DOCSLIB.ORG

Appendix: a Proof of De Rham's Theorem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Appendix: A Proof of De Rham's Theorem §1. Introduction In Chapter I we mentioned that some of the motivating ideas for the develop­ ment of homology theory in the Nineteenth century arose in connection with such topics as Stokes's theorem, Green's theorem, Gauss's divergence theorem, and the Cauchy integral theorem. De Rham's theorem may be looked on as the modern culmination of this particular line of thought. It relates the homology and cohomology of a differentiable manifold to the exterior differential forms on the manifold. Exterior differential forms are objects which can serve as integrands of line integrals, surface integrals, etc., such as occur in the statement of the classical Green's theorem and Stokes's theorem. De Rham's theorem is of obvious importance, because it is a connecting link between analysis on manifolds and the topological properties of manifolds. In this appendix we will assume that the reader is familiar with the basic properties of differentiable manifolds, differential forms on manifolds, and the integration of differential forms over (differentiable) singular cubes. These topics are explained in many current textbooks, and there would be little point in our repeating such an exposition here. As examples of such texts, we list the following: M. Spivak [6], Flanders [3], Warner [9], and Whitney [10]. The first part of this chapter is devoted to using differentiable singular cubes to define the homology and cohomology groups of a differentiable manifold. We prove that in studying the homology and cohomology groups of such a manifold, it suffices to consider only differentiable singular cubes; the nondifferentiable ones can be ignored. 251 252 Appendix: A Proof of De Rham's Theorem Next, we introduce what may be called the De Rham cacha in complex of a differentiable manifold. This cochain complex consists of the exterior differential forms, with the exterior derivative serving as the coboundary operator. There is a natural homomorphism from this De Rham complex to the cochain complex (with coefficient group R, the real numbers) based on differentiable singular cubes. This homomorphism is defined on any exterior differential form of degree p by integrating that form over differ­ entiable singular p-cubes. The general form of Stokes's theorem is precisely the assertion that this natural homomorphism is a cochain map. De Rham's theorem asserts that this natural cochain map induces an isomorphism on cohomology. The proof we give of De Rham's theorem is modelled on Milnor's proof of the Poincare duality theorem in Chapter IX. The reader who has worked through that proof should have no trouble grasping the structure of our proof of De Rham's theorem. Curtis and Dugundji [11] have also given a 'proof of De Rham's theorem along somewhat similar lines. §2. Differentiable Singular Chains Let M be an n-dimensional differentiable manifold of class COO (we assume the reader is familiar with this concept). In order to define a differentiable singular cube, we must make use of the fact that the standard unit p-cube, IP = {(Xl" .. ,Xp) E RPI0:s; Xi:s; 1, i = 1,2, ... ,p} is a subset of Euclidean space RP. For p > 0, a singular p-cube T:JP -+ M will be called differentiable if there exists an open neighborhood U of IP in RP and an extension T': U -+ M of T such that T' is differentiable (of class COO). We complete this definition by defining any singular O-cube to be differentiable. Remark: If a singular p-cube T:IP -+ M is differentiable, there will, in general, be many different choices for the open neighborhood U and the extension T': U -+ M. We now introduce the following notation: Q~(M) = subgroup of Qp(M) generated by the differentiable singular p-cubes, D~(M) = Dp(M) n Q~(M), C~(M) = Q~(M)/D~(M). The superscript S in the above notation is intended to suggest the word "smooth." We will refer to C~(M) as the group of differentiable or smooth p-chains of M. Note that C~(M) = Q~(M) = Qo(M) = Co(M). §2. Differentiable Singular Chains 253 Next, observe that if T:1P ~ M is a differentiable singular p-cube, then the faces AiT and BiT, 1 ::; i::; p, are all obviously differentiable singular (p - i)-cubes. It follows that op(T) E Q~_1(M). Thus QS(M) = {Q~(M),op} is a subcomplex of Q(M), and CS(M) = {C~(M)} is a subcomplex of C(M). We will also introduce the following notation: for any abelian group