DOCSLIB.ORG

HUREWICZ HOMOMORPHISMS by ANH-CHI LE B.Sc.,University Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
HUREWICZ HOMOMORPHISMS by ANH-CHI LE B.Sc.,University Of HUREWICZ HOMOMORPHISMS BY ANH-CHI LE B.Sc.,University of British Columbia,1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of MATHEMATICS We accept this as conforming to the required standard, THE UNIVERSITY OF BRITISH COLUMBIA May,197^ In presenting this thesis in partial fulfilment of the requiremenrs for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Depa rtment The University of British Columbia Vancouver 8, Canada ii Supervisor s Dr Roy Douglas ABSTRACT ; Theorem : Let X be simply connected . Hq(X) be finitely generated for each q JCq(X) be finite for each q ^. n . n > 1 Then , hq ,7Tq(X) > Hq(X) has finite kernel for q 4 2n has finite cokernel for q < 2n+l ker h2n+1 8 Q = ker KJ where , u is the cup product or n+1 the square free cup product on R (X) depending on whether n+1 is even or odd , respectively . ( RN+1(X) is a quotient group of H2+1(X)® HQ+1(X) to be defined in this thesis ) TABLE OF CONTENTS PAGE I Introduction 1 II The Meta-theorem 8 III The kernels and cokernels of Hurewicz homomorphisms 10 Bibliography Zk iv ACKNOWLEDGEMENTS The author thanks his supervisor , Dr Roy Rene Douglas , without whom the realization of this thesis would have been impossible , The author is also very happy to thank the University of British Columbia for their financial support (1972-197^) and Dr Rene' Held for his many helpful suggestions . 1 Chapter I INTRODUCTION The Mod ^Hurewicz Theorem provides a vital link between homotopy groups and homology groups . In particular , its importance lies in the fact that homology groups are more readily computable . The well-known theorem asserts that if X is simply connected and 7Tq(X) e^.a perfect and weakly complete class (Serre) of abelian groups , for q i- n ;then , fi^+1(X) is ^-isomorphic to Hn+1(X) by the Hurewicz n a homomorphism hn+^ , and at one level higher , n+2 ^-epimorphism . This thesis extends the theorem to higher dimensions for the case \f = ft , the Serre class of finite abelian groups . We establish that hq , q 4 2n , are £ isomorphisms and h2n+l is. epimorphism . We also compute kernel h2n+l (Mod ft ) . Counter-examples are provided to conclude' that further extensions' are not possible , in general . 2 l.Qn fi brat ions and the process of killing homotopy groups. (1.1)The definition of a fibration.- Let p :E ^ B be a map between topological spaces. Then , the map p is called a fibration (in the sense of Serre) if given any finite CW complex K,a map g :K > E and a map F :KxI B such that F(x,0)=p(g(x)) for all x in K ; then there exists a map G:KxI—E such that G(x,0)=g(x) and pG(x,t)=F(x,t) Let * be the base point of B. Then,F=p (*) is called the fibre of the fibration p We have a Hurewicz fibration if the above description holds when K is an arbitrary topological space. In this thesis,we shall use the term "fibration" without specifying "Serre" or "Hurewicz" when the two are both admissable. (1.2)Remark.- Any map A =»B can be identified up to homotopy with a fibration (See [s]p.99 ) E p > B such that the following diagram is commutative : (homotopy equivalences) By the(artificial)fibre of f , we mean the fibre of p . 3 (1.3)Annihilating homotopy groups .- Let X be a CW-complex and denote its n-skeleton by Xn For the following construction , let us assume that X is a connected CW-complex with X°= pt . Let L*^ifei be a set of generators of the homotopy group 7^+1(X) . By cellular approximation , we may assume that : n+1 n+1 *. :s » x l We now attach an n+2 -cell corresponding to each<^ =inclusion X C > Cc< Clearly , 71^(X)S K±(C* ) for (KUn and •^n+1(Cc<)=0 In order to "kill off" all the homotopy groups -^(X) for q>n , we iterate this process until we arrive at a space X(l,...,n) . where : the inclusion X ^ > X(l,...,n) induces; K_(X)=/T(X(1,...,n))for q*n and , moreover , 7T (X(l,...,n))=0 for q>n Now , the Cartan-Serre -Whitehead construction is to take the fibre (See (1.2)) of X C »X(l,...,n) and , we obtain : X(n+1,.. ,-) > X > X(l,..,,n) where TC (X(n+1,... ,°»)£0 if qfen = 7T(X) if q^n+1 Taking the fibre X(n+1,...,«) and killing off its homotopy groups in dimensions greater than m gives a space denote by X(n+l,...,m) Clearly , TT (X(n,...,m))=0 if q^n-1 Stt_(X) if n*q*=m =0 if q>m Observation.