Borsuk-Ulam Property of Finite Group Actions on Manifolds and Applications

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02007 Yuri A. Turygin

To my parents


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

alv-i-, able to find time to discuss topology with me. His influence on my mathematical

education has also been substantial.



ACKNOWLEDGMENTS .................

ABSTRACT .........................




Introduction .. .............
Borsuk-Ulam Theorem for (Zp)k-actions
Calculation of i,_ -1 (l) .. .....
Euler Class of rlc: EG x Ic(G) BG .
Proof of Theorem 1.2.1 .. ........
Borsuk-Ulam Theorem for Z2-actions .
Necessary Lemmas .. .........
Computation of Norms .. ........
Computation of Euler Class e(( mp) .
Proof of Theorem 1.6.2 .. ........


2.1 Introduction . . . . . . . .
2.2 Proof of Theorem 2.1.7 ........................
2.3 Proofs of the Approximation Theorems ...............


Bula's Property ..........
Lipschitz Compactification ...
The Construction .........


OF SPHERES .............

REFERENCES ................




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



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 Borsuk-Ulam theorems. We viewed the history of

the subject, stated a few classical results in this area and described a general approach to

proving Borsuk-Ulam 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 0-dimensional

maps of k-dimensional compact into k-dimensional cube and a conjecture by V.V.

U-i ni-1:ii about approximation of k-dimensional maps between compact by k-dimensional

simplicial maps of polyhedra.

In the third chapter we outline a general geometric construction which shows how

it might be possible to use Borsuk-Ulam type theorems for constructing an example of a

1-dimensional map between compact which cannot be approximated by 1-dimensional

simplicial maps of polyhedra.


1.1 Introduction

The famous Borsuk-Ulam 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 Borsuk-Ulam

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 Borsuk-Ulam

type theorems. Its proof helped to shape up the general approach to proving generalizations

of the Borsuk-Ulam 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'/.

n-sphere 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 Borsuk-Ulam Theorem for (Zp)-actions

The purpose of this section is to -,I -.- -1 another generalization of the Borsuk-Ulam

theorem which initially appeared in [24]. Further and until the rest of the dissertation p is

alv--v- 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 i-spheres

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 Borsuk-Ulam 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(xg-1)g)G.
Observe that E = F E where Em is a trivial m-dimensional 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 G-equivariant map s: M

M x R7[G] which is due to its equivariance must be of the form s(x) (x, ZcG f(xg-1)g)

for some f : M --R R, and the lemma follows. E

Usually, to prove a Borsuk-Ulam 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 non-trivial. 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

Borsuk-Ulam type theorem for Z, 11i i, p is odd, on odd dimensional spheres using a

KU-theory Euler class. The remaining case of Z2k-actions 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 non-triviality 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 G-bundle. 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 Stiefel-Whitney 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 1-dimensional 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 1-dimensional 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 1-dimensional bundle. Recall the isomorphism of R-modules:

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
S (N0(Ai A)A e (A a ... 0A ).
i-1 (al,...,ak)EG
It is a well known that the first Stiefel-Whitney class of a tensor product of

1-dimensional real vector bundles equals to the sum of the first Stiefel-Whitney 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
=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 1-dimensional generators and Zp[x, ..., xk is

a polynomial algebra on 2-dimensional 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


Definition 1.4.1. Lk = 1i Ha z/p(axll ... i+ ai-lx1 + xi)

The polynomial defined above is called the kth Dickson's polynomial (see [10] for more


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
which we will denote by L. Let pri: BG -B BZP be a projection on the ith coordinate.

Then let Ai be a 1-dimensional complex vector bundle obtained from the following

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,]

BG ri


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 C-modules: 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:



1 "'" '" 1 L ,


... 0 lk.


From a formula by Whitney and the fact that the first ('!., i class of a tensor product

of 1-dimensional 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
I J[(P -i)!]P ni (aixi + ... + ai-i-1 + x)Pl -
i-1 (al,..., i_ 1,1,0,...,0)
[(p )!]kLp1 (-l)kLP-1

The last equality follows from a theorem of Wilson which states that (p 1)!

(-1)(mod p). Thus e(T]c) = Cpk_ (c) = (-k1)L-1.