G, CS(M;G) = CS(M) ® G, Ct(M;G) = Hom(CS(M),G), H~(M;G) = Hp(CS(M;G», H~(M;G) = HP(Ct(M;G». We can now state the main theorem of this section: Theorem 2.1. Let M be a differentiable manifold. The inclusion map of chain complexes, induces an isomorphism of homology groups, H~(M) ~ HiM). Corollary 2.2. For any abelian group G, we have the following isomorphisms of homology and cohomology groups: H~(M;G) ~ Hp(M;G), H~(M;G) ~ HP(M;G). The corollary follows from the theorem by use of standard techniques (cf. Theorem V.2.3). Before we can prove the theorem, it is necessary to discuss to what extent the methods and results of Chapters II and IlIon homology theory carryover to the homology groups H~(M; G) for any differentiable manifold M. We will now do this in a brief but systematic fashion. (a) Let M1 and M2 be differentiable manifolds, and let f:M 1 ~M2 be a differentiable maps of class Coo. If T: 1P ~ M 1 is a differentiable singular p cube, in M b then fT: 1P ~ M 2 is also differentiable. Hence we get an induced chain map f# :CS(M 1) ~ CS(M2 ) with all the usual properties. (b) Two differentiable maps fO'/1:M 1 ~ M 2 will be called differentiably homotopic if there exists a map f:1 x M1 ~ M2 such that fo(x) = f(O,x) and f1(X) = f(i,x) for any x E M 1, and in addition, there exists an open neighborhood U of 1 x M 1 in R x M 1 and a map f': U ~ M 2 which is an extension of f, and is differentiable of class Coo. The technique of §I1.4 can now be applied verbatim to prove that the induced chain maps fo#, 254 Appendix: A Proof of De Rham's Theorem 11 # : CS(M 1) -+ CS(M 2) are chain homotopic. This has all the usual conse­ quences; in particular, the induced homomorphisms on homology and cohomology groups are the same. (c) An open, convex subset if Rn is differentiably contractible to a point; in fact, the standard formulas for proving that such a subset is contractible are differentiable homotopies in the sense of the preceding definition. From this it follows that if U is an open, convex subset of Rn, then HS(U;G) = {G for p = 0, P ° for p =1= 0, with similar formulas for H~( U; G). (d) Let M be a differentiable manifold, and let A be a subspace of M which is a differentiable submanifold. For example, A could be an arbitrary open subset of M, or A could be a closed submanifold of M. Then we can consider CS(A) as a subcomplex of CS(M); hence we can consider the quotient complex CS(M)/CS(A) = CS(M,A) and we obtain exact homology and coho­ mology sequences for the pair (M,A) using differentiable singular cubes. (e) If T:JR -+ M is a differentiable singular cube, the subdivision of T, Sdn(T) as defined in §II.7, is readily seen to be a linear combination of differentiable singular cubes. Hence the subdivision operator defines a chain map just as in §II.7. Unfortunately, the chain homotopy CPn:Cn(M) -+ Cn+ 1(M) defined in§II.7 does not map C~(M) into C~+ l(M). This is because thefunction 111:[2 -+ [t,1] is not differentiable (the function 110:[2 -+ [is differentiable). However, it is not difficult to get around this obstacle. Consider the real­ valued function 11'1 defined by , ( ) _ 1 + Xl - XIX2 111 Xl,X2 - 2 . - X2 It is readily verified that 11'1 maps [2 into the interval [t,1], and that 111 and 11'1 are equal along the boundary of the square [2. bbviously, 11'1 is differentiable in a neighborhood of [2. Thus if we substitute 11~ for 111 in the formula for Ge(T) in §II.7, then GiT) will be a linear combination of differentiable singular cubes whenever T is a differentiable singular cube. Moreover, the operator Ge will continue to satisfy identities (f.1) to (fA) of §II.7. Thus we can define a chain homotopy IPn: C~(M) -+ C~+ l(M) using the modified definition of Ge . From this point on, everything proceeds exactly as in §II.7. The net result is that we can prove an analog of Theorem II.6.3 for singular homology based on differentiable singular cubes, and the excision property (Theorem II.6.2) holds for this kind of homology theory. §2. Differentiable Singular Chains 255 (f) Suppose that the differentiable manifold M is the union of two open subsets, M= Uu V. Then we can obtain an exact Mayer-Vietoris sequence for this situation by the method described in §III.5. (g) Finally, we note that an analog of Proposition III.6.1 must hold for homology groups based on differentiable singular cubes; this is practically obvious.
Recommended publications
  • Arxiv:0704.2891V3 [Math.AG] 5 Dec 2007 Schwartz Functions on Nash Manifolds