- Given any map Y: X => X(i,...,m) «* B we can pull back through the path space PB of B , and have the fibration : SIB ^ W=X(m+l,.. ,co) P X 2.The Serre - Leray Spectral Sequence of a fibration . (2.1) A-spectral sequence is a sequence of abelian groups En together with a sequence of differential endomorphisms : dn 5 En >E n dndn=° » ^ E =H(E ,d ker d im d such that n+l n n^ * n / n (2.2) We shall describe the Serre-Leray spectral sequence which arises from a Serre fibration 0 = ( F * X P >B ) We assume B to be a simply - connected CW -complex with finite skeleta and X of the homotopy type of a finite CW-complex Remark.- Simply - connectednesss of the base space B is imposed in order-to avoid the discussion of "local coeffici -ents" in this general setting . The total space is filtered by the inverse p-images of the n-skeleton Bn of the base space B . 1 n Set Xn=p" (B ) Let H* stand for the ordinary "cellular" cohomology theory . We are now going to define the Serre - Leray Spectral Sequence (Er($£)tdr) of j£ in terms of the following exact couple : p+q DP,q= H (xp) The exactnesss axiom of H : h +( f ^P^X^X^) l , HP ^(Xp) 1 > HP^Xp^) +( +1 >HP i (xp,xp_1) , . enables us to derive (inductively) the following exact couple : r ' / r "r where , Dr+1=im fr , fr = r h E =H(E d ) dr=Sr r » r+l r! r deg(dr)=(r,l-r) Proposition.- a) Ep'q =Hp(BjHq(F)) P q p q p+q E ' = ker (H + (X) >H (X ^ ) _ Gr Hp+q (X) ker (Hp+q(X) > Hp+q(X„) ) b) H = H ( ;G) is a multiplicative cohomology theory if G is a commutative ring with unit . Therefore , the spectral sequence inherits a canonical multiplicative structure .With respect to this multiplication , the differentials dr turn out to be derivations , i.e. p+q dr(ab) = dr(a)b +(-l) adr(b) where a eEp'q r c)The edge homomorphisms f , p can be computed as follows Hn(B) -> Hn(X) ^ l£'0<$) Hn (X) JL__^ Hn ( F) (2.3)Pictorially , the Eg term of the spectral sequence can be represented as follows : H (F) • r-l 2 H (B) 7 The following proposition can be shown using the Serre -Leray spectral sequence Proposition .- Let F •—£-» X —>B be a fibration B be m-connected , F n-connected Then , we have Serre homology and cohomology exact sequences of finite lengths : H P H B F m+n+l< '-^— Vn+l^ m+n+1< '—^ W > > . ... „ Hm+n(F) , Hm+n+1(B)_P!^Hm+n+1(X)_£l Hm+n+1(F) Chapter II THE META-THEOREM Let us define a homomorphism X) (n X) hn(X) : 7Tn(X) >V - relating homotopy and homology groups of a space X . This homomorphism is given by : hn(X)([fl)=hn([fl>)=f^(gn) 11 for any homotopy class [f] € JT^CS ) where gnis a generator n of Hn(S ;Z) hn is natural and is called "Hurewicz*homomorphism" . The following is a crucial observation as far as studying the Hurewicz homomorphisms is concerned . We shall call it "THE META-THEOREM" 9 Let P(k,n) stand for : "h^^has finite kernel and cokernel" . This notation is used to formulate the theorem (3.1) The Meta-Theorem.- Let k £ 1 , n >, max(k-l,l) . If P(k,n) is true for any n-connected space , then , P(k,n+1) is true for any n+1 -connected space . PROOF : Suppose Y is n+1 -connected . We get , from the Serre exact sequence of the path fibration ftY > PY > Y , the following commutative diagram , the rows of which being exact s {PY) » \+kUY) rrn+k(PY) ren+k + n+k+ h n+k+1 H {PY) H. -> H. (ay) PY ^ n+k+ n+k+ n+k * W >: 0' ^ o Since n+k+1 4 n+l+n+1 = 2n+2 , the above commutative diagram is always true given the hypothesis of the theorem. In particular , by the square [L].hn+k(£Y) is equivalent to hn+jc+^(Y) .
