Introduction to Algebraic Topology and Algebraic
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Equivariant Geometric K-Homology with Coefficients
Equivariant geometric K-homology with coefficients Michael Walter Equivariant geometric K-homology with coefficients Diplomarbeit vorgelegt von Michael Walter geboren in Lahr angefertigt am Mathematischen Institut der Georg-August-Universität zu Göttingen 2010 v Equivariant geometric K-homology with coefficients Michael Walter Abstract K-homology is the dual of K-theory. Kasparov’s analytic version, where cycles are given by (ab- stract) elliptic operators over (not necessarily commutative) spaces, has proved to be an extremely powerful tool which, together with its bivariant generalization KK-theory, lies at the heart of many important results at the intersection of algebraic topology, functional analysis and geome- try. Independently, Baum and Douglas have proposed a geometric version of K-homology inspired by singular bordism. Cycles for this theory are given by vector bundles over compact Spinc- manifolds with boundary which map to the target, i.e. E f (M, BM) / (X, Y). There is a natural transformation to analytic K-homology defined by sending such a cycle to the pushforward of the class determined by the twisted Dirac operator. It is well-known to be an isomorphism, although a rigorous proof has appeared only recently. While both theories have obvious generalizations to the equivariant case and coefficients, the question whether these remain isomorphic is far from trivial (and has negative answer in the general case). In their work on equivariant correspondences Emerson and Meyer have isolated a useful sufficient condition for their theory which, while vastly more general, only deals with the absolute case. Our focus is not so much to construct a geometric theory in the most general situation, but to show that in the presence of a group action and coefficients the above picture still gives a generalized homology theory in a very geometrical way, isomorphic to Kasparov’s theory. -
HODGE THEORY LEFSCHETZ PENCILS 1. More on Lefschetz
HODGE THEORY LEFSCHETZ PENCILS JACOB KELLER 1. More on Lefschetz Pencils When studying the vanishing homology of a hyperplane section, we want to view the hyperplane as one element of a Lefschetz pencil. This is because associated to each singular divisor in the pencil, there is an element of the homology of our hyperplane that is not detected by the Lefschetz hyperplane theorem. In order to do this we should know more about Lefschetz pencils, and that’s how we will start this lecture. We want to be able to produce Lefschetz pencils containing a given hyperplane section. So we should get a better characterization of a Lefschetz pencil that is easier to compute. Let X be a projective variety embedded in PN that is not contained in a hyperplane. We consider the variety N∗ Z = {(x, H)|x ∈ X ∩ H is a singular point} ⊆ X × P To better understand the geometry of Z, we consider it’s projection π1 : Z → X. For any x ∈ X and N hyperplane H ⊆ P , the tangent plane to H ∩ X at x is the intersection of H and TxX. Thus X ∩ H will −1 be singular at x if and only if H contains TxX. Thus for x ∈ X, π1 (Z) is the set of hyperplanes that contain TxX. These hyperplanes form an N − n − 1 dimensional projective space, and thus π1 exhibits Z as a PN − n − 1-bundle over X, and we deduce that Z is smooth. N∗ Now we consider the projection π2 : Z → P . The image of this map is denoted DX and is called the discriminant of X. -
Cohomological Descent on the Overconvergent Site
COHOMOLOGICAL DESCENT ON THE OVERCONVERGENT SITE DAVID ZUREICK-BROWN Abstract. We prove that cohomological descent holds for finitely presented crystals on the overconvergent site with respect to proper or fppf hypercovers. 1. Introduction Cohomological descent is a robust computational and theoretical tool, central to p-adic cohomology and its applications. On one hand, it facilitates explicit calculations (analo- gous to the computation of coherent cohomology in scheme theory via Cechˇ cohomology); on another, it allows one to deduce results about singular schemes (e.g., finiteness of the cohomology of overconvergent isocrystals on singular schemes [Ked06]) from results about smooth schemes, and, in a pinch, sometimes allows one to bootstrap global definitions from local ones (for example, for a scheme X which fails to embed into a formal scheme smooth near X, one actually defines rigid cohomology via cohomological descent; see [lS07, comment after Proposition 8.2.17]). The main result of the series of papers [CT03,Tsu03,Tsu04] is that cohomological descent for the rigid cohomology of overconvergent isocrystals holds with respect to both flat and proper hypercovers. The barrage of choices in the definition of rigid cohomology is burden- some and makes their proofs of cohomological descent very difficult, totaling to over 200 pages. Even after the main cohomological descent theorems [CT03, Theorems 7.3.1 and 7.4.1] are proved one still has to work a bit to get a spectral sequence [CT03, Theorem 11.7.1]. Actually, even to state what one means by cohomological descent (without a site) is subtle. The situation is now more favorable. -
