property is wellknown in continuum theory and was first stated by
Bula. The first example of such a map was given by:
p: fJS2' fJ p2i
i=0 i=0
and was first Ii.i 1. 1 by A. Dranishnikov in [4]. The following theorem is a generalization
of his construction and heavily relies on Theorem 1.2.1.
Theorem 3.1.1. Let n, > pil(p 1) for I,'; i and let p: (Zp,) 0 CH, S"i be a product
of free actions of ,Z on S"i. Then the projection
00
i 1 i 1
does not ',ify Bula's p ,'/'p Il
The proof of the previous theorem is almost straightforward and therefore omitted in
this dissertation.
Interesting examples of open maps between compact without Bula's property were
constructed in [9]. Recall the following theorem from that paper:
Theorem 3.1.2. Let M be an ndimensional compact i,,, ..:./..1.1 with n > 3. Then there
exists a surjective open monotone map on M with nontrivial fibers which does not ,ri. fi
Bulas 1* 'p /
The author's interest in BorsukUlam theorems and Bula's property was stimulated
by Hurewicz theorem for maps and the conjecture by V. Usp, i1:ii about approximation of
kdimensional maps between compact which were discussed in the previous chapter. The
author believes that the answer to the V. Usp, i,1:ii's conjecture is "y' In other words,
there exist a kdimensional map between compact which cannot be approximated by
kdimensional simplicial maps of polyhedra or, equivalently, for which Hurewicz theorem
for maps does not hold.
There is enough evidence to believe that such an example exists. For instance, the
maps produced by Theorem 3.1.2 almost satisfy the requirements of being such examples.
Proposition 3.1.1. Let f: X + Y be an open ;, ii. l. , map such that for each y E Y we
have dim f(y) = 1. Then the following statements are equivalent:
(i) f: X Y does not .'li.fy Bula's p," I'/'i,/
(ii) for ni, p: X  [o, 1] there exists y E Y such that (fl(y)) = pt,
(iii) Hurewicz theorem does not hold for f,
(iv) The map f cannot be approximated by 1dimensional simplicial maps.
Proof. Suppose (i) holds and there exists a map p: X [0, 1] which does not collapse
any fiber of f into a point. Then define a map p: X + [0, 1] by a formula: ip(x) =
p(x)/diam(f (f())). Then define A 1(0) and B = '(1). Obviously, A n B = 0
and f(A) = f(B) Y which contradicts (i).
The implication (ii)=(i) is obvious.
By Theorem 2.1.4 the statements (ii) and (iv) are equivalent. To see how (i)>(ii)
note that if the map g: X [0, 1] exists, then it does not map any fiber of f to a
point. E
The previous proposition shows how Bula's property is related to Hurewicz theorem
for maps and V.Ui. i,:;ii's conjecture. Our goal now would be to construct an example
of a strictly 1dimensional map between compact which would satisfy at least one of the
properties (i) (iv).
The examples of maps produced by Theorem 3.1.2, although supply evidence that the
desired example exists, do not have uniform dimensionality of the fibers, i.e. in case n = 1
the fibers of the maps produced by Theorem 3.1.2 are not strictly 1dimensional.
In the next sections we will outline the idea how to construct an example of an open,
strictly 1dimensional map between metric compact without Bula's property.
3.2 Lipschitz Compactification
In this section we will discuss compactifications of proper metric spaces with respect
to a family of certain types of Lipschitz functions. Recall that a metric space is called
proper if and only if every closed ball in it is compact.
It is a wellknown fact in general topology that each compactification of a sufficiently
good topological space can be described as the set of maximal ideals of a Banach algebra
of functions. Lipschitz maps on a proper compact metric space do not form a Banach
algebra, so we will consider the smallest Banach algebra containing all Lipschitz functions.
Let X be a proper metric space. Denote by CL(X) the closure of the set of all
bounded Lipschitz functions f: X + R with Lip(f) < oc. For x E X define Q(x) =
(f(x))fECL(x) E RoL(X). It is easy to prove that ': X RoL(X) is an embedding. Define
the Lipschitz compactification LX of the proper metric space X as
LX = (X).
Proposition 3.2.1. Let (X, d) be a proper metric space and let X be the Lipschitz
, **'1,,'. iU. ailln. of X. Let U C X be an open subset of X and let F C U be a closed
subset of U. Let U C X be the unique maximal open subset of X such that U n X = U and
let F be the closure of F in X. Then
F cU i dist(X \ U, F) > 0. (.)
Proof. (w) There exists a function g: X [0; 1] such that g(F) = 0 and g(X \ U) =1.
As long as g is continuous on X there exist a sequence of Lipschitz functions {gj(x) i E N}
such that g(x) = limit gi(x).
We claim that there exist io E N for which gi0 (F) < and gi0 (X \U) > This can be
proven as follows. Assume there is no such io E N. Then there exist a sequence x, E X \ U
with lim,,, dist(x,, F) = 0 and a sequence y, E F with lim,,, d(x,, y,) = 0. Thus,
there exist the following double inequality:
1
3 < d(gi(x,), gi(y.)) < Ad(x,, y,),
from which is follows that gi(x) is not a Lipschitz function.
() Define a function f: X [0; 1] by a formula f(x) = d(x,X \ U). By
assumption, there exists 6 > 0 such that f(F) > J. Note that f(U) = 0. It is easy to check
that f: X + [0; 1] is a Lipschitz function. Therefore, there exists a function f: X + [0; 1]
such that fx = f. Obviously, f(F) > 6. E
An easy consequence of the previous proposition is the following
Proposition 3.2.2. Let f: X + Y ne a continuous Lipschitz map between proper metric
spaces. The the map f can be extended to an open continuous map Lf : LX + LY between
Lipschitz *,,/,i, /I'7 ,I/. ,mns.
Proof. Let x E LX and let U C LX be an open neighborhood of x E LX. We need
to prove that there exists an open neighborhood V C LY of the point Lf(x) such that
V cLf(U).
Let F C F C U be a subset of X such that the closure of F in LX contains x e LX.
Then by Proposition 3.2.1, dist(F,X \ U) > 0. Then dist(f(F),Y \ f(U)) > 0, because f
is locally a projection. Therefore, by Proposition 3.2.1, we have
f(F) c f(U n S).
3.3 The Construction
Let G be a finite group acting freely on a manifold M. We i that the action of G
on M has mBorsukUlam property if for every continuous map f: M  R"m there exists
an orbit of G in M which is collapsed by f to a point.
Let {pi} be a sequence of prime numbers such that limit, pi = oc. Recall that by
Theorem 1.1.3 we can choose a sequence {ni} of odd numbers such that a free Zp,action
on S"' will have 1BorsukUlam property, but will not have 2BorsukUlam property.