Load more

Recommended publications
  • Part III 3-Manifolds Lecture Notes C Sarah Rasmussen, 2019
    Part III 3-manifolds Lecture Notes c Sarah Rasmussen, 2019 Contents Lecture 0 (not lectured): Preliminaries2 Lecture 1: Why not ≥ 5?9 Lecture 2: Why 3-manifolds? + Intro to Knots and Embeddings 13 Lecture 3: Link Diagrams and Alexander Polynomial Skein Relations 17 Lecture 4: Handle Decompositions from Morse critical points 20 Lecture 5: Handles as Cells; Handle-bodies and Heegard splittings 24 Lecture 6: Handle-bodies and Heegaard Diagrams 28 Lecture 7: Fundamental group presentations from handles and Heegaard Diagrams 36 Lecture 8: Alexander Polynomials from Fundamental Groups 39 Lecture 9: Fox Calculus 43 Lecture 10: Dehn presentations and Kauffman states 48 Lecture 11: Mapping tori and Mapping Class Groups 54 Lecture 12: Nielsen-Thurston Classification for Mapping class groups 58 Lecture 13: Dehn filling 61 Lecture 14: Dehn Surgery 64 Lecture 15: 3-manifolds from Dehn Surgery 68 Lecture 16: Seifert fibered spaces 69 Lecture 17: Hyperbolic 3-manifolds and Mostow Rigidity 70 Lecture 18: Dehn's Lemma and Essential/Incompressible Surfaces 71 Lecture 19: Sphere Decompositions and Knot Connected Sum 74 Lecture 20: JSJ Decomposition, Geometrization, Splice Maps, and Satellites 78 Lecture 21: Turaev torsion and Alexander polynomial of unions 81 Lecture 22: Foliations 84 Lecture 23: The Thurston Norm 88 Lecture 24: Taut foliations on Seifert fibered spaces 89 References 92 1 2 Lecture 0 (not lectured): Preliminaries 0. Notation and conventions. Notation. @X { (the manifold given by) the boundary of X, for X a manifold with boundary. th @iX { the i connected component of @X. ν(X) { a tubular (or collared) neighborhood of X in Y , for an embedding X ⊂ Y .
    [Show full text]
  • Math 231Br: Algebraic Topology
    algebraic topology Lectures delivered by Michael Hopkins Notes by Eva Belmont and Akhil Mathew Spring 2011, Harvard fLast updated August 14, 2012g Contents Lecture 1 January 24, 2010 x1 Introduction 6 x2 Homotopy groups. 6 Lecture 2 1/26 x1 Introduction 8 x2 Relative homotopy groups 9 x3 Relative homotopy groups as absolute homotopy groups 10 Lecture 3 1/28 x1 Fibrations 12 x2 Long exact sequence 13 x3 Replacing maps 14 Lecture 4 1/31 x1 Motivation 15 x2 Exact couples 16 x3 Important examples 17 x4 Spectral sequences 19 1 Lecture 5 February 2, 2010 x1 Serre spectral sequence, special case 21 Lecture 6 2/4 x1 More structure in the spectral sequence 23 x2 The cohomology ring of ΩSn+1 24 x3 Complex projective space 25 Lecture 7 January 7, 2010 x1 Application I: Long exact sequence in H∗ through a range for a fibration 27 x2 Application II: Hurewicz Theorem 28 Lecture 8 2/9 x1 The relative Hurewicz theorem 30 x2 Moore and Eilenberg-Maclane spaces 31 x3 Postnikov towers 33 Lecture 9 February 11, 2010 x1 Eilenberg-Maclane Spaces 34 Lecture 10 2/14 x1 Local systems 39 x2 Homology in local systems 41 Lecture 11 February 16, 2010 x1 Applications of the Serre spectral sequence 45 Lecture 12 2/18 x1 Serre classes 50 Lecture 13 February 23, 2011 Lecture 14 2/25 Lecture 15 February 28, 2011 Lecture 16 3/2/2011 Lecture 17 March 4, 2011 Lecture 18 3/7 x1 Localization 74 x2 The homotopy category 75 x3 Morphisms between model categories 77 Lecture 19 March 11, 2011 x1 Model Category on Simplicial Sets 79 Lecture 20 3/11 x1 The Yoneda embedding 81 x2 82