    Arxiv:0704.2891V3 [Math.AG] 5 Dec 2007 Schwartz Functions on Nash Manifolds

    Schwartz functions on Nash manifolds Avraham Aizenbud and Dmitry Gourevitch ∗ July 11, 2011 Abstract In this paper we extend the notions of Schwartz functions, tempered func- tions and generalized Schwartz functions to Nash (i.e. smooth semi-algebraic) manifolds. We reprove for this case classically known properties of Schwartz functions on Rn and build some additional tools which are important in rep- resentation theory. Contents 1 Introduction 2 1.1 Mainresults................................ 3 1.2 Schwartz sections of Nash bundles . 4 1.3 Restricted topologyand sheaf properties . .... 4 1.4 Possibleapplications ........................... 5 1.5 Summary ................................. 6 1.6 Remarks.................................. 6 2 Semi-algebraic geometry 8 2.1 Basicnotions ............................... 8 arXiv:0704.2891v3 [math.AG] 5 Dec 2007 2.2 Tarski-Seidenberg principle of quantifier elimination anditsapplications............................ 8 2.3 Additional preliminary results . .. 10 ∗Avraham Aizenbud and Dmitry Gourevitch, Faculty of Mathematics and Computer Science, The Weizmann Institute of Science POB 26, Rehovot 76100, ISRAEL. E-mails: [email protected], [email protected]. Keywords: Schwartz functions, tempered functions, generalized functions, distributions, Nash man- ifolds. 1 3 Nash manifolds 11 3.1 Nash submanifolds of Rn ......................... 11 3.2 Restricted topological spaces and sheaf theory over them........ 12 3.3 AbstractNashmanifolds . 14 3.3.1 ExamplesandRemarks. 14 3.4 Nashvectorbundles ........................... 15 3.5 Nashdifferentialoperators . 16 3.5.1 Algebraic differential operators on a Nash manifold . .... 17 3.6 Nashtubularneighborhood . 18 4 Schwartz and tempered functions on affine Nash manifolds 19 4.1 Schwartzfunctions ............................ 19 4.2 Temperedfunctions. .. .. .. 20 4.3 Extension by zero of Schwartz functions . .. 20 4.4 Partitionofunity ............................. 21 4.5 Restriction and sheaf property of tempered functions .
  • Categories of Sets with a Group Action

    Categories of Sets with a Group Action

    Categories of sets with a group action Bachelor Thesis of Joris Weimar under supervision of Professor S.J. Edixhoven Mathematisch Instituut, Universiteit Leiden Leiden, 13 June 2008 Contents 1 Introduction 1 1.1 Abstract . .1 1.2 Working method . .1 1.2.1 Notation . .1 2 Categories 3 2.1 Basics . .3 2.1.1 Functors . .4 2.1.2 Natural transformations . .5 2.2 Categorical constructions . .6 2.2.1 Products and coproducts . .6 2.2.2 Fibered products and fibered coproducts . .9 3 An equivalence of categories 13 3.1 G-sets . 13 3.2 Covering spaces . 15 3.2.1 The fundamental group . 15 3.2.2 Covering spaces and the homotopy lifting property . 16 3.2.3 Induced homomorphisms . 18 3.2.4 Classifying covering spaces through the fundamental group . 19 3.3 The equivalence . 24 3.3.1 The functors . 25 4 Applications and examples 31 4.1 Automorphisms and recovering the fundamental group . 31 4.2 The Seifert-van Kampen theorem . 32 4.2.1 The categories C1, C2, and πP -Set ................... 33 4.2.2 The functors . 34 4.2.3 Example . 36 Bibliography 38 Index 40 iii 1 Introduction 1.1 Abstract In the 40s, Mac Lane and Eilenberg introduced categories. Although by some referred to as abstract nonsense, the idea of categories allows one to talk about mathematical objects and their relationions in a general setting. Its origins lie in the field of algebraic topology, one of the topics that will be explored in this thesis. First, a concise introduction to categories will be given.
  • Homology Homomorphism