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

ni-spheres. 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 non-trivial.
By universality property there exists the following commutative diagram:

M x, IRm(G) EG Xc Im(G)

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
q-1 l_

-= xi (x, +... + xj,) + Rk,
q=1 where Rk contains monomials in powers less than 2k-1. 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


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) = C2k-1(C-) = C2k-1(c/)T. (2)

By Lemma 1.4.1 we have
eP(rcC) = (- )kL' (- )kL -' (a1X1 + ... + k-1X-1 + Xk)

S(-1)k xk-1 + Rk,

where Rk contains Xk in powers less than pk-l(p 1). Thus

ep(C) (- l)(kmr mp 1lp-1)(p-1) +k,
() k- !C 1 + @ k,

where Rk contains Xk in powers less than mpk-l(p 1). Then by induction it follows that

ep(C() (-l)kmnX (P1) mp(p-1) mp. 1(p1)+
... X2k + Qk,

where Qk contains no monomials of the form

bxm(P-l) mp(p-l) p 1p--1), b O, b Z.

Therefore from the previous and (2) it follows that

m(p-1) mp(p-1) mpk-(p-1)
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(p-1) mp(p-1) mpk-1(p-1)
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 > mpi-l(p 1) for all i(1 < i < k) it follows that

e,(M) = c*(e,()) / 0.
Since A(f) is closed and G-invariant, the set M \ A(f) is also G-invariant, 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 non-vanishing section over M \ A(f). Thus

/3(ep({M)) = 0. Therefore there exists a non-trivial 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 non-trivial 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 Borsuk-Ulam Theorem for Z2k-actions

In [14] H. Munkholm and M. Nakaoka proved the following generalization of the

Borsuk-Ulam 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 G-action is given. Let M

be a differentiable m-in,,,,,':;'..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)pk-1 + m + 3].
The proof of the theorem above is essentially based on non-triviality of a KU-theoretic

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.

One can easily show that functions {f: E -+ M} which do not collapse any orbit of G to a

point are in one-to-one 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

KU-theoretic 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 K-theory 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 G-module IRm (G) is a sum of all non-trivial irreducible complex

G-modules, which makes ( a sum of one-dimensional complex vector bundles. This

decomposition allows to compute the KU-theoretic 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 K-theory in order to

prove that ( does not have a non-vanishing 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,, if one assumes = S2n+1, the action G C E

to be free and the in,,,:. .1./ M to be an m-dimensional tej'l''.I1-j:./ .l i ,,,,./'1

To the best knowledge of the author a Borsuk-Ulam 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 m-dimensional 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 E|f(x)

f(gx) Vg e 2k has dimension at least (2n + 1) [2(m 1) + m2k-1(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


1.7 Necessary Lemmas

Let G = Z2k and suppose that a free action of G on S2n+1 is given. Then consider a


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 -


Lemma 1.7.2. ( = m(p A A2 2 ... A2 1-1)

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


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
KU (S2n+1/G). In the following sections we will prove that e( E mp) is non-trivial,

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


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*),


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 Pk-1
[Q(7) Q(-,' -)] [Q) pk-1 k-1-1


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


Therefore, the

p. Also, note that


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)
In this section we will compute the Euler class e( E mp) E KU (S2"+1/G) and will

prove that it is non-trivial 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 one-dimensional complex vector bundle over CP". It is a

well-known fact that the total space of the spherization of the m-fold 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) ....

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.

Therefore, KU (S2+1/) [x]/ ((x + 1) 1,xn+1). The equality x = e(A) in
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) m2k-2(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+ 2k-1 [ I ]

h(x)(x 1)n1 + g(x)(x

1)j f 2jX.

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+l-d-n2k + 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:

,,(7)[ ]m-'

h(7)(7 1)n+1-d-nm2k-


))n+l-d-mn2k 1

So, it follows that

N(( 1))n+l-d-mr2k- I J N(4y))[2]nm-

By Lemma 1.8.1 we have:

N(4 )(7)) ]m- =

- H 2(2J-2- 1)(m2 -- 1-1)

2Z j25- 1(m2h 1-1

N(()2J (7)) 2 j 1-1

S2E -(2_-25 )(m2 -- 1-1)

2) m2k 2(k-1)-2 -1+1

k- (7) kI
N(@y(7)[ m-I

S(X) [ .]mn-

Note that according to Lemma 1.8.1, N(7 1) = p. Thus, we have

n + 1 d m2k-1 < m2k-2(k 1) 2k-1 + 1


d > n 2k-l(m 1) m2-2(k 1).

