Positive Alexander Duality for Pursuit and Evasion
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Universal Coefficient Theorem for Homology
Universal Coefficient Theorem for Homology We present a direct proof of the universal coefficient theorem for homology that is simpler and shorter than the standard proof. Theorem 1 Given a chain complex C in which each Cn is free abelian, and a coefficient group G, we have for each n the natural short exact sequence 0 −−→ Hn(C) ⊗ G −−→ Hn(C ⊗ G) −−→ Tor(Hn−1(C),G) −−→ 0, (2) which splits (but not naturally). In particular, this applies immediately to singular homology. Theorem 3 Given a pair of spaces (X, A) and a coefficient group G, we have for each n the natural short exact sequence 0 −−→ Hn(X, A) ⊗ G −−→ Hn(X, A; G) −−→ Tor(Hn−1(X, A),G) −−→ 0, which splits (but not naturally). We shall derive diagram (2) as an instance of the following elementary result. Lemma 4 Given homomorphisms f: K → L and g: L → M of abelian groups, with a splitting homomorphism s: L → K such that f ◦ s = idL, we have the split short exact sequence ⊂ f 0 0 −−→ Ker f −−→ Ker(g ◦ f) −−→ Ker g −−→ 0, (5) where f 0 = f| Ker(g ◦ f), with the splitting s0 = s| Ker g: Ker g → Ker(g ◦ f). Proof We note that f 0 and s0 are defined, as f(Ker(g ◦ f)) ⊂ Ker g and s(Ker g) ⊂ Ker(g ◦f). (In detail, if l ∈ Ker g,(g ◦f)sl = gfsl = gl = 0 shows that sl ∈ Ker(g ◦f).) 0 0 0 Then f ◦ s = idL restricts to f ◦ s = id. Since Ker f ⊂ Ker(g ◦ f), we have Ker f = Ker f ∩ Ker(g ◦ f) = Ker f. -
Introduction to Homology
Introduction to Homology Matthew Lerner-Brecher and Koh Yamakawa March 28, 2019 Contents 1 Homology 1 1.1 Simplices: a Review . .2 1.2 ∆ Simplices: not a Review . .2 1.3 Boundary Operator . .3 1.4 Simplicial Homology: DEF not a Review . .4 1.5 Singular Homology . .5 2 Higher Homotopy Groups and Hurweicz Theorem 5 3 Exact Sequences 5 3.1 Key Definitions . .5 3.2 Recreating Groups From Exact Sequences . .6 4 Long Exact Homology Sequences 7 4.1 Exact Sequences of Chain Complexes . .7 4.2 Relative Homology Groups . .8 4.3 The Excision Theorems . .8 4.4 Mayer-Vietoris Sequence . .9 4.5 Application . .9 1 Homology What is Homology? To put it simply, we use Homology to count the number of n dimensional holes in a topological space! In general, our approach will be to add a structure on a space or object ( and thus a topology ) and figure out what subsets of the space are cycles, then sort through those subsets that are holes. Of course, as many properties we care about in topology, this property is invariant under homotopy equivalence. This is the slightly weaker than homeomorphism which we before said gave us the same fundamental group. 1 Figure 1: Hatcher p.100 Just for reference to you, I will simply define the nth Homology of a topological space X. Hn(X) = ker @n=Im@n−1 which, as we have said before, is the group of n-holes. 1.1 Simplices: a Review k+1 Just for your sake, we review what standard K simplices are, as embedded inside ( or living in ) R ( n ) k X X ∆ = [v0; : : : ; vk] = xivi such that xk = 1 i=0 For example, the 0 simplex is a point, the 1 simplex is a line, the 2 simplex is a triangle, the 3 simplex is a tetrahedron. -
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. -
The Decomposition Theorem, Perverse Sheaves and the Topology Of
The decomposition theorem, perverse sheaves and the topology of algebraic maps Mark Andrea A. de Cataldo and Luca Migliorini∗ Abstract We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem, indicate some important applications and examples. Contents 1 Overview 3 1.1 The topology of complex projective manifolds: Lefschetz and Hodge theorems 4 1.2 Families of smooth projective varieties . ........ 5 1.3 Singular algebraic varieties . ..... 7 1.4 Decomposition and hard Lefschetz in intersection cohomology . 8 1.5 Crash course on sheaves and derived categories . ........ 9 1.6 Decomposition, semisimplicity and relative hard Lefschetz theorems . 13 1.7 InvariantCycletheorems . 15 1.8 Afewexamples.................................. 16 1.9 The decomposition theorem and mixed Hodge structures . ......... 17 1.10 Historicalandotherremarks . 18 arXiv:0712.0349v2 [math.AG] 16 Apr 2009 2 Perverse sheaves 20 2.1 Intersection cohomology . 21 2.2 Examples of intersection cohomology . ...... 22 2.3 Definition and first properties of perverse sheaves . .......... 24 2.4 Theperversefiltration . .. .. .. .. .. .. .. 28 2.5 Perversecohomology .............................. 28 2.6 t-exactness and the Lefschetz hyperplane theorem . ...... 30 2.7 Intermediateextensions . 31 ∗Partially supported by GNSAGA and PRIN 2007 project “Spazi di moduli e teoria di Lie” 1 3 Three approaches to the decomposition theorem 33 3.1 The proof of Beilinson, Bernstein, Deligne and Gabber . -