    [Show full text]
  • INTRODUCTION to ALGEBRAIC TOPOLOGY 1 Category And
    INTRODUCTION TO ALGEBRAIC TOPOLOGY (UPDATED June 2, 2020) SI LI AND YU QIU CONTENTS 1 Category and Functor 2 Fundamental Groupoid 3 Covering and fibration 4 Classification of covering 5 Limit and colimit 6 Seifert-van Kampen Theorem 7 A Convenient category of spaces 8 Group object and Loop space 9 Fiber homotopy and homotopy fiber 10 Exact Puppe sequence 11 Cofibration 12 CW complex 13 Whitehead Theorem and CW Approximation 14 Eilenberg-MacLane Space 15 Singular Homology 16 Exact homology sequence 17 Barycentric Subdivision and Excision 18 Cellular homology 19 Cohomology and Universal Coefficient Theorem 20 Hurewicz Theorem 21 Spectral sequence 22 Eilenberg-Zilber Theorem and Kunneth¨ formula 23 Cup and Cap product 24 Poincare´ duality 25 Lefschetz Fixed Point Theorem 1 1 CATEGORY AND FUNCTOR 1 CATEGORY AND FUNCTOR Category In category theory, we will encounter many presentations in terms of diagrams. Roughly speaking, a diagram is a collection of ‘objects’ denoted by A, B, C, X, Y, ··· , and ‘arrows‘ between them denoted by f , g, ··· , as in the examples f f1 A / B X / Y g g1 f2 h g2 C Z / W We will always have an operation ◦ to compose arrows. The diagram is called commutative if all the composite paths between two objects ultimately compose to give the same arrow. For the above examples, they are commutative if h = g ◦ f f2 ◦ f1 = g2 ◦ g1. Definition 1.1. A category C consists of 1◦. A class of objects: Obj(C) (a category is called small if its objects form a set). We will write both A 2 Obj(C) and A 2 C for an object A in C.
    [Show full text]
  • Lecture Notes on Homotopy Theory and Applications
    LAURENTIUMAXIM UNIVERSITYOFWISCONSIN-MADISON LECTURENOTES ONHOMOTOPYTHEORY ANDAPPLICATIONS i Contents 1 Basics of Homotopy Theory 1 1.1 Homotopy Groups 1 1.2 Relative Homotopy Groups 7 1.3 Homotopy Extension Property 10 1.4 Cellular Approximation 11 1.5 Excision for homotopy groups. The Suspension Theorem 13 1.6 Homotopy Groups of Spheres 13 1.7 Whitehead’s Theorem 16 1.8 CW approximation 20 1.9 Eilenberg-MacLane spaces 25 1.10 Hurewicz Theorem 28 1.11 Fibrations. Fiber bundles 29 1.12 More examples of fiber bundles 34 1.13 Turning maps into fibration 38 1.14 Exercises 39 2 Spectral Sequences. Applications 41 2.1 Homological spectral sequences. Definitions 41 2.2 Immediate Applications: Hurewicz Theorem Redux 44 2.3 Leray-Serre Spectral Sequence 46 2.4 Hurewicz Theorem, continued 50 2.5 Gysin and Wang sequences 52 ii 2.6 Suspension Theorem for Homotopy Groups of Spheres 54 2.7 Cohomology Spectral Sequences 57 2.8 Elementary computations 59 n 2.9 Computation of pn+1(S ) 63 3 2.10 Whitehead tower approximation and p5(S ) 66 Whitehead tower 66 3 3 Calculation of p4(S ) and p5(S ) 67 2.11 Serre’s theorem on finiteness of homotopy groups of spheres 70 2.12 Computing cohomology rings via spectral sequences 74 2.13 Exercises 76 3 Fiber bundles. Classifying spaces. Applications 79 3.1 Fiber bundles 79 3.2 Principal Bundles 86 3.3 Classification of principal G-bundles 92 3.4 Exercises 97 4 Vector Bundles. Characteristic classes. Cobordism. Applications. 99 4.1 Chern classes of complex vector bundles 99 4.2 Stiefel-Whitney classes of real vector bundles 102 4.3 Stiefel-Whitney classes of manifolds and applications 103 The embedding problem 103 Boundary Problem.