Corollary 1.9.1. Let d > 0 and suppose that the class e(dA ( mp) is trivial in
KU (S2+1/22k). Then d > n 2k-1(m 1) m2k-2(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 one-dimensional 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

non-vanishing 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) m22-2(k 1).

It follows now that:

dim A(f) > (2n + 1) [2k(m 1) + m2-l1(k 1) + 1].



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 C-space 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 finite-dimensional

paracompact space and each countable-dimensional metrizable space has property C. By a

C-compactum we mean a compact C-space.

In [26] V.V. U-i. i-1:;ii introduced the notion of a map admitting an approximation

by k-dimensional simplicial maps. Following him we -zi- that a map f: X Y admits

approximation by k-dimensional 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 wx-map, Ky is an wy-map and p is a k-dimensional simplicial map

between polyhedra |II and I I.

In that paper V.V. Usp, ii-1:ii proposed the following question and conjectured that in

the general case the answer to it is "no".

(Q1) Does every k-dimensional map f: X Y between compact admit
approximation by k-dimensional simplicial maps?

In [6] A.N. Dranishnikov and V.V. Usp, ii-l:;i proved that light maps admit approximation

by finite-to-one simplicial maps. In this paper we give some partial results answering the

question of V.V. U-p. i-l1:ii in affirmative.

Theorem 2.1.1. Let f: X Y be a k-dimensional map between C-compacta. 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 wx-map, K y is an uwy-map and p is a k-dimensional simplicial map

between compact py1;,,. Ji,i,, I|/C and L1. Furthermore, one can il;,. n, assume that

Theorem 2.1.2. k-dimensional maps between compact admit approximation by (k + 1)-

dimensional simplicial maps.

Theorem 2.1.3. k-dimensional maps of Bing compact (i.e. compact with each compo-

nent her, .:/.r, :,7;; indecomposable) admit approximation by

k-dimensional simplicial maps.

It turned out that the question (Qi) is closely related to the next question proposed

by B.A. P i-Lkov 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 k-dimensional 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 k-dimensional 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 k-dimensional 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 k-dimensional map between finite dimensional

compact. Then there exists a a-compact A C X such that dim A < k 1 and dim f IX\A


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 k-dimensional map between C-compacta. Then

(i) there exists a a-compact 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 k-dimensional map between C-compacta. Then

e- dim(X) < e- dim(Y x Ik).

Proof. From [6] it follows that the extensional dimension cannot be lowered by 0-dimensional

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 f-l(y) between B and C. Then there exists a

a-compact 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 u-compact 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 f-l(y) there exists a pair (7, p) E A such

that G c U, and H C U,. Then T(,,,) C A separates f-l(y) between G and H. So,

dim(f-l(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

(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) = f-1(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 f-l(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 f-l(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}.


Lemma 2.2.3. Let f: X Y be a k-dimensional map between C-compacta 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 f-1(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,,:;/ {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 f-l(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 f-1([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 C-compactum 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 f-l(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 k-dimensional map between C-compacta. 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
k-dimensional 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 c-small 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 c-small 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.

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 k-dimensional 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 a-compact 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') {p-1(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 1|A| 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
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 p-1((). 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 non-negative 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| -+ I|C' by

requiring that the vertex 3 E Ba goes to a. Clearly, the map p' is finite-to-one. 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 k-dimensional simplicial map

p: I|C | 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 k-dimensional.

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 k-dimensional 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 k-dimensional 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


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 well-known 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, > pi-l(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

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 n-dimensional 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 Borsuk-Ulam theorems and Bula's property was stimulated

by Hurewicz theorem for maps and the conjecture by V. Usp, i-1:ii about approximation of

k-dimensional 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 k-dimensional map between compact which cannot be approximated by

k-dimensional 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 (f-l(y)) = pt,
(iii) Hurewicz theorem does not hold for f,

(iv) The map f cannot be approximated by 1-dimensional 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.U-i. i,-:;ii's conjecture. Our goal now would be to construct an example

of a strictly 1-dimensional 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 1-dimensional.

In the next sections we will outline the idea how to construct an example of an open,

strictly 1-dimensional 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 well-known 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:

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 m-Borsuk-Ulam 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 1-Borsuk-Ulam property, but will not have 2-Borsuk-Ulam 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 non-degenerate

connected compact space.