Singular Homology of Arithmetic Schemes Alexander Schmidt
AlgebraAlgebraAlgebraAlgebra & & & & NumberNumberNumberNumber TheoryTheoryTheoryTheory Volume 1 2007 No. 2 Singular homology of arithmetic schemes Alexander Schmidt mathematicalmathematicalmathematicalmathematicalmathematicalmathematicalmathematical sciences sciences sciences sciences sciences sciences sciences publishers publishers publishers publishers publishers publishers publishers 1 ALGEBRA AND NUMBER THEORY 1:2(2007) Singular homology of arithmetic schemes Alexander Schmidt We construct a singular homology theory on the category of schemes of finite type over a Dedekind domain and verify several basic properties. For arithmetic schemes we construct a reciprocity isomorphism between the integral singular homology in degree zero and the abelianized modified tame fundamental group. 1. Introduction The objective of this paper is to construct a reasonable singular homology theory on the category of schemes of finite type over a Dedekind domain. Our main criterion for “reasonable” was to find a theory which satisfies the usual properties of a singular homology theory and which has the additional property that, for schemes of finite type over Spec(ޚ), the group h0 serves as the source of a reciprocity map for tame class field theory. In the case of schemes of finite type over finite fields this role was taken over by Suslin’s singular homology; see [Schmidt and Spieß 2000]. In this article we motivate and give the definition of the singular homology groups of schemes of finite type over a Dedekind domain and we verify basic properties, e.g. homotopy -
Algebraic Topology Is the Usage of Algebraic Tools to Study Topological Spaces
Chapter 1 Singular homology 1 Introduction: singular simplices and chains This is a course on algebraic topology. We’ll discuss the following topics. 1. Singular homology 2. CW-complexes 3. Basics of category theory 4. Homological algebra 5. The Künneth theorem 6. Cohomology 7. Universal coefficient theorems 8. Cup and cap products 9. Poincaré duality. The objects of study are of course topological spaces, and the machinery we develop in this course is designed to be applicable to a general space. But we are really mainly interested in geometrically important spaces. Here are some examples. • The most basic example is n-dimensional Euclidean space, Rn. • The n-sphere Sn = fx 2 Rn+1 : jxj = 1g, topologized as a subspace of Rn+1. • Identifying antipodal points in Sn gives real projective space RPn = Sn=(x ∼ −x), i.e. the space of lines through the origin in Rn+1. • Call an ordered collection of k orthonormal vectors an orthonormal k-frame. The space of n n orthonormal k-frames in R forms the Stiefel manifold Vk(R ), topologized as a subspace of (Sn−1)k. n n • The Grassmannian Grk(R ) is the space of k-dimensional linear subspaces of R . Forming n n the span gives us a surjection Vk(R ) ! Grk(R ), and the Grassmannian is given the quotient n n−1 topology. For example, Gr1(R ) = RP . 1 2 CHAPTER 1. SINGULAR HOMOLOGY All these examples are manifolds; that is, they are Hausdorff spaces locally homeomorphic to Eu- clidean space. Aside from Rn itself, the preceding examples are also compact. -
Singular Homology Groups and Homotopy Groups By
SINGULAR HOMOLOGY GROUPS AND HOMOTOPY GROUPS OF FINITE TOPOLOGICAL SPACES BY MICHAEL C. McCoRD 1. Introduction. Finite topological spaces have more interesting topological properties than one might suspect at first. Without thinking about it very long, one might guess that the singular homology groups and homotopy groups of finite spaces vanish in dimension greater than zero. (One might jump to the conclusion that continuous maps of simplexes and spheres into a finite space must be constant.) However, we shall show (see Theorem 1) that exactly the same singular homology groups and homotopy groups occur for finite spaces as occur for finite simplicial complexes. A map J X Y is a weatc homotopy equivalence if the induced maps (1.1) -- J, -(X, x) ---> -( Y, Jx) are isomorphisms for all x in X aIld all i >_ 0. (Of course in dimension 0, "iso- morphism" is understood