Homological Algebra
Homological Algebra Donu Arapura April 1, 2020 Contents 1 Some module theory3 1.1 Modules................................3 1.6 Projective modules..........................5 1.12 Projective modules versus free modules..............7 1.15 Injective modules...........................8 1.21 Tensor products............................9 2 Homology 13 2.1 Simplicial complexes......................... 13 2.8 Complexes............................... 15 2.15 Homotopy............................... 18 2.23 Mapping cones............................ 19 3 Ext groups 21 3.1 Extensions............................... 21 3.11 Projective resolutions........................ 24 3.16 Higher Ext groups.......................... 26 3.22 Characterization of projectives and injectives........... 28 4 Cohomology of groups 32 4.1 Group cohomology.......................... 32 4.6 Bar resolution............................. 33 4.11 Low degree cohomology....................... 34 4.16 Applications to finite groups..................... 36 4.20 Topological interpretation...................... 38 5 Derived Functors and Tor 39 5.1 Abelian categories.......................... 39 5.13 Derived functors........................... 41 5.23 Tor functors.............................. 44 5.28 Homology of a group......................... 45 1 6 Further techniques 47 6.1 Double complexes........................... 47 6.7 Koszul complexes........................... 49 7 Applications to commutative algebra 52 7.1 Global dimensions.......................... 52 7.9 Global dimension of -
Homology Theory on Algebraic Varieties
HOMOLOGY THEORYon ALGEBRAIC VARIETIES HOMOLOGY THEORY ON ALGEBRAIC VARIETIES by ANDREW H. WALLACE Assistant Professor of Mathematics University of Toronto PERGAMON PRESS LONDON NEW YORK PARIS LOS ANGELES 1958 . PERGAMON PRESS LTD. 4 and 5 Fitzroy Square, London W.I. PERGAMON PRESS INC. 122 East 55th Street, New York, N.Y. 10638 South Wilton Place, Los Angeles 47, California PERGAMON PRESS S.A.R.L. 24 Rue des Ecoles, Paris V" Copyright 0 1958 A. H. Wallace Library of Congress Card Number 57-14497 Printed in Northern Ireland at The Universities Press, Be fast CONTENTS INTRODUCTION vu 1. -LINEAR SECTIONS OF AN ALGEBRAIC VARIETY 1. Hyperplane sections of a non-singular variety 1 2. A family of linear sections of W 2 3. The fibring of a variety defined over the complex numbers 7. 4. Homology groups related to V(K) 17 II. THE SINGULAR SECTIONS 1. Statement of the results 23 2. Proof of Theorem 11 25 III. A PENCIL OF HYPERPLANE SECTIONS 1. The choice of a pencil 34 2. Notation 37 3. Reduction to local theorems 38. IV. LEFSCHETZ'S FIRST AND SECOND THEOREMS 1. Lefschetz's first main theorem 43 2. Statement of Lefschetz's second main theorem 49 3. Sketch proof of Theorem 19 49- .4. Some immediate consequences _ 84 V. PROOF OF LEFSCHETZ'S SECOND THEOREM 1. Deformation theorems 56 2. Some remarks dh Theorem 19 80 3. Formal verification of Theorem 19; the vanishing cycle62 4. Proof of Theorem 19, parts (1) and (2) 64 5. Proof of Theorem 19, part (3) 87 v Vi CONTENTS VI. -
Homology Stratifications and Intersection Homology 1 Introduction
ISSN 1464-8997 (on line) 1464-8989 (printed) 455 Geometry & Topology Monographs Volume 2: Proceedings of the Kirbyfest Pages 455–472 Homology stratifications and intersection homology Colin Rourke Brian Sanderson Abstract A homology stratification is a filtered space with local ho- mology groups constant on strata. Despite being used by Goresky and MacPherson [3] in their proof of topological invariance of intersection ho- mology, homology stratifications do not appear to have been studied in any detail and their properties remain obscure. Here we use them to present a simplified version of the Goresky–MacPherson proof valid for PL spaces, and we ask a number of questions. The proof uses a new technique, homology general position, which sheds light on the (open) problem of defining generalised intersection homology. AMS Classification 55N33, 57Q25, 57Q65; 18G35, 18G60, 54E20, 55N10, 57N80, 57P05 Keywords Permutation homology, intersection homology, homology stratification, homology general position Rob Kirby has been a great source of encouragement. His help in founding the new electronic journal Geometry & Topology has been invaluable. It is a great pleasure to dedicate this paper to him. 1 Introduction Homology stratifications are filtered spaces with local homology groups constant on strata; they include stratified sets as special cases. Despite being used by Goresky and MacPherson [3] in their proof of topological invariance of intersec- tion homology, they do not appear to have been studied in any detail and their properties remain obscure. It is the purpose of this paper is to publicise these neglected but powerful tools. The main result is that the intersection homology groups of a PL homology stratification are given by singular cycles meeting the strata with appropriate dimension restrictions. -