Proposition 3.3.1. There exists a sequence of continuous Lipschitz functions fi: S'i
R2 such that:
(i) For ,,, x C S'i we have diamfi(Zp, x) > 1,
(ii) limit" L(f)= 0, where L(fi) is the Lipschitz constant of fi.
Now consider the following obvious map:
Upi: US'i UiCPL1.
This is an open Lipschitz map between proper metric spaces. By Proposition 3.2.2, we
have an open map
L(Upi): L(US') L(ULCP i])
between Lipschitz compactifications.
Proposition 3.3.2. Every fiber of L(Upi): L(US"i) L(LUCP1 ) is a nondegenerate
connected compact space.
Proof. The space CP" is a countable union of finite dimensional spaces. Therefore, it is a
Cspace. In [22] it was proved that there exists a map g: S" [0; 1] such that the map
pAg : S CP x [0; 1]
is 0dimensional. It follows from the last statement that g(pl(y)) =[a, b] with a / b for
every y E CP. Let Ly: [a, b] [0,1] be a linear transformation of [a, b] into [0, 1]. Define
a map p: S  [0, 1] as a composition p = Ly o g. The continuity of the map p essentially
follows from the fact that p is an open map. Now, define A = 1(0) and B = p1(1).
The sets A and B are closed and disjoint and each one of them intersects each fiber of p.
Also, note that dist(A, B) > 0, therefore, the closures of A and B in are disjoint LS" by
Lemma 3.2.1. Now, nontriviality of each fiber of the map Lp follows from Proposition
??. O
Proposition 3.3.3. The map L(Up,): L(US"T) + L(ULCP[ ]) has 1BorsukUlam
,, '/'. ,i ;, i.e. for every continuous function : L(US"i) I = [0; 1] there exists y
L(ULCPl]) such that o(L(ULp)'(y)) = pt.
Proof. Suppose there exists a function p: Lp: LS"  I = [0; 1] such that for every y E
LCP we have p(Lpl(y)) / pt. Then there exists E > 0 such that diam p(Lpl(y)) > E
for every y e LCPc.
Let {O~,a c A} be an open covering of LS" such that for any a c A and any
2,i
x, x' e O, it follows that Ip(x) p(x')l < Let e P be the generator of the group
Zpk. We can choose k E N to be large enough, so that the points x e S1 = P1(y) and
e7p x S1 = P(y) are {O1,a c A}close for any choice of y e CP", i.e. there exists
27i
c E A such that x, e x C O,.
Now, choose n large enough, so that Theorem ?? guarantees existence of an orbit of
Zp~ which will be collapsed to a point under the following composition:
S2n1 iLSQ .
Here i: S2n1 LS is an inclusion. As p o i is a Lipschitz map, the image of such an
orbit will have a diameter less than E, because of our pervious assumptions. The later
conclusion contradicts our initial assumption. O
To produce an example of an open strictly 1dimensional map it remains to prove
Proposition 3.3.1 and the fact that the fibers of our map are are all strictly 1dimensional.
The later fact intuitively seems obvious, since a fiber in the corona of our map is being
approached by circles in a Lipschitz manner. So, it seems, that the fibers in the corona
should also be 1dimensional, since dimension cannot be raised by a Lipschitz map. Once
we have proved all these, we have produced the desired example which remains to have
only one downside, namely, the spaces involved in the map are nonmetarizable spaces. To
get rid of this downside, one needs to use Scepin's Spectral Theorem.
After applying Scepin's Spectral Theorem we have produced an example of a map
which gives a positive solution to the V.Usp, nilii's conjecture and answers the question of
B.A.P ti'nkov in the negative.
APPENDIX A
BORSUKULAM THEOREM FOR Zp x ... x ZpKACTIONS ON PRODUCTS OF
SPHERES
Set G = Zp x ... x Zp, for a fixed odd prime p and k > 1 and let M = S21 x ... x
S2nK1. In this section we will discuss a failed attempt to prove a BorsukUlam theorem
for Gactions on products of spheres and maps into Euclidean spaces. This case is more
difficult than the case of (Zp)kactions, because the Euler class of the corresponding vector
bundle in the ordinary cohomology theory turns out to be trivial. Below we will also show
that the Euler class of the vector bundle, which need to be considered, in the complex
Ktheory is trivial as well. Therefore, this particular case calls for a more sophisticated
cohomology theory in order to extract a BorsukUlam theorem.
Let Li be a 1dimensional complex Zpimodule, in which Zpi acts by multiplication by
2,i
e p and let
A': S2ni1 pi L L n(pi)
be the associated complex vector bundle. Here L (pi) is the lens space S2nl1/G. Denote
by
Ti: L'(p) x ... x LT(pk) L(pi)
the projection on the ith factor and set A = 7r*(A'). Then it follows from the isomorphism
C[G] C[Zp] ... 0 C[Zp,]
that
= (M xPo C[G] M/G) ... 0 k,
where i = (M xz C[Zpi] LTj(pi)). From elementary representation theory we have
isomorphism C[Zpi] Li D L2 D ... L Lti and therefore
SA A ... AAr.
Thus we have
k
t`(g(A, A? ... ED Af') (A D ... 0D A)
ti (ie,...,ak)EG
(ai,...,ac)70
where e, is 1dimensional trivial complex vector bundle. It is easy to see that
S= (M X Ic(G) M/G) (A Ak).
(ai,...,ac))o
To prove a BorsukUlam type theorem for maps from M to R1 one needs to prove that the
Euler class e( ) of the vector bubdle
b = (M XG I(G) M/G)
is nontrivial. Obviously, '_ = ], therefore, it is sufficient to prove that e(Q) / 0. This
probably can be done by means of some extraordinary cohomology theory. Let us see what
happens in the case of complex Ktheory.
First, recall that
Ko(L (p) x ... x L" (pk)) Z[x,,...,xk]l,
where
I = ( ...," (x + 1)P ..., (k + 1) 1).
Here x = [Ai 1] (1 < i < k). This facts follow easily from the existence of
homeomorphisms
(AP)r, _ Li (i) (t
from the Gysin sequence and the Kunneth formula for Ktheory. By (Ai )s we mean the
spherization of the vector bundle A< .
From the structure of the formal group law for Kc given by a formula F1K (x, y)=
x + y + xy we are able to conclude that e(A') = (x, + 1)"a 1 and therefore
k
(A D... 0 A ) i((Xi + 1)&1 1,..., (Xk + 1)k 1),
i=
where ac is the ith symmetric polynomial. Thus we have
k
T(,) n i((XI +) , (X + ) ) 
(a1,...,ak)#0 i 1
In ((Xi + 1)a(X2 + )a2 ... (Xk + 1)a ).