    Homology Homomorphism

    || Computations Induced (Co)homology homomorphism Daher Al Baydli Department of Mathematics National University of Ireland,Galway Feberuary, 2017 Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 1 / 11 Overview Resolution Chain and cochain complex homology cohomology Induce (Co)homology homomorphism Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 2 / 11 Resolution Definition Let G be a group and Z be the group of integers considered as a trivial ZG-module. The map : ZG −! Z from the integral group ring to Z, given by Σmg g 7−! Σmg , is called the augmentation. Definition Let G be a group. A free ZG-resolution of Z is an exact sequence of free G G @n+1 G @n G @n−1 @2 G @1 ZG-modules R∗ : · · · −! Rn+1 −! Rn −! Rn−1 −! · · · −! R1 −! G G R0 −! R−1 = Z −! 0 G with each Ri a free ZG-module for all i ≥ 0. Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 3 / 11 Definition Let C = (Cn;@n)n2Z be a chain complex of Z-modules. For each n 2 Z, the nth homology module of C is defined to be the quotient module Ker@n Hn(C) = Im@n+1 G We can also construct an induced cochain complex HomZG (R∗ ; A) of abelian groups: G G δn G δn−1 HomZG (R∗ ; A): · · · − HomZG (Rn+1; A) − HomZG (Rn ; A) − G δn−2 δ1 G δ0 G HomZG (Rn−1; A) − · · · − HomZG (R1 ; A) − HomZG (R0 ; A) Chain and cochain complex G Given a ZG-resolution R∗ of Z and any ZG-module A we can use the G tensor product to construct an induced chain complex R∗ ⊗ZG A of G G @n+1 G @n abelian groups: R∗ ⊗ZG A : · · · −! Rn+1 ⊗ZG A −! Rn ⊗ZG A −! G @n−1 @2 G @1 G Rn−1 ⊗ZG A −! · · · −! R1 ⊗ZG A −! R0 ⊗ZG A Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 4 / 11 Definition Let C = (Cn;@n)n2Z be a chain complex of Z-modules.
  • University of Alberta

    University of Alberta

    University of Alberta Extensions of Skorohod’s almost sure representation theorem by Nancy Hernandez Ceron A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Department of Mathematical and Statistical Sciences ©Nancy Hernandez Ceron Fall 2010 Edmonton, Alberta Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission. Library and Archives Bibliothèque et Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l’édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre référence ISBN:978-0-494-62500-2 Our file Notre référence ISBN: 978-0-494-62500-2 NOTICE: AVIS: The author has granted a non- L’auteur a accordé une licence non exclusive exclusive license allowing Library and permettant à la Bibliothèque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l’Internet, prêter, telecommunication or on the Internet, distribuer et vendre des thèses partout dans le loan, distribute and sell theses monde, à des fins commerciales ou autres, sur worldwide, for commercial or non- support microforme, papier, électronique et/ou commercial purposes, in microform, autres formats.
  • Homology Groups of Homeomorphic Topological Spaces

    Homology Groups of Homeomorphic Topological Spaces

    An Introduction to Homology Prerna Nadathur August 16, 2007 Abstract This paper explores the basic ideas of simplicial structures that lead to simplicial homology theory, and introduces singular homology in order to demonstrate the equivalence of homology groups of homeomorphic topological spaces. It concludes with a proof of the equivalence of simplicial and singular homology groups. Contents 1 Simplices and Simplicial Complexes 1 2 Homology Groups 2 3 Singular Homology 8 4 Chain Complexes, Exact Sequences, and Relative Homology Groups 9 ∆ 5 The Equivalence of H n and Hn 13 1 Simplices and Simplicial Complexes Definition 1.1. The n-simplex, ∆n, is the simplest geometric figure determined by a collection of n n + 1 points in Euclidean space R . Geometrically, it can be thought of as the complete graph on (n + 1) vertices, which is solid in n dimensions. Figure 1: Some simplices Extrapolating from Figure 1, we see that the 3-simplex is a tetrahedron. Note: The n-simplex is topologically equivalent to Dn, the n-ball. Definition 1.2. An n-face of a simplex is a subset of the set of vertices of the simplex with order n + 1. The faces of an n-simplex with dimension less than n are called its proper faces. 1 Two simplices are said to be properly situated if their intersection is either empty or a face of both simplices (i.e., a simplex itself). By \gluing" (identifying) simplices along entire faces, we get what are known as simplicial complexes. More formally: Definition 1.3. A simplicial complex K is a finite set of simplices satisfying the following condi- tions: 1 For all simplices A 2 K with α a face of A, we have α 2 K.
  • Algebraic Topology