Proof. The space CP" is a countable union of finite dimensional spaces. Therefore, it is a

C-space. 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 0-dimensional. It follows from the last statement that g(p-l(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 = p-1(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, non-triviality 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 1-Borsuk-Ulam

,, '/'. ,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(Lp-l(y)) / pt. Then there exists E > 0 such that diam p(Lp-l(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
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 = P-1(y) and
e7p x S1 = P-(y) are {O1,a c A}-close for any choice of y e CP", i.e. there exists
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:

S2n-1 iLSQ .

Here i: S2n-1 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 1-dimensional map it remains to prove

Proposition 3.3.1 and the fact that the fibers of our map are are all strictly 1-dimensional.

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 1-dimensional, 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 non-metarizable 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, ni-lii's conjecture and answers the question of

B.A.P ti-'nkov in the negative.


Set G = Zp x ... x Zp, for a fixed odd prime p and k > 1 and let M = S2-1 x ... x

S2nK-1. In this section we will discuss a failed attempt to prove a Borsuk-Ulam theorem

for G-actions on products of spheres and maps into Euclidean spaces. This case is more

difficult than the case of (Zp)k-actions, 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

K-theory is trivial as well. Therefore, this particular case calls for a more sophisticated

cohomology theory in order to extract a Borsuk-Ulam theorem.

Let Li be a 1-dimensional complex Zpi-module, in which Zpi acts by multiplication by
e p and let

A': S2ni-1 pi L L n(pi)

be the associated complex vector bundle. Here L (pi) is the lens space S2-nl1/G. Denote


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,]


= (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

t`(g(A, A? ... ED Af') (A D ... 0D A)
ti (ie,...,ak)EG

where e, is 1-dimensional trivial complex vector bundle. It is easy to see that

S= (M X Ic(G) M/G) (A Ak).

To prove a Borsuk-Ulam 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 non-trivial. 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 K-theory.

First, recall that

Ko(L (p) x ... x L" (pk)) Z[x,,...,xk]l,


I = ( ...," (x + 1)P ..., (k + 1) 1).

Here x = [Ai 1] (1 < i < k). This facts follow easily from the existence of


(AP)r, -_ Li (i) (t
from the Gysin sequence and the Kunneth formula for K-theory. 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
(A D... 0 A ) i((Xi + 1)&1 1,..., (Xk + 1)k 1),

where ac is the ith symmetric polynomial. Thus we have
T(,) n i((XI +) -, (X + ) ) -
(a1,...,ak)#0 i 1

In ((Xi + 1)a(X2 + )a2 ... (Xk + 1)a ).
To prove a Borsuk-Ulam 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

alv--i- belong to the ideal I which make the complex K-theory approach inefficient when

trying to prove a Borsuk-Ulam theorem for these kinds of actions.


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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 t-i-, kov. In summer 1998 Professor P t-i-, 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.

Full Text








IwouldliketothanksmyadvisorAlexanderDranishnikovformanyencouragingconversationsovertheyearsonvarioustopicsintopologywhichwereofagreatinuenceonmymathematicaleducation.IalsowouldliketothankYuliRudyakforbeingalwaysabletondtimetodiscusstopologywithme.Hisinuenceonmymathematicaleducationhasalsobeensubstantial. 4


page ACKNOWLEDGMENTS ................................. 4 ABSTRACT ........................................ 6 CHAPTER 1BORSUK-ULAMTHEOREMS ........................... 7 1.1Introduction ................................... 7 1.2Borsuk-UlamTheoremfor(Zp)k-actions ................... 8 1.3Calculationofw2k1() ............................. 10 1.4EulerClassofC:EGGIC(G)!BG 11 1.5ProofofTheorem 1.2.1 ............................. 13 1.6Borsuk-UlamTheoremforZ2k-actions .................... 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 3BORSUK-ULAMTHEOREMSFORMAPSWITHINFINITEFIBERS .... 38 3.1Bula'sProperty ................................. 38 3.2LipschitzCompactication ........................... 40 3.3TheConstruction ................................ 41 APPENDIX ABORSUK-ULAMTHEOREMFORZP:::ZPK-ACTIONSONPRODUCTSOFSPHERES .................................... 45 REFERENCES ....................................... 48 BIOGRAPHICALSKETCH ................................ 50 5