(ai,...,ac))o
To prove a BorsukUlam theorem we need to find conditions under which the polynomial
e(qr) does not belong to the ideal
I = ( ..., (x + 1) 1,..., (xk + ) 1)
from above. The author is being able to prove that in fact the polynomial e(qr) almost
alvi belong to the ideal I which make the complex Ktheory approach inefficient when
trying to prove a BorsukUlam theorem for these kinds of actions.
REFERENCES
[1] D.G. Bourgin, Multiplicity of solutions in frame mappings, Illinois J. Math., vol. 9
(1965), 169177
[2] P.E. Conner and E.E. Floyd, Differentiable periodic maps, SpringerV 1 I Berlin,
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[3] A.N. Dranishnikov, On Qfibrations without disjoint sections, Funct. Anal. Appl., 22
(1988) no.2, 151152
[4] A. N. Dranishnikov, A vibration that does not accept two disjoint I,,i. i\ i.,l. 1
sections, Topology Appl. 35 (1990) 7173
[5] A. N. Dranishnikov, The EilenbergBorsuk theorem for maps into arbitrary com
plexes, Math. Sbornik vol. 185(1994), no. 4, 8190.
[6] A. N. Dranishnikov and V. V. Usp, il;ii, Light maps and extensional dimension,
Topology Appl., 80 (1997) 9199.
[7] A.N. Dranishnikov, Cohomological dimension of compact metric spaces, 6 issue 1
(2001), Topology Atlas Invited Contributions
[8] M. Levin, Bing maps and finite dimensional maps, Fund. Math. 151(1) (1996) 4752.
[9] M. Levin and W. Lewis, Some mapping theorems for extensional dimension,
arXiv:math.GN/0103199
[10] B.M. Mann and R.J. Milgram On the Chern classes of the regular representations of
some finite groups, Proc. Edinburgh Math. Soc (1982) 25, 259268
[11] J. Milnor, Groups which act on S" without fixed points, Amer. J. Math., vol. 79, n.
3 (1957), pp. 623630
[12] H.J. Munkholm BorsukUlam type theorems for proper Zpactions on (mod p)
homology nspheres, Math. Scand. 24 (1969) 167185
[13] H.J. Munkholm On the BorsukUlam theorem for the Zp, actions on S2"1 and
maps S2n1  Rm, Osaka J. Math 7 (1970) 451456
[14] H.J. Munkholm and M. Nakaoka, The BorsukUlam theorem and formal groups,
Osaka J. Math. 9(1972), 337349
[15] M. Nakaoka Generalizations of BorsukUlam theorem, Osaka J. Math. 7(1970),
423441
[16] B. A. P .inkov, The dimension and geometry of mappings, (Russian) Doklad. Akad.
Nauk. SSSR 221(1975), 543546.
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Nauk. vol.39 (1984), no. 5(239), 107130.
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[18] B. A. P .inkov, On the geometry of continuous mappings of finite dimensional
metrizable compact, Proc. Steklov Inst. Math. vol. 212(1996), 138162.
[19] J.E. Roberts, A stronger BorsukUlam type theorem for proper Zpactions on mod p
homology nsheres, Proc. Amer. Math. Soc., vol. 72, n. 2 (1978), pp. 381386
[20] Y. Sternfeld, On finitedimensional maps and other maps with "small" fibers, Fund.
Math., 147 (1995), 127133.
[21] H. Torunczyk, Finite to one restrictions of continuous functions, Fund. Math. 125
(1985), 237249.
[22] H. Murat Tuncali, V. Valov, On dimensionally restricted maps, Fund. Math. 175
(1)(2002), 3552.
[23] Yuri A. Turygin, Approximation of kdimensional maps, Topology and It's Applica
tions, Topology And It's Applications, 139 (2004) 227235
[24] Yuri A. Turygin, A BorsukUlam theorem for (Zp)kactions on products of (mod p)
homology spheres, Topology And It's Applications, 154 (2007) 455461
[25] Yuri A. Turygin, A BorsukUlam theorem for Z_ ., tions on spheres, preprint
[26] V. V. Usp, nl;ii, A selection theorem for Cspases, Topology Appl. 85 (1998)
351374.
[27] V. Volovikov, BourginYang theorem for Lpactions, Mat. Sb., vol. 183, n. 7 (1992),
pp. 115144
[28] C.T. Yang, On maps from spheres to Euclidean spaces, Amer. J. Math. vol. 79, no.
4(1957), 725732
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Ann. Math. vol. 60, no. 2 (1954), 262282
BIOGRAPHICAL SKETCH
I was born in Troitsk, Moscow Region, on September 4, 1978. After graduation from
High School 5 at the age of 16, I enrolled into a program in mathematics at People's
Friendship University. In a couple of years I realized that for me to have a realistic chance
of becoming a mathematician I need to get into a better school. In fall 1997, I started
attending a research seminar on General Topology at Moscow State University organized
by Boris Alekseevich P ti, kov. In summer 1998 Professor P ti, kov helped me transfer
to the Mathematics Department of Moscow State University, where I continued studying
topology under his supervision. I graduated from Moscow State University in June 2002,
with a bachelors degree in Mathematics. In August 2002 I started in the PhD program at
the University of Florida which I completed in May 2007.