    Algebraic Topology

    Algebraic Topology Len Evens Rob Thompson Northwestern University City University of New York Contents Chapter 1. Introduction 5 1. Introduction 5 2. Point Set Topology, Brief Review 7 Chapter 2. Homotopy and the Fundamental Group 11 1. Homotopy 11 2. The Fundamental Group 12 3. Homotopy Equivalence 18 4. Categories and Functors 20 5. The fundamental group of S1 22 6. Some Applications 25 Chapter 3. Quotient Spaces and Covering Spaces 33 1. The Quotient Topology 33 2. Covering Spaces 40 3. Action of the Fundamental Group on Covering Spaces 44 4. Existence of Coverings and the Covering Group 48 5. Covering Groups 56 Chapter 4. Group Theory and the Seifert{Van Kampen Theorem 59 1. Some Group Theory 59 2. The Seifert{Van Kampen Theorem 66 Chapter 5. Manifolds and Surfaces 73 1. Manifolds and Surfaces 73 2. Outline of the Proof of the Classification Theorem 80 3. Some Remarks about Higher Dimensional Manifolds 83 4. An Introduction to Knot Theory 84 Chapter 6. Singular Homology 91 1. Homology, Introduction 91 2. Singular Homology 94 3. Properties of Singular Homology 100 4. The Exact Homology Sequence{ the Jill Clayburgh Lemma 109 5. Excision and Applications 116 6. Proof of the Excision Axiom 120 3 4 CONTENTS 7. Relation between π1 and H1 126 8. The Mayer-Vietoris Sequence 128 9. Some Important Applications 131 Chapter 7. Simplicial Complexes 137 1. Simplicial Complexes 137 2. Abstract Simplicial Complexes 141 3. Homology of Simplicial Complexes 143 4. The Relation of Simplicial to Singular Homology 147 5. Some Algebra. The Tensor Product 152 6.
  • A Radial Basis Function Partition of Unity Method for Transport on the Sphere

    A Radial Basis Function Partition of Unity Method for Transport on the Sphere

    A RADIAL BASIS FUNCTION PARTITION OF UNITY METHOD FOR TRANSPORT ON THE SPHERE by Kevin Aiton A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Boise State University May 2014 c 2014 Kevin Aiton ALL RIGHTS RESERVED BOISE STATE UNIVERSITY GRADUATE COLLEGE DEFENSE COMMITTEE AND FINAL READING APPROVALS of the thesis submitted by Kevin Aiton Thesis Title: A Radial Basis Function Partition of Unity Method for Transport on the Sphere Date of Final Oral Examination: 06 December 2013 The following individuals read and discussed the thesis submitted by student Kevin Aiton, and they evaluated his presentation and response to questions during the final oral examination. They found that the student passed the final oral examination. Grady Wright, Ph.D. Chair, Supervisory Committee Donna Calhoun, Ph.D. Member, Supervisory Committee Inanc Senocak, Ph.D. Member, Supervisory Committee The final reading approval of the thesis was granted by Grady Wright, Ph.D., Chair of the Supervisory Committee. The thesis was approved for the Graduate College by John R. Pelton, Ph.D., Dean of the Graduate College. ACKNOWLEDGMENTS This work was supported, in part, by the National Science Foundation (NSF) under grant DMS-0934581 and also by the Boise State Mathematics Department under the 2013 summer graduate fellowship. I would like to express gratitude to Professor Grady Wright. He has been both kind and patient. I would also like to thank the MOOSE development team at Idaho National Laboratory for letting me work remotely so I could work on my thesis. Finally, I would like to give gratitude to my family.
  • Chapter 9 Partitions of Unity, Covering Maps ~