Thisdissertationisdevotedtoseveraltopicsingeometrictopologyanddimensiontheory.IntherstchapterwediscussBorsuk-Ulamtheorems.Weviewedthehistoryofthesubject,statedafewclassicalresultsinthisareaanddescribedageneralapproachtoprovingBorsuk-Ulamtypetheorems.Theresultsoftheauthorinthisareaarealsostatedandprovedinthischapter. Inthesecondchapterwediscusstwocloselyrelatedquestionsindimensiontheory.Namely,aberwiseversionoftheclassicaltheorembyHurewiczabout0-dimensionalmapsofk-dimensionalcompactaintok-dimensionalcubeandaconjecturebyV.V.Uspenskijaboutapproximationofk-dimensionalmapsbetweencompactabyk-dimensionalsimplicialmapsofpolyhedra. InthethirdchapterweoutlineageneralgeometricconstructionwhichshowshowitmightbepossibletouseBorsuk-Ulamtypetheoremsforconstructinganexampleofa1-dimensionalmapbetweencompactawhichcannotbeapproximatedby1-dimensionalsimplicialmapsofpolyhedra. 6


29 ]andD.G.Bourgin[ 1 ]: 2 ]andithasrstappearedintheirfamousbook. 1.1.2 becameacornerstoneinthedevelopmentoftheBorsuk-Ulamtypetheorems.ItsproofhelpedtoshapeupthegeneralapproachtoprovinggeneralizationsoftheBorsuk-Ulamtheorem.ItsimportancecanhardlybeoverestimatedalsoduetothefactthatithasthefamoustheorembyJ.Milnor[ 11 ]asoneofitscorollaries.ThetheoremofJ.Milnorassertsthateveryelementofordertwoinagroupwhichactsfreelyonaspheremustbecentral(see[ 15 ]fordetails).Thelatertheoremplayedanimportantroleinthesolutionofthesocalled"sphericalspaceformproblem"whichaimwastogiveaclassicationofallnitegroupswhichadmitafreeactiononasphere. IntheirconsequentworksH.Munkholm[ 12 ]andM.Nakaoka[ 15 ]showedthatthedierentiabilityconditionontheinvolutionTintheformulationofTheorem 1.1.2 can 7


24 ].Furtheranduntiltherestofthedissertationpisalwaysassumedtobeaprimenumber. 3 ].Inthecasenim(pk1)foralli(1ik)thetheoremabovewasprovedbyV.V.Volovikovin[ 27 ].Moreover,intheVolovikov'stheoremtheactioncanbeassumedanarbitraryfreeaction. 8


Proof. Conversely,givenacontinuoussectionsof,itdenesaG-equivariantmaps:M!MRm[G]whichisduetoitsequivariancemustbeoftheforms(x)=(x;Pg2Gf(xg1)g)forsomef:M!Rm,andthelemmafollows. Usually,toproveaBorsuk-UlamtypetheoremformapsintoEuclideanspacesoneshowsthattheEulerclassofthevectorbundle:MGIRm(G)!M=Ginasuitablecohomologytheoryisnon-trivial.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.InthatpaperheprovesaBorsuk-UlamtypetheoremforZpa-actions,pisodd,onodddimensionalspheresusinga 9


25 ]andinthisdissertation(seeTheorem 1.6.2 ). TheprooftheTheorem 1.2.1 isbasedonthenon-trivialityofthe(modp)Eulerclassofacorrespondingvectorbundle.ThenexttwosectionswillbedevotedtothecalculationofEulerclassesofrelevantvectorbundles. InthissectionassumethatG=(Z2)k.AsusualBGstandsfortheclassifyingspaceofGandEGstandsforthetotalspaceoftheuniversalG-bundle.Thissectionisdevotedtothecalculationofthe(mod2)Eulerclassofavectorbundle:EGGIR(G)!BG,i.e.itsStiefel-Whitneyclassw2k1().ThesecalculationsarethenneededintheproofofTheorem 1.2.1 incasep=2.RecallthatH(BG;Z2)isapolynomialalgebraZ2[x1;:::;xk]on1-dimensionalgenerators. Letibeavectorbundleobtainedfromthefollowingdiagram:E(i)!S1Z2R[Z2]i##BGpri!RP1