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IwouldliketothanksmyadvisorAlexanderDranishnikovformanyencouragingconversationsovertheyearsonvarioustopicsintopologywhichwereofagreatinuenceonmymathematicaleducation.IalsowouldliketothankYuliRudyakforbeingalwaysabletondtimetodiscusstopologywithme.Hisinuenceonmymathematicaleducationhasalsobeensubstantial. 4
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page ACKNOWLEDGMENTS ................................. 4 ABSTRACT ........................................ 6 CHAPTER 1BORSUKULAMTHEOREMS ........................... 7 1.1Introduction ................................... 7 1.2BorsukUlamTheoremfor(Zp)kactions ................... 8 1.3Calculationofw2k1() ............................. 10 1.4EulerClassofC:EGGIC(G)!BG 11 1.5ProofofTheorem 1.2.1 ............................. 13 1.6BorsukUlamTheoremforZ2kactions .................... 16 1.7NecessaryLemmas ............................... 18 1.8ComputationofNorms ............................. 21 1.9ComputationofEulerClasse(m) .................... 22 1.10ProofofTheorem 1.6.2 ............................. 25 2HUREWICZTHEOREMANDAPPROXIMATIONOFMAPS ......... 27 2.1Introduction ................................... 27 2.2ProofofTheorem 2.1.7 ............................. 30 2.3ProofsoftheApproximationTheorems .................... 34 3BORSUKULAMTHEOREMSFORMAPSWITHINFINITEFIBERS .... 38 3.1Bula'sProperty ................................. 38 3.2LipschitzCompactication ........................... 40 3.3TheConstruction ................................ 41 APPENDIX ABORSUKULAMTHEOREMFORZP:::ZPKACTIONSONPRODUCTSOFSPHERES .................................... 45 REFERENCES ....................................... 48 BIOGRAPHICALSKETCH ................................ 50 5
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Thisdissertationisdevotedtoseveraltopicsingeometrictopologyanddimensiontheory.IntherstchapterwediscussBorsukUlamtheorems.Weviewedthehistoryofthesubject,statedafewclassicalresultsinthisareaanddescribedageneralapproachtoprovingBorsukUlamtypetheorems.Theresultsoftheauthorinthisareaarealsostatedandprovedinthischapter. Inthesecondchapterwediscusstwocloselyrelatedquestionsindimensiontheory.Namely,aberwiseversionoftheclassicaltheorembyHurewiczabout0dimensionalmapsofkdimensionalcompactaintokdimensionalcubeandaconjecturebyV.V.Uspenskijaboutapproximationofkdimensionalmapsbetweencompactabykdimensionalsimplicialmapsofpolyhedra. InthethirdchapterweoutlineageneralgeometricconstructionwhichshowshowitmightbepossibletouseBorsukUlamtypetheoremsforconstructinganexampleofa1dimensionalmapbetweencompactawhichcannotbeapproximatedby1dimensionalsimplicialmapsofpolyhedra. 6
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29 ]andD.G.Bourgin[ 1 ]: 2 ]andithasrstappearedintheirfamousbook. 1.1.2 becameacornerstoneinthedevelopmentoftheBorsukUlamtypetheorems.ItsproofhelpedtoshapeupthegeneralapproachtoprovinggeneralizationsoftheBorsukUlamtheorem.ItsimportancecanhardlybeoverestimatedalsoduetothefactthatithasthefamoustheorembyJ.Milnor[ 11 ]asoneofitscorollaries.ThetheoremofJ.Milnorassertsthateveryelementofordertwoinagroupwhichactsfreelyonaspheremustbecentral(see[ 15 ]fordetails).Thelatertheoremplayedanimportantroleinthesolutionofthesocalled"sphericalspaceformproblem"whichaimwastogiveaclassicationofallnitegroupswhichadmitafreeactiononasphere. IntheirconsequentworksH.Munkholm[ 12 ]andM.Nakaoka[ 15 ]showedthatthedierentiabilityconditionontheinvolutionTintheformulationofTheorem 1.1.2 can 7
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24 ].Furtheranduntiltherestofthedissertationpisalwaysassumedtobeaprimenumber. 3 ].Inthecasenim(pk1)foralli(1ik)thetheoremabovewasprovedbyV.V.Volovikovin[ 27 ].Moreover,intheVolovikov'stheoremtheactioncanbeassumedanarbitraryfreeaction. 8
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Proof. Conversely,givenacontinuoussectionsof,itdenesaGequivariantmaps:M!MRm[G]whichisduetoitsequivariancemustbeoftheforms(x)=(x;Pg2Gf(xg1)g)forsomef:M!Rm,andthelemmafollows. Usually,toproveaBorsukUlamtypetheoremformapsintoEuclideanspacesoneshowsthattheEulerclassofthevectorbundle:MGIRm(G)!M=Ginasuitablecohomologytheoryisnontrivial.ThenthedimensionrestrictionsonthecoincidencesetA(f)follow(seetheproofofTheorem 1.2.1 ).Thetheoremsfrom[ 12 13 ]wereprovedinthisway.Unfortunately,whenoneusesordinarycohomologytheory,Eulerclassofveryoftenturnsouttobetrivial(see[ 12 ]).This,infact,isthereasonwhyallavailableresultsintheareaarerestrictedtotheactionsofsofewgroups.InthissettingtheresultsofH.Munkholmfrom[ 13 ](alsosee[ 14 ])areespeciallyinteresting.InthatpaperheprovesaBorsukUlamtypetheoremforZpaactions,pisodd,onodddimensionalspheresusinga 9
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25 ]andinthisdissertation(seeTheorem 1.6.2 ). TheprooftheTheorem 1.2.1 isbasedonthenontrivialityofthe(modp)Eulerclassofacorrespondingvectorbundle.ThenexttwosectionswillbedevotedtothecalculationofEulerclassesofrelevantvectorbundles. InthissectionassumethatG=(Z2)k.AsusualBGstandsfortheclassifyingspaceofGandEGstandsforthetotalspaceoftheuniversalGbundle.Thissectionisdevotedtothecalculationofthe(mod2)Eulerclassofavectorbundle:EGGIR(G)!BG,i.e.itsStiefelWhitneyclassw2k1().ThesecalculationsarethenneededintheproofofTheorem 1.2.1 incasep=2.RecallthatH(BG;Z2)isapolynomialalgebraZ2[x1;:::;xk]on1dimensionalgenerators. Letibeavectorbundleobtainedfromthefollowingdiagram:E(i)!S1Z2R[Z2]i##BGpri!RP1