    Chapter 9 Partitions of Unity, Covering Maps ~

    Chapter 9 Partitions of Unity, Covering Maps ~ 9.1 Partitions of Unity To study manifolds, it is often necessary to construct var- ious objects such as functions, vector fields, Riemannian metrics, volume forms, etc., by gluing together items con- structed on the domains of charts. Partitions of unity are a crucial technical tool in this glu- ing process. 505 506 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ The first step is to define “bump functions”(alsocalled plateau functions). For any r>0, we denote by B(r) the open ball n 2 2 B(r)= (x1,...,xn) R x + + x <r , { 2 | 1 ··· n } n 2 2 and by B(r)= (x1,...,xn) R x1 + + xn r , its closure. { 2 | ··· } Given a topological space, X,foranyfunction, f : X R,thesupport of f,denotedsuppf,isthe closed set! supp f = x X f(x) =0 . { 2 | 6 } 9.1. PARTITIONS OF UNITY 507 Proposition 9.1. There is a smooth function, b: Rn R, so that ! 1 if x B(1) b(x)= 2 0 if x Rn B(2). ⇢ 2 − See Figures 9.1 and 9.2. 1 0.8 0.6 0.4 0.2 K3 K2 K1 0 1 2 3 Figure 9.1: The graph of b: R R used in Proposition 9.1. ! 508 CHAPTER 9. PARTITIONS OF UNITY, COVERING MAPS ~ > Figure 9.2: The graph of b: R2 R used in Proposition 9.1. ! Proposition 9.1 yields the following useful technical result: 9.1. PARTITIONS OF UNITY 509 Proposition 9.2. Let M be a smooth manifold.
  • Manifolds with Non-Stable Fundamental Groups at Infinity, II

    Manifolds with Non-Stable Fundamental Groups at Infinity, II

    ISSN 1364-0380 (on line) 1465-3060 (printed) 255 Geometry & Topology G T GG T T T Volume 7 (2003) 255–286 G T G T G T G T Published: 31 March 2003 T G T G T G T G T G G G G T T Manifolds with non-stable fundamental groups at infinity, II C R Guilbault FC Tinsley Department of Mathematical Sciences, University of Wisconsin-Milwaukee Milwaukee, Wisconsin 53201, USA and Department of Mathematics, The Colorado College Colorado Springs, Colorado 80903, USA Email: [email protected], [email protected] Abstract In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann’s famous collaring theorem that may be applied to manifolds having non-stable fundamental group systems at infinity. In this paper we show that, for mani- folds with compact boundary, the condition of inward tameness has substatial implications for the algebraic topology at infinity. In particular, every inward tame manifold with compact boundary has stable homology (in all dimensions) and semistable fundamental group at each of its ends. In contrast, we also con- struct examples of this sort which fail to have perfectly semistable fundamental group at infinity. In doing so, we exhibit the first known examples of open manifolds that are inward tame and have vanishing Wall finiteness obstruction at infinity, but are not pseudo-collarable. AMS Classification numbers Primary: 57N15, 57Q12 Secondary: 57R65, 57Q10 Keywords End, tame, inward tame, open collar, pseudo-collar, semistable, Mittag-Leffler, perfect group, perfectly semistable, Z-compactification Proposed:SteveFerry Received:6September2002 Seconded: BensonFarb,RobionKirby Accepted: 12March2003 c Geometry & Topology Publications 256 C R Guilbault and F C Tinsley 1 Introduction In [7] we presented a program for generalizing Siebenmann’s famous collaring theorem (see [15]) to include open manifolds with non-stable fundamental group systems at infinity.
  • Fibrations of Classifying Spaces (1) 577