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


10 ].Thelemmawhichisstatedafterthenextdenitionisessentiallyborrowedfromtheirpaper. ThepolynomialdenedaboveiscalledthekthDickson'spolynomial(see[ 10 ]formoredetails). pinducesonCastructureofaC[Zp]-modulewhichwewilldenotebyL.Letpri:BG!BZpbeaprojectionontheithcoordinate.Thenletibea1-dimensionalcomplexvectorbundleobtainedfromthefollowingdiagram:E(i)!S1ZpLi##BGpri!BZp Letibeavectorbundleobtainedfromthefollowingdiagram:E(i)!S1ZpC[Zp]i##BGpri!BZp


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<:::

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;


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)isclosedandG-invariant,thesetMnA(f)isalsoG-invariant,andthereforewecanconsiderthefollowingexactsequenceofapair::::!Hl(M=G;(MnA(f))=G)!Hl(M=G)!Hl((MnA(f))=G)!:::: 1.2.1 thevectorbundleMhasanon-vanishingsectionoverMnA(f).Thus(ep(M))=0.Thereforethereexistsanon-trivialelement2Hm(pk1)(M=G;(MnA(f))=G) 15


7 ]).SincethegroupGisniteiteasilyfollowsthatdimA(f)dimZpA(f)dimMm(pk1); 14 ]H.MunkholmandM.NakaokaprovedthefollowinggeneralizationoftheBorsuk-Ulamtheorem: First,considerabundle^:GMG!=G,whereMG=QjGji=1MandGactsonMGbypermutingthecoordinates.Everycontinuousmapf:!Minducesasections(f)ofthebundle^givenbytheformula:s(f)(xG)=(x;Xg2Gf(xg1)g)G:


1.2.1 fordetails. Nowconsiderthenormalbundleof=GMinGMG.Itwasshownin[ 14 ]thathasastructureofacomplexvectorbundle(Proposition2[ 14 ]).MainlyitfollowsfromthefactthatallirreduciblerepresentationsofGarecomplex.LetbethegKU-theoreticThomclassof.Itturnsoutthatforanycontinuousmapf:!Mtheinducedhomomorphisms(f):gKU(GMG;=GM)!gKU(=G) inK-theorymapstotheEulerclassofthevectorbundle:GIRm(G)!=G,whereIRmisthekerneloftheaugmentationhomomorphismRm[G]!Rm(seeProposition3of[ 14 ]).HereandthroughoutthepaperRmisassumedtobearingwithmultiplicationgivenbythemultiplicationofthecoordinates. Itfollowsfromourconstructionsthatiff:!MisamapwhichdoesnotcollapseanyorbitofGtoapoint,thens(f)()=e()=0. ThecomplexG-moduleIRm(G)isasumofallnon-trivialirreduciblecomplexG-modules,whichmakesasumofone-dimensionalcomplexvectorbundles.ThisdecompositionallowstocomputethegKU-theoreticEulerclassof.Fromcertainconsiderationsinelementaryalgebraicnumbertheoryitfollowsthate()6=0provideddimissucientlylarge,whichcompletestheproofofthetheorem. IncaseG=Z2k;k>1;neithernorhaveacomplexstructure,simply,becausethedimensionofandisodd.ThisfactdoesnotallowtousecomplexK-theoryinordertoprovethatdoesnothaveanon-vanishingsectionprovideddimensionofissucientlylarge.Presumably,thisisthereasontheaboveresultofH.MunkholmandM.NakaokaisrestrictedtothecaseofoddordergroupsG=Zpk.NotethattheEulerclassofinordinarycohomologytheoryistrivialifk>2(see[ 12 ],[ 14 ]fordetails). 17


1.6.1 donotneedtobeassumeddieren-tiable.Theproofworkswithoutanychangesifoneassumes=S2n+1,theactionGtobefreeandthemanifoldMtobeanm-dimensionaltopologicalmanifold. 1.6.2 whichrstappearedin[ 25 ]coversthisgap. 1.6.2 untilsection 1.10 .InthenextsectionswewillstateandproveallthenecessaryresultswhichareneededfortheproofofTheorem 1.6.2 18


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




isanobviousinclusion.ThefollowingpropositionplaysanimportantroleintheproofofTheorem 1.6.2 : (a)N(pl1)=ppl,0l