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1.2.1 incaseofoddprimes.Recallthat:H(BG;Zp)=Zp(y1;:::;yk)Zp[x1;:::;xk]; ChernclassesofaregularrepresentationofG,i.e.ChernclassesofthevectorbundleC"1C:EGGC[G]!BG,wererstcomputedbyB.M.MannandR.J.Milgramin 11
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10 ].Thelemmawhichisstatedafterthenextdenitionisessentiallyborrowedfromtheirpaper. ThepolynomialdenedaboveiscalledthekthDickson'spolynomial(see[ 10 ]formoredetails). pinducesonCastructureofaC[Zp]modulewhichwewilldenotebyL.Letpri:BG!BZpbeaprojectionontheithcoordinate.Thenletibea1dimensionalcomplexvectorbundleobtainedfromthefollowingdiagram:E(i)!S1ZpLi##BGpri!BZp Letibeavectorbundleobtainedfromthefollowingdiagram:E(i)!S1ZpC[Zp]i##BGpri!BZp
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1.2.1 1.2.1 .HereassumethatpisanyprimenumberandthatG=(Zp)k. 1.2.1 Byuniversalitypropertythereexiststhefollowingcommutativediagram:MIRm(G)!EGGIRm(G)M##M=G'!BG: 1.3 .ThenfromtheisomorphismIRm=IR:::IR=mIRitfollowsthat=:::=m.Thuse2()=w2k1()=w2k1()m.ByLemma 1.3.1 wehavew2k1()=kYq=1Y1i1<:::
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whereQkdoesnotcontainmonomialsoftheformxm1x2m2:::x2k1mk.ItiseasytoverifythatH(M=G;Z2)=Z2[x1;:::;xk]=(xn1+11;:::;xnk+1k); 1.4 .ThenfromtheisomorphismICm=IC:::IC=mICitfollowsthatC=:::=m,whereCisacomplexicationofthevectorbundle.Wehavethefollowingchainofequalities:ep()2=ep(C)=c2k1(C)=c2k1(C)m:(2) ByLemma 1.4.1 wehaveep(C)=(1)kLp1k=(1)kLp1k124Yj2Zp(1x1+:::+k1xk1+xk)35p1==(1)kxpk1(p1)kLp1k1+Rk;
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21xmp(p1) 22xmpk1(p1) 2k+^Qk;(3) wherea2(1)q(modp)forsomeq0and^Qkcontainsnomonomialsoftheformbxm(p1) 21xmp(p1) 22xmpk1(p1) 2k;b6=0;b2Zp: 21;:::;xnk+1 2k); 21;:::;xnk+1 2k): SinceA(f)isclosedandGinvariant,thesetMnA(f)isalsoGinvariant,andthereforewecanconsiderthefollowingexactsequenceofapair::::!Hl(M=G;(MnA(f))=G)!Hl(M=G)!Hl((MnA(f))=G)!:::: 1.2.1 thevectorbundleMhasanonvanishingsectionoverMnA(f).Thus(ep(M))=0.Thereforethereexistsanontrivialelement2Hm(pk1)(M=G;(MnA(f))=G) 15
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7 ]).SincethegroupGisniteiteasilyfollowsthatdimA(f)dimZpA(f)dimMm(pk1); 14 ]H.MunkholmandM.NakaokaprovedthefollowinggeneralizationoftheBorsukUlamtheorem: First,considerabundle^:GMG!=G,whereMG=QjGji=1MandGactsonMGbypermutingthecoordinates.Everycontinuousmapf:!Minducesasections(f)ofthebundle^givenbytheformula:s(f)(xG)=(x;Xg2Gf(xg1)g)G:
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1.2.1 fordetails. Nowconsiderthenormalbundleof=GMinGMG.Itwasshownin[ 14 ]thathasastructureofacomplexvectorbundle(Proposition2[ 14 ]).MainlyitfollowsfromthefactthatallirreduciblerepresentationsofGarecomplex.LetbethegKUtheoreticThomclassof.Itturnsoutthatforanycontinuousmapf:!Mtheinducedhomomorphisms(f):gKU(GMG;=GM)!gKU(=G) inKtheorymapstotheEulerclassofthevectorbundle:GIRm(G)!=G,whereIRmisthekerneloftheaugmentationhomomorphismRm[G]!Rm(seeProposition3of[ 14 ]).HereandthroughoutthepaperRmisassumedtobearingwithmultiplicationgivenbythemultiplicationofthecoordinates. Itfollowsfromourconstructionsthatiff:!MisamapwhichdoesnotcollapseanyorbitofGtoapoint,thens(f)()=e()=0. ThecomplexGmoduleIRm(G)isasumofallnontrivialirreduciblecomplexGmodules,whichmakesasumofonedimensionalcomplexvectorbundles.ThisdecompositionallowstocomputethegKUtheoreticEulerclassof.Fromcertainconsiderationsinelementaryalgebraicnumbertheoryitfollowsthate()6=0provideddimissucientlylarge,whichcompletestheproofofthetheorem. IncaseG=Z2k;k>1;neithernorhaveacomplexstructure,simply,becausethedimensionofandisodd.ThisfactdoesnotallowtousecomplexKtheoryinordertoprovethatdoesnothaveanonvanishingsectionprovideddimensionofissucientlylarge.Presumably,thisisthereasontheaboveresultofH.MunkholmandM.NakaokaisrestrictedtothecaseofoddordergroupsG=Zpk.NotethattheEulerclassofinordinarycohomologytheoryistrivialifk>2(see[ 12 ],[ 14 ]fordetails). 17
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1.6.1 donotneedtobeassumeddierentiable.Theproofworkswithoutanychangesifoneassumes=S2n+1,theactionGtobefreeandthemanifoldMtobeanmdimensionaltopologicalmanifold. 1.6.2 whichrstappearedin[ 25 ]coversthisgap. 1.6.2 untilsection 1.10 .InthenextsectionswewillstateandproveallthenecessaryresultswhichareneededfortheproofofTheorem 1.6.2 18
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1.6 ,theproofofTheorem 1.6.2 heavilyreliesonthegeometryofthevectorbundle.Inthenextlemmawewillgiveafulldescriptionofintermsof"smaller"vectorbundleswhosegeometryisfairlysimple. LetGactonCbyrotationsby2 1.7.2 andthefollowingobviousequality:=2k1: Let:E()!S2n+1=Gbeavectorbundleinducedfromavectorbundle:S2n+1GR!S2n+1=Gviaaprojectionpr:S2n+1=GM!S2n+1.Itiseasy 19
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Itiseasytoseethatthereexiststhefollowingcommutativediagram:S2n+1G(MGRm)!S2n+1GRm#"j#mS2n+1GMG^!S2n+1=G;
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isanobviousinclusion.ThefollowingpropositionplaysanimportantroleintheproofofTheorem 1.6.2 : (a)N(pl1)=ppl,0l
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[Q(pl1):Q]=pkpk1 InthissectionwewillcomputetheEulerclasse(m)2gKU0(S2n+1=G)andwillprovethatitisnontrivialprovidedthedimensionofthesphereS2n+1issucientlylarge. mandlet:S2n+1ZmC!S2n+1=Zm Proof.