    [Show full text]
  • Spectra and Stable Homotopy Theory
    Spectra and stable homotopy theory Lectures delivered by Michael Hopkins Notes by Akhil Mathew Fall 2012, Harvard Contents Lecture 1 9/5 x1 Administrative announcements 5 x2 Introduction 5 x3 The EHP sequence 7 Lecture 2 9/7 x1 Suspension and loops 9 x2 Homotopy fibers 10 x3 Shifting the sequence 11 x4 The James construction 11 x5 Relation with the loopspace on a suspension 13 x6 Moore loops 13 Lecture 3 9/12 x1 Recap of the James construction 15 x2 The homology on ΩΣX 16 x3 To be fixed later 20 Lecture 4 9/14 x1 Recap 21 x2 James-Hopf maps 21 x3 The induced map in homology 22 x4 Coalgebras 23 Lecture 5 9/17 x1 Recap 26 x2 Goals 27 Lecture 6 9/19 x1 The EHPss 31 x2 The spectral sequence for a double complex 32 x3 Back to the EHPss 33 Lecture 7 9/21 x1 A fix 35 x2 The EHP sequence 36 Lecture 8 9/24 1 Lecture 9 9/26 x1 Hilton-Milnor again 44 x2 Hopf invariant one problem 46 x3 The K-theoretic proof (after Atiyah-Adams) 47 Lecture 10 9/28 x1 Splitting principle 50 x2 The Chern character 52 x3 The Adams operations 53 x4 Chern character and the Hopf invariant 53 Lecture 11 8/1 x1 The e-invariant 54 x2 Ext's in the category of groups with Adams operations 56 Lecture 12 10/3 x1 Hopf invariant one 58 Lecture 13 10/5 x1 Suspension 63 x2 The J-homomorphism 65 Lecture 14 10/10 x1 Vector fields problem 66 x2 Constructing vector fields 70 Lecture 15 10/12 x1 Clifford algebras 71 x2 Z=2-graded algebras 73 x3 Working out Clifford algebras 74 Lecture 16 10/15 x1 Radon-Hurwitz numbers 77 x2 Algebraic topology of the vector field problem 79 x3 The homology of
    [Show full text]
  • Digital Hurewicz Theorem and Digital Homology Theory
    Turkish Journal of Mathematics Turk J Math (2020) 44: 739 – 759 http://journals.tubitak.gov.tr/math/ © TÜBİTAK Research Article doi:10.3906/mat-2002-52 Digital Hurewicz theorem and digital homology theory Samira Sahar JAMIL1;2;∗, Danish ALI2 1Department of Mathematics, University of Karachi, Karachi, Pakistan 2Department of Mathematical Sciences, Institute of Business Administration, Karachi, Pakistan Received: 14.02.2020 • Accepted/Published Online: 15.03.2020 • Final Version: 08.05.2020 Abstract: In this paper, we develop homology groups for digital images based on cubical singular homology theory for topological spaces. Using this homology, we obtain two main results that make this homology different from already- existing homologies of digital images. We prove digital analog of Hurewicz theorem for digital cubical singular homology. We also show that the homology functors developed in this paper satisfy properties that resemble the Eilenberg–Steenrod axioms of homology theory, in particular, the homotopy and the excision axioms. We finally define axioms of digital homology theory. Key words: Digital topology, digital homology theory, digital Hurewicz theorem, cubical singular homology for digital images, digital excision 1. Introduction Digital topology is a developing research area, where topological properties of digital images are explored. In this area, digital images are mostly defined as subsets of Zd , equipped with certain adjacency relations. Though digital images are discrete in nature, they model continuous objects of the real world. Researchers are trying to understand whether or not digital images show similar properties as their continuous counterparts. The main motivation behind such studies is to develop a theory for digital images that is similar to the theory of topological spaces in classical topology.