[Q(pl1):Q]=pkpk1 InthissectionwewillcomputetheEulerclasse(m)2gKU0(S2n+1=G)andwillprovethatitisnon-trivialprovidedthedimensionofthesphereS2n+1issucientlylarge. mandlet:S2n+1ZmC!S2n+1=Zm Proof.


maptherstChernclassoftotherstChernclassof. 1.7.2 andProposition 1.9.1 23


1.8.1 wehave:2k1Yj=2N(j())[2k1


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:


1.9.3 ,wemusthave:dn2k1(m1)m2k2(k1):


In[ 26 ]V.V.Uspenskijintroducedthenotionofamapadmittinganapproximationbyk-dimensionalsimplicialmaps.Followinghimwesaythatamapf:X!Yadmitsapproximationbyk-dimensionalsimplicialmapsifforeverypairofopencovers!XofthespaceXand!YofthespaceYthereexistsacommutativediagramofthefollowingformXX!jKjf##pYY!jLj; InthatpaperV.V.Uspenskijproposedthefollowingquestionandconjecturedthatinthegeneralcasetheanswertoitis"no". (Q1)Doeseveryk-dimensionalmapf:X!Ybetweencompactaadmitapproximationbyk-dimensionalsimplicialmaps? 27


6 ]A.N.DranishnikovandV.V.Uspenskijprovedthatlightmapsadmitapproximationbynite-to-onesimplicialmaps.InthispaperwegivesomepartialresultsansweringthequestionofV.V.Uspenskijinarmative. 16 ].Werecallthatthediagonalproductoftwomapsf:X!Yandg:X!Zisamapf4g:X!YZdenedbyf4g(x)=(f(x);g(x)). (Q2)Letf:X!Ybeak-dimensionalmapbetweencompacta.Doesthereexistamapg:X!Iksuchthatdim(f4g)0? Inthispaperweprovethefollowingtheoremwhichstatesthatthequestions(Q1)and(Q2)areequivalent. 28


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 ].


6 ]itfollowsthattheextensionaldimensioncannotbeloweredby0-dimensionalmapssothecorollaryisanimmediateconsequenceoftheorem 2.1.7 OnecanunderstandthestatementofthepreviouscorollaryasageneralizationoftheclassicalHurewiczformula. 2.1.7 Proof. LetF=N0[SfNk:k1gbetheunionofallnitesequencesofpositiveintegersplusemptysequenceN0=fg.Foreveryi2Fletusdenotebyjijthelengthofthesequenceiandby(i;p)thesequenceobtainedbyaddingtoiapositiveintegerp. 30


(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.


(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


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;


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)0itissacienttonda-compactsubsetAinXofdimensionatmostk1suchthatdimfjXnA0.TheexistenceofsuchsubsetAfollowsimmediatelyfromLemmas 2.2.1 2.2.2 and 2.2.5 34


(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:


Recallthat'isanfO(y;t)g-map.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


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


AquestionaboutexistenceofopenmapsbetweencompactawithinniteberswhichdoesnotsatisfyBula'spropertyiswell-knownincontinuumtheoryandwasrststatedbyBula.Therstexampleofsuchamapwasgivenby:p:1Yi=0S2i!1Yi=0RP2i 4 ].ThefollowingtheoremisageneralizationofhisconstructionandheavilyreliesonTheorem 1.2.1 InterestingexamplesofopenmapsbetweencompactawithoutBula'spropertywereconstructedin[ 9 ].Recallthefollowingtheoremfromthatpaper: 38


Thereisenoughevidencetobelievethatsuchanexampleexists.Forinstance,themapsproducedbyTheorem 3.1.2 almostsatisfytherequirementsofbeingsuchexamples. (i)f:X!YdoesnotsatisfyBula'sproperty, (ii)forany':X![o;1]thereexistsy2Ysuchthat'(f1(y))=pt; (iv)Themapfcannotbeapproximatedby1-dimensionalsimplicialmaps. 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.Ourgoalnowwouldbetoconstructanexampleofastrictly1-dimensionalmapbetweencompactawhichwouldsatisfyatleastoneoftheproperties(i)(iv). TheexamplesofmapsproducedbyTheorem 3.1.2 ,althoughsupplyevidencethatthedesiredexampleexists,donothaveuniformdimensionalityofthebers,i.e.incasen=1thebersofthemapsproducedbyTheorem 3.1.2 arenotstrictly1-dimensional. 39