    Fibrations of Classifying Spaces (1) 577

    transactions of the american mathematical society Volume 343, Number 1, May 1994 FIBRATIONS OF CLASSIFYINGSPACES KENSHIISHIGURO AND DIETRICH NOTBOHM Abstract. We investigate fibrations of the form Z —»Y —►X, where two of the three spaces are classifying spaces of compact connected Lie groups. We obtain certain finiteness conditions on the third space which make it also a classifying space. Our results allow to express some of the basic notions in group theory in terms of homotopy theory, i.e., in terms of classifying spaces. As an application we prove that every retract of the classifying space of a compact connected Lie group is again a classifying space. 1. Introduction In group theory one has the notions of a subgroup, i.e., a monomorphism, of a normal subgroup, of a kernel, and of an epimorphism. Following the idea of Rector [24] and Rector and Stasheff [26] to describe group theory or Lie group theory from the homotopy point of view, one may look for definitions of these notions in terms of classifying spaces. For a monomorphism, results of Quillen [23], Dwyer and Wilkerson [6], and the second author [20] give a strong hint to say that a map / : BH -> BG between the classifying spaces of compact Lie groups is a monomorphism if H*(BH; Fp) is a finitely generated module over the ring H*(BG; Fp) for any prime p. A similiar definition is already given in [24]. The notion of a kernel is discussed in [20]. In order to find homotopy theoretical definitions for a normal subgroup and an epimorphism, we are led to study fibrations of classifying spaces.
  • On Phi-Families

    On Phi-Families

    ON PHI-FAMILIES CHARLES E. WATTS 1. Introduction. The purpose of this note is to show that the notion of sections with support in a phi-family in the Cartan version of the Leray theory of sheaves can be avoided by the following expedient. One uses the phi-family to construct a new space in the manner of a one-point compactification. Then a sheaf on the original space is shown to yield a new sheaf on the new space whose cohomology (with unrestricted supports) is that of the original sheaf with restricted supports. A partial generalization to the Grothendieck cohomology theory is given. 2. Phi-families. A family 5 of subsets of a topological space X is a family of supports [l ] provided: (I) each member of 5 is closed; (II) if FE$, then each closed subset of F is E$; (III) if Fi, F2E5, then Fi\JF2E5- Given a family of supports 5, we choose an object co (^X, and define X' = \}$, X*=X'VJ{ oo }. We then topologize X* by saying that a subset U oi X* is open iff either U is open in X' or else X*— UE%- It is readily verified that these open sets in fact form a topology for X* and that the inclusion X'EX* is a topological imbedding. The family of supports JF is a phi-family [3] provided also: (IV) each member of fJ has a closed neighborhood in J; (V) each member of J is paracompact. Proposition 1. If ff is a phi-family, then X* is paracompact.
  • Uniformly Continuous Partitions of Unity on a Metric Space

    Uniformly Continuous Partitions of Unity on a Metric Space

    Canad. Math. Bull. Vol. 26 (1), 1983 UNIFORMLY CONTINUOUS PARTITIONS OF UNITY ON A METRIC SPACE BY STEWART M. ROBINSON AND ZACHARY ROBINSON ABSTRACT. In this paper, we construct, for any open cover of a metric space, a partition of unity consisting of a family of uniformly continuous functions. It is a well-known characterization of paracompact spaces that every open covering admits a partition of unity subordinate to it. In a series of lectures on "Non-Linear Analysis on Banach Spaces" given at both Dalhousie University and Cleveland State University, K. Sundaresan posed the following question: If (X, d) is a metric space, may one select this partition of unity to consist of uniformly continuous functions? In this note we provide an affirmative answer to this question. THEOREM. Let X be a paracompact space in which each open set is the support of a Lipschitz continuous function. Then there is a Lipschitz continuous partition of unity subordinate to any open cover of X. Proof. Let X be a space which satisfies the hypotheses and let °U be an open cover of X. Since X is paracompact, °U admits an open refinement that is both o--discrete and locally finite; denote this refinement by Unez+ °ttn> where each %n is a discrete family of open sets. Let Sn = UAG^^- We may select a gn : X —» [0,1] to be a Lipschitz continuous function with support Sn. For / e Z+, define k, : [0,1]-> [0,1] by fc(jc)f/-x if x e [0,1//] J 11 otherwise, and define gnj : X —» [0, 1] by g„,y(x) = ^.(gn(x)).