to mean simply "1-1 correspondence," since ro(X, x), the set of path components of X, is not in general endowed with a group struc- ture.) It is a well-known theorem of J.H.C. Whitehead (see [4; 167]) that every weak homotopy equivalence induces isomorphisms on singular homology groups (hence also on singular cohomology rings.) Note that the general case is reduced to the case where X and Y are path connected by the assumption that (1.1) is a 1-1 correspondence for i 0. THEOnnM 1. (i) For each finite topological space X there exist a finite simplicial complex K and a wealc homotopy equivalence f IK[ X. (ii) For each finite sim- plicial complex K there exist a finite topological space-- X and a wealc homotopy equivalence ] "[K[ X. -
A Note on the Homotopy Type of the Alexander Dual
A NOTE ON THE HOMOTOPY TYPE OF THE ALEXANDER DUAL EL´IAS GABRIEL MINIAN AND JORGE TOMAS´ RODR´IGUEZ Abstract. We investigate the homotopy type of the Alexander dual of a simplicial complex. In general the homotopy type of K does not determine the homotopy type of its dual K∗. Moreover, one can construct for each finitely presented group G, a simply ∗ connected simplicial complex K such that π1(K ) = G. We study sufficient conditions on K for K∗ to have the homotopy type of a sphere. We also extend the simplicial Alexander duality to the context of reduced lattices. 1. Introduction Let A be a compact and locally contractible proper subspace of Sn. The classical n Alexander duality theorem asserts that the reduced homology groups Hi(S − A) are isomorphic to the reduced cohomology groups Hn−i−1(A) (see for example [6, Thm. 3.44]). The combinatorial (or simplicial) Alexander duality is a special case of the classical duality: if K is a finite simplicial complex and K∗ is the Alexander dual with respect to a ground set V ⊇ K0, then for any i ∼ n−i−3 ∗ Hi(K) = H (K ): Here K0 denotes the set of vertices (i.e. 0-simplices) of K and n is the size of V .A very nice and simple proof of the combinatorial Alexander duality can be found in [4]. An alternative proof of this combinatorial duality can be found in [3]. In these notes we relate the homotopy type of K with that of K∗. We show first that, even though the homology of K determines the homology of K∗ (and vice versa), the homotopy type of K does not determine the homotopy type of K∗. -
Notation and Basic Facts in Knot Theory
Appendix A Notation and Basic Facts in Knot Theory In this appendix, our aim is to provide a quick review of basic terminology and some facts in knot theory, which we use or need in this book. This appendix lists them item by item (without proof); so we do no attempt to give a full rigorous treatments; Instead, we work somewhat intuitively. The reader with interest in the details could consult the textbooks [BZ, R, Lic, KawBook]. • First, we fix notation on the circle S1 := {(x, y) ∈ R2 | x2 + y2 = 1}, and consider a finite disjoint union S1. A link is a C∞-embedding of L :S1 → S3 in the 3-sphere. We denote often by #L the number of the disjoint union, and denote the image Im(L) by only L for short. If #L is 1, L is usually called a knot, and is written K instead. This book discusses embeddings together with orientation. For an oriented link L, we denote by −L the link with its orientation reversed, and by L∗ the mirror image of L. • (Notations of link components). Given a link L :S1 → S3, let us fix an open tubular neighborhood νL ⊂ S3. Throughout this book, we denote the complement S3 \ νL by S3 \ L for short. Since we mainly discuss isotopy classes of S3 \ L,we may ignore the choice of νL. 2 • For example, for integers s, t ∈ Z , the torus link Ts,t (of type (s, t)) is defined by 3 2 s t 2 2 S Ts,t := (z,w)∈ C z + w = 0, |z| +|w| = 1 . -
Equivariant Singular Homology and Cohomology I
