Characterization and the Topological Rigidity of Nobeling Manifolds

106 Pages·2013·1.472 MB·English

by  Andrzej Nagorko

MEMOIRS
of the
American Mathematical Society
Volume 223 • Number 1048 (second of 5 numbers) • May 2013
Characterization and Topological
Rigidity of No¨beling Manifolds
Andrzej Nago´rko
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Number 1048
Characterization and Topological
Rigidity of No¨beling Manifolds
Andrzej Nago´rko
May2013 • Volume223 • Number1048(secondof5numbers) • ISSN0065-9266
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Contents
Part 1. Introduction and preliminaries 1
Chapter 1. Introduction 3
Chapter 2. Preliminaries 7
2.1. Covers and interior covers 7
2.2. Absolute extensors 8
2.3. Nerves of covers and barycentric stars 10
2.4. Strong universality 12
2.5. n-Homotopy equivalence 15
2.6. Z-sets 20
Part 2. Reducing the proof of the main results to the construction
of n-regular and n-semiregular N -covers 23
n
Chapter 3. Approximation within an N -cover 25
n
3.1. Z(F)-sets 25
3.2. Approximation within a cover 26
3.3. Z-collections1 27
Chapter 4. Constructing closed N -covers 29
n
4.1. Adjustment of a collection 29
4.2. Limits of sequences of adjustments 30
4.3. Construction of a closed N -swelling 30
n
Chapter 5. Carrier and nerve theorems 33
5.1. Regular covers 33
5.2. Carrier theorem 33
5.3. Nerve theorem 35
Chapter 6. Anticanonical maps and semiregularity 37
6.1. A nerve theorem and the notion of semiregularity 37
6.2. A construction of regular covers 39
6.3. A construction of semiregular covers 42
Chapter 7. Extending homeomorphisms by the use of a “brick partitionings”
technique 45
Chapter 8. Proof of the main results 47
1weshallnotusethetheoremprovedinthssectionuntilthethirdpartofthepaper.
v
vi CONTENTS
Part 3. Constructing n-semiregular and n-regular N -covers 53
n
Chapter 9. Basic constructions in N -spaces 55
n
9.1. Adjustment to a Z-collection 55
9.2. Fitting closed N -neighborhoods 56
n
9.3. Patching of holes 58
Chapter 10. Core of a cover 59
10.1. The existence of an n-core 60
10.2. An n-core of a limit of a sequence of deformations 62
10.3. Proof of theorem 10.1 64
10.4. Retraction onto a core and a proof of theorem 6.17 69
Chapter 11. Proof of theorem 6.7 71
11.1. Patching of small holes 71
11.2. (cid:2)-Contractibility 75
11.3. Proof of theorem 6.7 for k =0 79
11.4. (cid:3)-Contractibility 80
11.5. Patching of large holes 82
11.6. Proof of theorem 6.7 for k >0 85
Bibliography 89
Index 91
Abstract
We develop a theory of N¨obeling manifolds similar to the theory of Hilbert
spacemanifolds. WeshowthatitreflectsthetheoryofMengermanifoldsdeveloped
by M. Bestvina [10] and is its counterpart in the realm of complete spaces. In
particular we prove the N¨obeling manifold characterization conjecture.
We define the n-dimensional universal No¨beling space νn to be the subset of
R2n+1 consisting of all points with at most n rational coordinates. To enable com-
parison with the infinite dimensional case we let ν∞ denote the Hilbert space. We
define an n-dimensional No¨beling manifold to be a Polish space locally homeomor-
phic to νn. The following theorem for n=∞ is the characterization theorem of H.
Torun´czyk [41]. We establish it for n < ∞, where it was known as the N¨obeling
manifold characterization conjecture.
Characterization theorem. An n-dimensional Polish ANE(n)-space is a
N¨obeling manifold if and only if it is strongly universal in dimension n.
The following theorem was proved by D. W. Henderson and R. Schori [27] for
n=∞. We establish it in the finite dimensional case.
Topological rigidity theorem. Two n-dimensional N¨obeling manifolds are
homeomorphic if and only if they are n-homotopy equivalent.
Wealsoestablish theopenembeddingtheorem, theZ-set unknottingtheorem,
the local Z-set unknotting theorem and the sum theorem for N¨obeling manifolds.
ReceivedbytheeditorJune8,2007,and,inrevisedform,January25,2011.
ArticleelectronicallypublishedonOctober9,2012;S0065-9266(2012)00643-5.
2010 MathematicsSubjectClassification. Primary55M10,54F45;Secondary54C20.
Key words and phrases. N¨obeling manifold characterization, No¨beling space characteriza-
tion,Z-setunknotting theorem,openembedding theorem,carrier,nerve theorem,regularcover,
semiregularcover.
Affiliation at time of publication: University of Warsaw, Department of Mathematics, ul.
Banacha2,02-097Warszawa;email: [email protected].
(cid:2)c2012 American Mathematical Society
vii

Part 1
Introduction and preliminaries