    [Show full text]
  • Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties: Formality and Splitting
    REAL NON-ABELIAN MIXED HODGE STRUCTURES FOR QUASI-PROJECTIVE VARIETIES: FORMALITY AND SPLITTING J.P.PRIDHAM Abstract. We define and construct mixed Hodge structures on real schematic homo- topy types of complex quasi-projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. We also show that these split on tensoring with the ring R[x] equipped with the Hodge filtration given by powers of (x − i), giving new results even for simply connected varieties. The mixed Hodge struc- tures can thus be recovered from the Gysin spectral sequence of cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, these structures all become real variations of mixed Hodge structure. Introduction The main aims of this paper are to construct mixed Hodge structures on the real relative Malcev homotopy types of complex varieties, and to investigate how far these can be recovered from the structures on cohomology groups of local systems, and in particular from the Gysin spectral sequence. In [Mor], Morgan established the existence of natural mixed Hodge structures on the minimal model of the rational homotopy type of a smooth variety X, and used this to define natural mixed Hodge structures on the rational homotopy groups π∗(X ⊗ Q) of X. This construction was extended to singular varieties by Hain in [Hai2]. When X is also projective, [DGMS] showed that its rational homotopy type is formal; in particular, this means that the rational homotopy groups can be recovered from the cohomology ring H∗(X; Q).
    [Show full text]
  • Introduction
    Introduction The theory we describe in this book was developed over a long period, starting about 1965, and always with the aim of developing groupoid methods in homotopy theory of dimension greater than 1. Algebraic work made substantial progress in the early 1970s, in work with Chris Spencer. A substantial step forward in 1974 by Brown and Higgins led us over the years into many fruitful areas of homotopy theory and what is now called ‘higher dimensional algebra’2. We published detailed reports on all we found as the journey proceeded, but the overall picture of the theory is still not well known. So the aim of this book is to give a full, connected account of this work in one place, so that it can be more readily evaluated, used appropriately, and, we hope, developed. Structure of the subject There are several features of the theory and so of our exposition which divert from standard practice in algebraic topology, but are essential for the full success of our methods. Sets of base points: Enter groupoids The notion of a ‘space with base point’ is standard in algebraic topology and homotopy theory, but in many situations we are unsure which base point to choose. One example is if p W Y ! X is a covering map of spaces. Then X may have a chosen base point x, but it is not clear which base point to choose in the discrete inverse image space p1.x/. It makes sense then to take p1.x/ as a set of base points. Choosing a set of base points according to the geometry of the situation has the implication that we deal with fundamental groupoids 1.X; X0/ on a set X0 of base points rather than with the family of fundamental groups 1.X; x/, x 2 X0.
    [Show full text]
  • The Hurewicz Theorem April 25, 2011
    The Hurewicz Theorem April 25, 2011 1 Introduction The fundamental group and homology groups both give extremely useful information, particularly about path-connected spaces. Both can be considered as functors, so we can use these constructional invariants as convenient guides to classifying spaces. However, though homology groups are often easy to compute, the fundamental group sometimes is not. In fact, it is often not even obvious when a space is simply connected. In particular, noncontractible simply connected spaces are difficult to identify, as contractibility is often the most geometrically intuitive way to determine if a space is simply connected. As such, we would like to know if there is a connection between these two seemingly disjoint geometric concepts. The answer lies in the Hurewicz theorem, which in general gives us a connection between generalizations of the fundamental group (called homotopy groups) and the homology groups. As we will show, there exists a \Hurewicz homomorphism" from the nth homotopy group into the nth homology group for each n, and the Hurewicz theorem gives us information about this homomorphism for specific values of n. For the particular case of the fundamental group, the Hurewicz theorem indicates that the Hurewicz homomorphism induces an isomorphism between a quotient of the fundamental group and the first homology group, which provides us with a lot of information about the fundamental group. In many cases, the Hurewicz theorem tells us that the Hurewicz homomorphism is actually an isomorphism between the lowest nontrivial homotopy group and the lowest nontrivial reduced homology group. Moreover, it gives us a method for determining which homotopy group is the lowest nontrivial one given the homology groups.