Licensed to Univ of Rochester. Prepared on Tue Jul 28 10:51:47 EDT 2015for download from IP 128.151.13.18. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms MEMOIRS of the American Mathematical Society This journal is designed particularly for long research papers (and groups of cognate papers) in pure and applied mathematics. It includes, in general, longer papers than those in the TRANSACTIONS. Mathematical papers intended for publication in the Memoirs should be addressed to one of the editors. Subjects, and the editors associated with them, follow: Real analysis (excluding harmonic analysis) and applied mathematics to FRANCOIS TREVES, Depart• ment of Mathematics, Rutgers University, New Brunswick, NJ 08903. Harmonic and complex analysis to HUGO ROSSI, Department of Mathematics, University of Utah, Salt Lake City, UT 84112. Abstract analysis to ALEXANDRA IONESCU TULCEA, Department of Mathematics, Northwestern University, Evanston, IL 60201. Algebra and number theory (excluding universal algebras) to STEPHEN S. SHATZ, Department of Mathe• matics, University of Pennsylvania, Philadelphia, PA 19174. Logic, foundations, universal algebras and combinatorics to ALISTAIR H. LACHLAN, Department of Mathematics, Simon Fraser University, Burnaby, 2, B. C, Canada. Topology to PHILIP T. CHURCH, Department of Mathematics, Syracuse University, Syracuse, NY 13210. Global analysis and differential geometry to VICTOR W. GUILLEMIN, c/o Ms. M. McQuillin, Depart• ment of Mathematics, Harvard University, Cambridge, MA 02138. Probability and statistics to DANIEL W. STROOCK, Department of Mathematics, University of Colorado, Boulder, CO 80302 All other communications to the editors should be addressed to Managing Editor, ALISTAIR H. LACHLAN MEMOIRS are printed by photo-offset from camera-ready copy fully prepared by the authors. -
Arxiv:1607.08163V2 [Math.GT] 24 Sep 2016
LECTURES ON THE TRIANGULATION CONJECTURE CIPRIAN MANOLESCU Abstract. We outline the proof that non-triangulable manifolds exist in any dimension greater than four. The arguments involve homology cobordism invariants coming from the Pinp2q symmetry of the Seiberg-Witten equations. We also explore a related construction, of an involutive version of Heegaard Floer homology. 1. Introduction The triangulation conjecture stated that every topological manifold can be triangulated. The work of Casson [AM90] in the 1980's provided counterexamples in dimension 4. The main purpose of these notes is to describe the proof of the following theorem. Theorem 1.1 ([Man13]). There exist non-triangulable n-dimensional topological manifolds for every n ¥ 5. The proof relies on previous work by Galewski-Stern [GS80] and Matumoto [Mat78], who reduced this problem to a different one, about the homology cobordism group in three dimensions. Homology cobordism can be explored using the techniques of gauge theory, as was done, for example, by Fintushel and Stern [FS85, FS90], Furuta [Fur90], and Frøyshov [Fro10]. In [Man13], Pinp2q-equivariant Seiberg-Witten Floer homology is used to construct three new invariants of homology cobordism, called α, β and γ. The properties of β suffice to answer the question raised by Galewski-Stern and Matumoto, and thus prove Theorem 1.1. The paper is organized as follows. Section 2 contains background material about triangulating manifolds. In particular, we sketch the arguments of Galewski-Stern and Matumoto that reduced Theorem 1.1 to a problem about homology cobordism. In Section 3 we introduce the Seiberg-Witten equations, finite dimensional approxima- tion, and the Conley index. -
The Ways I Know How to Define the Alexander Polynomial
ALL THE WAYS I KNOW HOW TO DEFINE THE ALEXANDER POLYNOMIAL KYLE A. MILLER Abstract. These began as notes for a talk given at the Student 3-manifold seminar, Spring 2019. There seems to be many ways to define the Alexander polynomial, all of which are somehow interrelated, but sometimes there is not an obvious path between any two given definitions. As the title suggests, this is an exploration of all the ways I know how to define this knot invariant. While we will touch on a number of facts and properties, these notes are not meant to be a complete survey of the Alexander polynomial. Contents 1. Introduction2 2. Alexander’s definition2 2.1. The Dehn presentation2 2.2. Abelianization4 2.3. The associated matrix4 3. The Alexander modules6 3.1. Orders7 3.2. Elementary ideals8 3.3. The Fox calculus 10 3.4. Torus knots 13 3.5. The Wirtinger presentation 13 3.6. Generalization to links 15 3.7. Fibered knots 15 3.8. Seifert presentation 16 3.9. Duality 18 4. The Conway potential 18 4.1. Alexander’s three-term relation 20 5. The HOMFLY-PT polynomial 24 6. Kauffman state sum 26 6.1. Another state sum 29 7. The Burau representation 29 8. Vassiliev invariants 29 9. The Alexander quandle 29 9.1. Projective Alexander quandles 33 10. Reidemeister torsion 33 11. Knot Floer homology 33 References 33 Date: May 3, 2019. 1 DRAFT 2020/11/14 18:57:28 2 KYLE A. MILLER 1. Introduction Recall that a link is an embedded closed 1-manifold in S3, and a knot is a 1-component link.