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maptherstChernclassoftotherstChernclassof. 1.7.2 andProposition 1.9.1 23
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1.8.1 wehave:2k1Yj=2N(j())[2k1
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1.8.1 ,N(1)=p.Thus,wehaven+1dm2k1m2k2(k1)2k1+1 anddn2k1(m1)m2k2(k1): 1.7.2 itfollowsthat:e(dm)=((x+1)1)d[((x+1)1)((x+1)21):::((x+1)2k11)]m: 1.9.3 1.6.2 1.6.2 .Recallthatforamapf:S2n+1!Mwedeneacoincidenceset:A(f)=fx2S2n+1jf(x)=f(gx)8g2Z2kg: 1.6.2 1.7.1 wehavee(d)s(f)()=e(d)e(m)=0:
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1.9.3 ,wemusthave:dn2k1(m1)m2k2(k1):
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In[ 26 ]V.V.Uspenskijintroducedthenotionofamapadmittinganapproximationbykdimensionalsimplicialmaps.Followinghimwesaythatamapf:X!Yadmitsapproximationbykdimensionalsimplicialmapsifforeverypairofopencovers!XofthespaceXand!YofthespaceYthereexistsacommutativediagramofthefollowingformXX!jKjf##pYY!jLj; InthatpaperV.V.Uspenskijproposedthefollowingquestionandconjecturedthatinthegeneralcasetheanswertoitis"no". (Q1)Doeseverykdimensionalmapf:X!Ybetweencompactaadmitapproximationbykdimensionalsimplicialmaps? 27
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6 ]A.N.DranishnikovandV.V.Uspenskijprovedthatlightmapsadmitapproximationbynitetoonesimplicialmaps.InthispaperwegivesomepartialresultsansweringthequestionofV.V.Uspenskijinarmative. 16 ].Werecallthatthediagonalproductoftwomapsf:X!Yandg:X!Zisamapf4g:X!YZdenedbyf4g(x)=(f(x);g(x)). (Q2)Letf:X!Ybeakdimensionalmapbetweencompacta.Doesthereexistamapg:X!Iksuchthatdim(f4g)0? Inthispaperweprovethefollowingtheoremwhichstatesthatthequestions(Q1)and(Q2)areequivalent. 28
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16 ],[ 17 ],[ 18 ],[ 21 ],[ 20 ],[ 8 ],[ 9 ],[ 22 ]).In[ 16 ]Pasynkovannouncedthefollowingtheoremtowhichtheproofappearedmuchlaterin[ 17 ]and[ 18 ]. 21 ]TorunczykprovedthefollowingtheoremwhichiscloselyrelatedtothetheoremprovedbyPasynkov. 2.1.5 holds(forf)ifandonlyifthestatementoftheorem 2.1.6 holds(forf)(theproofcanbefoundin[ 8 ]). WeimprovetheargumentusedbyTorunzykin[ 21 ]toprovethenexttheoremandtheimplication"(="oftheorem 2.1.4 5 ].
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6 ]itfollowsthattheextensionaldimensioncannotbeloweredby0dimensionalmapssothecorollaryisanimmediateconsequenceoftheorem 2.1.7 OnecanunderstandthestatementofthepreviouscorollaryasageneralizationoftheclassicalHurewiczformula. 2.1.7 Proof. LetF=N0[SfNk:k1gbetheunionofallnitesequencesofpositiveintegersplusemptysequenceN0=fg.Foreveryi2Fletusdenotebyjijthelengthofthesequenceiandby(i;p)thesequenceobtainedbyaddingtoiapositiveintegerp. 30
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(a)F(i)isclosedinY,thesetsU(i)andV(i)areopensubsetsofXand[U(i)]\[V(i)]=?; (b)U()B;V()CandF()=Y; (c)U(i;p)U(i)\f1(F(i;p))andV(i;p)V(i)\f1(F(i;p))foreveryp2N; (d)F(i)[fF(i;p):p2NganddiamF(i)<1 (e)thesetE(i)=f1(F(i))n(U(i)[V(i))admitsanopencoveroforderkanddiameter1 1+jij; (f)innotationsof(e)thefamilyfE(i;p):p2NgisdiscreteinX. ThenthereexistsaclosedsubsetTofXsuchthatdimTk1andforanyy2YthesetTseparatesf1(y)betweenBandC. Proof.
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(1)FpisclosedinY,thesetsUpandVpareopensubsetsofXand[Up]\[Vp]=?; (2)UpU\f1(Fp),VpV\f1(Fp); (3)FSfFp:p2NganddiamFp<; (4)thesetEp=f1(Fp)n(Up[Vp)admitsanopencoveroforderkanddiameter; (5)innotationsof(4)thefamilyfEp:p2NgisdiscreteinX. Proof. andifGisamemberoflthenf1(G)n(U(G)[V(G))Ql(y) forsomey2F.LetfF(G):G2gbeaclosedshrinkingofthecoverandletE(G)=f1(F(G))n(U(G)[V(G)) 32
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2.2.2 Proof. 2.2.2 .NowtogetthesetsU(i;p),V(i;p)andF(i;p)forallp2NapplyLemma 2.2.3 tothesetsU=U(i),V=V(i),F=F(i)andto=1 1+jij. 2.2.2 Proof. 2.2.2 .Take=minf(U(i);V(i)) 4;1 1+jijg.Byassumption,thereexistsacommutativediagramofthefollowingform:XX!jKjf##pYY!jLj;
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2.2.3 tothesetsU,V,FandtotoproducethesetsUp,VpandFpforallp2Nsatisfyingconditions(1)(5)ofLemma 2.2.3 .NowsetU(i;p)=1X(Up),V(i;p)=1X(Vp)andF(i;p)=1Y(Fp).Sincetakingapreimagepreservesintersections,unionsandsubtractions,thesetsU(i;p),V(i;p)andF(i;p)satisfytheconditions(a)(f). 2.1.7 8 ]),so,itissucienttoproveonly(i).But(i)immediatelyfollowsfromLemmas 2.2.1 2.2.2 and 2.2.4 2.1.4 8 ]toshowthatthereexistsamapg:X!Ikwithdim(f4g)0itissacienttondacompactsubsetAinXofdimensionatmostk1suchthatdimfjXnA0.TheexistenceofsuchsubsetAfollowsimmediatelyfromLemmas 2.2.1 2.2.2 and 2.2.5 34
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(a1)&!Y; (b1)theorderof&doesnotexceeddimY+1; (c1)(&)=fUI:(;)2ADgfO(y;t):(y;t)2YIkg. Thepartitionofunityfu:2AgonYsubordinatedtothecover&givesrisetothecanonicalmap:Y!jN&j.Thenthemapid:YIk!jN&jIkisan(&)map.By:jN&jIk!jN&jwedenotetheprojection.LetandbesuchtriangulationsonpolyhedrajN&jIkandjN&jrespectivelysuchthatthefollowingconditionsaresatised: (10):jN&jIk!jN&jisasimplicialmaprelativetothetriangulationsand; (20)f(id)1(St(a;)):a2g&; (30)f1(St(b;)):b2g&. Letusdene=fSt(a;):a2gand=fSt(b;):b2g.Denethepartitionofunityfwa:a2gonjN&jIksubordinatedtothecoverbylettingwa(z)tobethebarycentriccoordinateofz2jN&jIkwithrespecttothevertexa2.Analogously,denethepartitionofunityfvb:b2gonjN&jsubordinatedtothecover.Notethattheprojection:jN&jIk!jN&jsendsthestarsoftheverticesofthetriangulationtothestarsoftheverticesofthetriangulation.Thatiswhythereisasimplicialmap$:N!Nbetweenthenervesofthecoversand.Moreover,thefollowingdiagramcommutes.jN&jIk!jNj##$jN&j!jNj:
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Recallthat'isanfO(y;t)gmap.Foreverya2pickapoint(y;t)asuchthatWaO(y;t)a.Letwa=wa(f4g)andBa=(y;t)a.Thensupp(wa)(f4g)1(Wa)(f4g)1(O(y;t)a): 36
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ThefollowingresultsofM.Levin[ 8 ]andY.Sternfeld[ 20 ]areneededtoprovetheorems 2.1.2 and 2.1.3 : 2.1.2 2.3.1 and 2.1.4 2.1.3 2.3.2 and 2.1.4 37
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AquestionaboutexistenceofopenmapsbetweencompactawithinniteberswhichdoesnotsatisfyBula'spropertyiswellknownincontinuumtheoryandwasrststatedbyBula.Therstexampleofsuchamapwasgivenby:p:1Yi=0S2i!1Yi=0RP2i 4 ].ThefollowingtheoremisageneralizationofhisconstructionandheavilyreliesonTheorem 1.2.1 InterestingexamplesofopenmapsbetweencompactawithoutBula'spropertywereconstructedin[ 9 ].Recallthefollowingtheoremfromthatpaper: 38
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Thereisenoughevidencetobelievethatsuchanexampleexists.Forinstance,themapsproducedbyTheorem 3.1.2 almostsatisfytherequirementsofbeingsuchexamples. (i)f:X!YdoesnotsatisfyBula'sproperty, (ii)forany':X![o;1]thereexistsy2Ysuchthat'(f1(y))=pt; (iv)Themapfcannotbeapproximatedby1dimensionalsimplicialmaps. Proof. Theimplication(ii))(i)isobvious. ByTheorem 2.1.4 thestatements(ii)and(iv)areequivalent.Toseehow(i),(ii)notethatifthemapg:X![0;1]exists,thenitdoesnotmapanyberofftoapoint. ThepreviouspropositionshowshowBula'spropertyisrelatedtoHurewicztheoremformapsandV.Uspenskij'sconjecture.Ourgoalnowwouldbetoconstructanexampleofastrictly1dimensionalmapbetweencompactawhichwouldsatisfyatleastoneoftheproperties(i)(iv). TheexamplesofmapsproducedbyTheorem 3.1.2 ,althoughsupplyevidencethatthedesiredexampleexists,donothaveuniformdimensionalityofthebers,i.e.incasen=1thebersofthemapsproducedbyTheorem 3.1.2 arenotstrictly1dimensional. 39
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ItisawellknownfactingeneraltopologythateachcompacticationofasucientlygoodtopologicalspacecanbedescribedasthesetofmaximalidealsofaBanachalgebraoffunctions.LipschitzmapsonapropercompactmetricspacedonotformaBanachalgebra,sowewillconsiderthesmallestBanachalgebracontainingallLipschitzfunctions. LetXbeapropermetricspace.DenotebyCL(X)theclosureofthesetofallboundedLipschitzfunctionsf:X!RwithLip(f)<1.Forx2Xdene(x)=(f(x))f2CL(X)2RCL(X).Itiseasytoprovethat:X,!RCL(X)isanembedding.DenetheLipschitzcompacticationLXofthepropermetricspaceXasLX= Weclaimthatthereexisti02Nforwhichgi0(F)<1 3andgi0(Xn~U)>2 3.Thiscanbeprovenasfollows.Assumethereisnosuchi02N.Thenthereexistasequencexn2XnU
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30suchthatf(F)>.Notethatf(U)=0.Itiseasytocheckthatf:X![0;1]isaLipschitzfunction.Therefore,thereexistsafunctionf:X![0;1]suchthatfjX=f.Obviously,f(F). Aneasyconsequenceofthepreviouspropositionisthefollowing Proof. LetFF~UbeasubsetofXsuchthattheclosureofFinLXcontainsx2LX.ThenbyProposition 3.2.1 ,dist(F;Xn~U)>0.Thendist(f(F);Ynf(~U))>0,becausefislocallyaprojection.Therefore,byProposition 3.2.1 ,wehave 41
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1.1.3 wecanchooseasequencefnigofoddnumberssuchthatafreeZpiactiononSniwillhave1BorsukUlamproperty,butwillnothave2BorsukUlamproperty. (i)Foranyx2Sniwehavediamfi(Zpix)1; 3.2.2 ,wehaveanopenmapL(tpi):L(tSni)!L(tCP[ni betweenLipschitzcompactications. Proof. 22 ]itwasprovedthatthereexistsamapg:cS1![0;1]suchthatthemap^p4g:cS1![CP1[0;1] is0dimensional.Itfollowsfromthelaststatementthatg(^p1(y))=[a;b]witha6=bforeveryy2[CP1.LetLy:[a;b]![0;1]bealineartransformationof[a;b]into[0;1].Deneamap':cS1![0;1]asacomposition'=Lyg.Thecontinuityofthemap'essentiallyfollowsfromthefactthat^pisanopenmap.Now,deneA='1(0)andB='1(1).ThesetsAandBareclosedanddisjointandeachoneofthemintersectseachberof^p.Also,notethatdist(A;B)>0,therefore,theclosuresofAandBinaredisjointLcS1by 42
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3.2.1 .Now,nontrivialityofeachberofthemapL^pfollowsfromProposition??. Proof. LetfOj2AgbeanopencoveringofLcS1suchthatforany2Aandanyx;x02Oitfollowsthatj'(x)'(x0)j< pkbethegeneratorofthegroupZpk.Wecanchoosek2Ntobelargeenough,sothatthepointsx2S1=^p1(y)ande2i pkx2S1=^p1(y)arefOj2Agcloseforanychoiceofy2[CP1,i.e.thereexists2Asuchthatx;e2i pkx2O. Now,choosenlargeenough,sothatTheorem??guaranteesexistenceofanorbitofZpkwhichwillbecollapsedtoapointunderthefollowingcomposition:S2n1i,!LcS1'!I: Toproduceanexampleofanopenstrictly1dimensionalmapitremainstoproveProposition 3.3.1 andthefactthatthebersofourmapareareallstrictly1dimensional.Thelaterfactintuitivelyseemsobvious,sinceaberinthecoronaofourmapisbeingapproachedbycirclesinaLipschitzmanner.So,itseems,thatthebersinthecoronashouldalsobe1dimensional,sincedimensioncannotberaisedbyaLipschitzmap.Oncewehaveprovedallthese,wehaveproducedthedesiredexamplewhichremainstohave 43