    [Show full text]
  • Notes on A^ 1-Contractibility and A^ 1-Excision
    Notes on A1-contractibility and A1-excision Yuri Shimizu∗ September 7, 2018 Abstract We prove that a smooth scheme of dimension n over a perfect field is A1-weakly equivalent to a point if it is A1-n-connected. We also prove an excision result for A1-homotopy sheaves over a perfect field. 1 Introduction In this paper, we discuss n-connectedness of schemes over a field k in the sense of A1-homotopy theory. Morel-Voevodsky [MV] defines an A1-homotopy version of homotopy groups or sets as sheaves over the large Nisnevich site on the 1 category of smooth k-schemes Smk. They are called the A -homotopy sheaves 1 A A1 and denoted by πi (X) for X ∈ Smk. A scheme X is called -n-connected if A1 πi (X) is isomorphic to the constant sheaf valued on a point for all 0 ≤ i ≤ n. A1-0-connected schemes are simply called A1-connected. In A1-homotopy theory, schemes are considered up to A1-weak equivalences. We say that a scheme over k is A1-contractible if it is A1-weak equivalence to Spec k. Morel-Voevodsky [MV] proved the Whitehead theorem for A1-homotopy, which says that a scheme is A1-contractible if it is A1-n-connected for all non- negative integer n. Our first result is an improvement of this theorem, showing that the range of n can be actually restricted to 0 ≤ n ≤ dim X. More precisely, we prove the following. arXiv:1809.01895v1 [math.AG] 6 Sep 2018 Theorem 1.1 (see Theorem 3.2).
    [Show full text]
  • The Hurewicz Theorem by CW Approximation
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Helsingin yliopiston digitaalinen arkisto The Hurewicz theorem by CW approximation Master's thesis, University of Helsinki Student: Supervisor: Jonas Westerlund Erik Elfving December 11, 2016 Tiedekunta/Osasto Fakultet/Sektion – Faculty Laitos/Institution– Department Faculty of Science Department of Mathematics and Statistics Tekijä/Författare – Author Jonas Westerlund Työn nimi / Arbetets titel – Title The Hurewicz theorem by CW approximation Oppiaine /Läroämne – Subject Mathematics Työn laji/Arbetets art – Level Aika/Datum – Month and year Sivumäärä/ Sidoantal – Number of pages Master’s thesis December 2016 30 pages Tiivistelmä/Referat – Abstract In this thesis we prove the Hurewicz theorem which states that the n-th homology and homotopy groups are isomorphic for an (n-1)-connected topological space. There exists proofs of the Hurewicz theorem in which one constructs a concrete isomorphism between the spaces, but in this thesis we avoid the construction by transferring the problem to the realm of CW complexes and cellular structures by a technique known as cellular approximation. Combined with the cellular homology groups and related results this technique allows us to analyse the space on a cell-by-cell basis. This reduces the problem significantly and gives rise to many methods not applicable otherwise. To prove the theorem we lay out the foundations of homotopy theory and homology theory. The singular homology theory is introduced, which in turn is used together with the concept of degree to define the cellular homology groups suitable for the analysis of CW complexes. Since CW complexes are built out of homeomorphic copies of the open unit disk extending to its boundary, it became crucial to prove various properties of these subspaces in both homotopy and homology.
    [Show full text]
  • The 2-Dimensional Stable Homotopy Hypothesis 2
    THE 2-DIMENSIONAL STABLE HOMOTOPY HYPOTHESIS NICK GURSKI, NILES JOHNSON, AND ANGÉLICA M. OSORNO ABSTRACT. We prove that the homotopy theory of Picard 2-categories is equivalent to that of stable 2-types. 1. INTRODUCTION Grothendieck’s Homotopy Hypothesis posits an equivalence of homotopy theories be- tween homotopy n-types and weak n-groupoids. We pursue a similar vision in the sta- ble setting. Inspiration for a stable version of the Homotopy Hypothesis begins with [Seg74, May72] which show, for 1-categories, that symmetric monoidal structures give rise to infinite loop space structures on their classifying spaces. Thomason [Tho95] proved this is an equivalence of homotopy theories, relative to stable homotopy equiva- lences. This suggests that the categorical counterpart to stabilization is the presence of a symmetric monoidal structure with all cells invertible, an intuition that is reinforced by a panoply of results from the group-completion theorem of May [May74] to the Baez- Dolan stabilization hypothesis [BD95, Bat17] and beyond. A stable homotopy n-type is a spectrum with nontrivial homotopy groups only in dimensions 0 through n. The cor- responding symmetric monoidal n-categories with invertible cells are known as Picard n-categories. We can thus formulate the Stable Homotopy Hypothesis. Stable Homotopy Hypothesis. There is an equivalence of homotopy theories between n n Pic , Picard n-categories equipped with categorical equivalences, and Sp0 , stable homo- topy n-types equipped with stable equivalences. For n = 0, the Stable Homotopy Hypothesis is the equivalence of homotopy theories between abelian groups and Eilenberg-Mac Lane spectra. The case n = 1 is described by the second two authors in [JO12].
    [Show full text]