Itisawell-knownfactingeneraltopologythateachcompacticationofasucientlygoodtopologicalspacecanbedescribedasthesetofmaximalidealsofaBanachalgebraoffunctions.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


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


1.1.3 wecanchooseasequencefnigofoddnumberssuchthatafreeZpi-actiononSniwillhave1-Borsuk-Ulamproperty,butwillnothave2-Borsuk-Ulamproperty. (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] is0-dimensional.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


3.2.1 .Now,non-trivialityofeachberofthemapL^pfollowsfromProposition??. Proof. LetfOj2AgbeanopencoveringofLcS1suchthatforany2Aandanyx;x02Oitfollowsthatj'(x)'(x0)j< pkbethegeneratorofthegroupZpk.Wecanchoosek2Ntobelargeenough,sothatthepointsx2S1=^p1(y)ande2i pkx2S1=^p1(y)arefOj2Ag-closeforanychoiceofy2[CP1,i.e.thereexists2Asuchthatx;e2i pkx2O. Now,choosenlargeenough,sothatTheorem??guaranteesexistenceofanorbitofZpkwhichwillbecollapsedtoapointunderthefollowingcomposition:S2n1i,!LcS1'!I: Toproduceanexampleofanopenstrictly1-dimensionalmapitremainstoproveProposition 3.3.1 andthefactthatthebersofourmapareareallstrictly1-dimensional.Thelaterfactintuitivelyseemsobvious,sinceaberinthecoronaofourmapisbeingapproachedbycirclesinaLipschitzmanner.So,itseems,thatthebersinthecoronashouldalsobe1-dimensional,sincedimensioncannotberaisedbyaLipschitzmap.Oncewehaveprovedallthese,wehaveproducedthedesiredexamplewhichremainstohave 43


AfterapplyingScepin'sSpectralTheoremwehaveproducedanexampleofamapwhichgivesapositivesolutiontotheV.Uspenskij'sconjectureandanswersthequestionofB.A.Pasynkovinthenegative. 44


SetG=Zp:::Zpkforaxedoddprimepandk1andletM=S2n11:::S2nk1:InthissectionwewilldiscussafailedattempttoproveaBorsuk-UlamtheoremforG-actionsonproductsofspheresandmapsintoEuclideanspaces.Thiscaseismoredicultthanthecaseof(Zp)k-actions,becausetheEulerclassofthecorrespondingvectorbundleintheordinarycohomologytheoryturnsouttobetrivial.BelowwewillalsoshowthattheEulerclassofthevectorbundle,whichneedtobeconsidered,inthecomplexK-theoryistrivialaswell.Therefore,thisparticularcasecallsforamoresophisticatedcohomologytheoryinordertoextractaBorsuk-Ulamtheorem. LetLibea1-dimensionalcomplexZpi-module,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;


isnon-trivial.Obviously,C=;therefore,itissucienttoprovethate()6=0.Thisprobablycanbedonebymeansofsomeextraordinarycohomologytheory.LetusseewhathappensinthecaseofcomplexK-theory. First,recallthat~K0C(Ln1(p):::Lnk(pk))wZ[x1;:::;xk]=I;


fromabove.Theauthorisbeingabletoprovethatinfactthepolynomiale()almostalwaysbelongtotheidealIwhichmakethecomplexK-theoryapproachinecientwhentryingtoproveaBorsuk-Ulamtheoremforthesekindsofactions. 47


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IwasborninTroitsk,MoscowRegion,onSeptember4,1978.AftergraduationfromHighSchool5attheageof16,IenrolledintoaprograminmathematicsatPeople'sFriendshipUniversity.InacoupleofyearsIrealizedthatformetohavearealisticchanceofbecomingamathematicianIneedtogetintoabetterschool.Infall1997,IstartedattendingaresearchseminaronGeneralTopologyatMoscowStateUniversityorganizedbyBorisAlekseevichPasynkov.Insummer1998ProfessorPasynkovhelpedmetransfertotheMathematicsDepartmentofMoscowStateUniversity,whereIcontinuedstudyingtopologyunderhissupervision.IgraduatedfromMoscowStateUniversityinJune2002,withabachelorsdegreeinMathematics.InAugust2002IstartedinthePhDprogramattheUniversityofFloridawhichIcompletedinMay2007. 50