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AfterapplyingScepin'sSpectralTheoremwehaveproducedanexampleofamapwhichgivesapositivesolutiontotheV.Uspenskij'sconjectureandanswersthequestionofB.A.Pasynkovinthenegative. 44
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SetG=Zp:::Zpkforaxedoddprimepandk1andletM=S2n11:::S2nk1:InthissectionwewilldiscussafailedattempttoproveaBorsukUlamtheoremforGactionsonproductsofspheresandmapsintoEuclideanspaces.Thiscaseismoredicultthanthecaseof(Zp)kactions,becausetheEulerclassofthecorrespondingvectorbundleintheordinarycohomologytheoryturnsouttobetrivial.BelowwewillalsoshowthattheEulerclassofthevectorbundle,whichneedtobeconsidered,inthecomplexKtheoryistrivialaswell.Therefore,thisparticularcasecallsforamoresophisticatedcohomologytheoryinordertoextractaBorsukUlamtheorem. LetLibea1dimensionalcomplexZpimodule,inwhichZpiactsbymultiplicationbye2i piandlet0:S2ni1ZpiLi!Lni(pi) betheassociatedcomplexvectorbundle.HereLni(pi)isthelensspaceS2n11=G:Denotebyi:Ln1(p):::Lnk(pk)!Lni(pi) theprojectionontheithfactorandseti=(0i):ThenitfollowsfromtheisomorphismC[G]wC[Zp]:::C[Zpk] that=(M0C[G]!M=G)w1:::k;
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isnontrivial.Obviously,C=;therefore,itissucienttoprovethate()6=0.Thisprobablycanbedonebymeansofsomeextraordinarycohomologytheory.LetusseewhathappensinthecaseofcomplexKtheory. First,recallthat~K0C(Ln1(p):::Lnk(pk))wZ[x1;:::;xk]=I;
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fromabove.Theauthorisbeingabletoprovethatinfactthepolynomiale()almostalwaysbelongtotheidealIwhichmakethecomplexKtheoryapproachinecientwhentryingtoproveaBorsukUlamtheoremforthesekindsofactions. 47
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[1] D.G.Bourgin,Multiplicityofsolutionsinframemappings,IllinoisJ.Math.,vol.9(1965),169177 [2] P.E.ConnerandE.E.Floyd,Dierentiableperiodicmaps,SpringerVerlag,Berlin,1964 [3] A.N.Dranishnikov,OnQbrationswithoutdisjointsections,Funct.Anal.Appl.,22(1988)no.2,151152 [4] [5] A.N.Dranishnikov,TheEilenbergBorsuktheoremformapsintoarbitrarycomplexes,Math.Sbornikvol.185(1994),no.4,8190. [6] A.N.DranishnikovandV.V.Uspenskij,Lightmapsandextensionaldimension,TopologyAppl.,80(1997)9199. [7] A.N.Dranishnikov,Cohomologicaldimensionofcompactmetricspaces,6issue1(2001),TopologyAtlasInvitedContributions [8] M.Levin,Bingmapsandnitedimensionalmaps,Fund.Math.151(1)(1996)4752. [9] M.LevinandW.Lewis,Somemappingtheoremsforextensionaldimension,arXiv:math.GN/0103199 [10] B.M.MannandR.J.MilgramOntheChernclassesoftheregularrepresentationsofsomenitegroups,Proc.EdinburghMath.Soc(1982)25,259268 [11] J.Milnor,GroupswhichactonSnwithoutxedpoints,Amer.J.Math.,vol.79,n.3(1957),pp.623630 [12] H.J.MunkholmBorsukUlamtypetheoremsforproperZpactionson(modp)homologynspheres,Math.Scand.24(1969)167185 [13] H.J.MunkholmOntheBorsukUlamtheoremfortheZpaactionsonS2n1andmapsS2n1!Rm,OsakaJ.Math7(1970)451456 [14] H.J.MunkholmandM.Nakaoka,TheBorsukUlamtheoremandformalgroups,OsakaJ.Math.9(1972),337349 [15] M.NakaokaGeneralizationsofBorsukUlamtheorem,OsakaJ.Math.7(1970),423441 [16] B.A.Pasynkov,Thedimensionandgeometryofmappings,(Russian)Doklad.Akad.Nauk.SSSR221(1975),543546. [17] B.A.Pasynkov,Theoremon!mappingsformappings,(Russian)Uspehi.Math.Nauk.vol.39(1984),no.5(239),107130. 48
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[18] B.A.Pasynkov,Onthegeometryofcontinuousmappingsofnitedimensionalmetrizablecompacta,Proc.SteklovInst.Math.vol.212(1996),138162. [19] J.E.Roberts,AstrongerBorsukUlamtypetheoremforproperZpactionsonmodphomologynsheres,Proc.Amer.Math.Soc.,vol.72,n.2(1978),pp.381386 [20] Y.Sternfeld,Onnitedimensionalmapsandothermapswith"small"bers,Fund.Math.,147(1995),127133. [21] H.Torunczyk,Finitetoonerestrictionsofcontinuousfunctions,Fund.Math.125(1985),237249. [22] H.MuratTuncali,V.Valov,Ondimensionallyrestrictedmaps,Fund.Math.175(1)(2002),35{52. [23] YuriA.Turygin,Approximationofkdimensionalmaps,TopologyandIt'sApplications,TopologyAndIt'sApplications,139(2004)227235 [24] YuriA.Turygin,ABorsukUlamtheoremfor(Zp)kactionsonproductsof(modp)homologyspheres,TopologyAndIt'sApplications,154(2007)455461 [25] YuriA.Turygin,ABorsukUlamtheoremforZ2kactionsonspheres,preprint [26] V.V.Uspenskij,AselectiontheoremforCspases,TopologyAppl.85(1998)351374. [27] V.Volovikov,BourginYangtheoremforZnpactions,Mat.Sb.,vol.183,n.7(1992),pp.115144 [28] C.T.Yang,OnmapsfromspherestoEuclideanspaces,Amer.J.Math.vol.79,no.4(1957),725732 [29] C.Y.Yang,OntheoremsofBorsukUlam,KakutaniYamabeYajoboandDyson,I,Ann.Math.vol.60,no.2(1954),262282
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IwasborninTroitsk,MoscowRegion,onSeptember4,1978.AftergraduationfromHighSchool5attheageof16,IenrolledintoaprograminmathematicsatPeople'sFriendshipUniversity.InacoupleofyearsIrealizedthatformetohavearealisticchanceofbecomingamathematicianIneedtogetintoabetterschool.Infall1997,IstartedattendingaresearchseminaronGeneralTopologyatMoscowStateUniversityorganizedbyBorisAlekseevichPasynkov.Insummer1998ProfessorPasynkovhelpedmetransfertotheMathematicsDepartmentofMoscowStateUniversity,whereIcontinuedstudyingtopologyunderhissupervision.IgraduatedfromMoscowStateUniversityinJune2002,withabachelorsdegreeinMathematics.InAugust2002IstartedinthePhDprogramattheUniversityofFloridawhichIcompletedinMay2007. 50