Math 6280 - Class 27
MATH 6280 - CLASS 27 Contents 1. Reduced and relative homology and cohomology 1 2. Eilenberg-Steenrod Axioms 2 2.1. Axioms for unreduced homology 2 2.2. Axioms for reduced homology 4 2.3. Axioms for cohomology 5 These notes are based on • Algebraic Topology from a Homotopical Viewpoint, M. Aguilar, S. Gitler, C. Prieto • A Concise Course in Algebraic Topology, J. Peter May • More Concise Algebraic Topology, J. Peter May and Kate Ponto • Algebraic Topology, A. Hatcher 1. Reduced and relative homology and cohomology Definition 1.1. For A ⊂ X a sub-complex, let Cn(A) ⊂ Cn(X) as the cells of A are a subset of the cells of X. • Let C∗(X; A) = C∗(X)=C∗(A) and H∗(X; A) = H∗(C∗(X; A)). • Let ∗ ∗ ∗ C (X; A) = ker(C (X) ! C (A)) = Hom(C∗(X; A); Z) and H∗(X) = H∗(C∗(X; A)). We can give similar definitions for H∗(X; A; M) and H∗(X; A; M): Definition 1.2. For X a based CW-complex with based point ∗ a zero cell, 1 2 MATH 6280 - CLASS 27 • Let Ce∗(X) = C∗(X)=C∗(∗) and He∗(X) = H∗(Ce∗(X)). • Let ∗ ∗ ∗ Ce (X) = ker(C (X) ! C (∗)) = Hom(Ce∗(X); Z) and He ∗(X) = H∗(Ce∗(X)). ∼ Remark 1.3. Note that C∗(X; A) = C∗(X)=C∗(A) = Ce∗(X=A). Indeed, X=A has a CW-structure all cells in A identified to the base point and one cell for each cell not in A. Therefore, we have a natural isomorphism ∼ H∗(X; A) = He∗(X=A): Similarly, H∗(X; A) =∼ He ∗(X=A) Unreduced cohomology can be though of as a functor from the homotopy category of pairs of topological spaces to abelian groups: H∗(−; −; M): hCWpairs ! Ab where H∗(X; M) = H∗(X; ;; M): 2. -
Lecture 15. De Rham Cohomology
Lecture 15. de Rham cohomology In this lecture we will show how differential forms can be used to define topo- logical invariants of manifolds. This is closely related to other constructions in algebraic topology such as simplicial homology and cohomology, singular homology and cohomology, and Cechˇ cohomology. 15.1 Cocycles and coboundaries Let us first note some applications of Stokes’ theorem: Let ω be a k-form on a differentiable manifold M.For any oriented k-dimensional compact sub- manifold Σ of M, this gives us a real number by integration: " ω : Σ → ω. Σ (Here we really mean the integral over Σ of the form obtained by pulling back ω under the inclusion map). Now suppose we have two such submanifolds, Σ0 and Σ1, which are (smoothly) homotopic. That is, we have a smooth map F : Σ × [0, 1] → M with F |Σ×{i} an immersion describing Σi for i =0, 1. Then d(F∗ω)isa (k + 1)-form on the (k + 1)-dimensional oriented manifold with boundary Σ × [0, 1], and Stokes’ theorem gives " " " d(F∗ω)= ω − ω. Σ×[0,1] Σ1 Σ1 In particular, if dω =0,then d(F∗ω)=F∗(dω)=0, and we deduce that ω = ω. Σ1 Σ0 This says that k-forms with exterior derivative zero give a well-defined functional on homotopy classes of compact oriented k-dimensional submani- folds of M. We know some examples of k-forms with exterior derivative zero, namely those of the form ω = dη for some (k − 1)-form η. But Stokes’ theorem then gives that Σ ω = Σ dη =0,sointhese cases the functional we defined on homotopy classes of submanifolds is trivial. -
Math 601 Algebraic Topology Hw 4 Selected Solutions Sketch/Hint
MATH 601 ALGEBRAIC TOPOLOGY HW 4 SELECTED SOLUTIONS SKETCH/HINT QINGYUN ZENG 1. The Seifert-van Kampen theorem 1.1. A refinement of the Seifert-van Kampen theorem. We are going to make a refinement of the theorem so that we don't have to worry about that openness problem. We first start with a definition. Definition 1.1 (Neighbourhood deformation retract). A subset A ⊆ X is a neighbourhood defor- mation retract if there is an open set A ⊂ U ⊂ X such that A is a strong deformation retract of U, i.e. there exists a retraction r : U ! A and r ' IdU relA. This is something that is true most of the time, in sufficiently sane spaces. Example 1.2. If Y is a subcomplex of a cell complex, then Y is a neighbourhood deformation retract. Theorem 1.3. Let X be a space, A; B ⊆ X closed subspaces. Suppose that A, B and A \ B are path connected, and A \ B is a neighbourhood deformation retract of A and B. Then for any x0 2 A \ B. π1(X; x0) = π1(A; x0) ∗ π1(B; x0): π1(A\B;x0) This is just like Seifert-van Kampen theorem, but usually easier to apply, since we no longer have to \fatten up" our A and B to make them open. If you know some sheaf theory, then what Seifert-van Kampen theorem really says is that the fundamental groupoid Π1(X) is a cosheaf on X. Here Π1(X) is a category with object pints in X and morphisms as homotopy classes of path in X, which can be regard as a global version of π1(X). -
The Fundamental Group and Seifert-Van Kampen's
THE FUNDAMENTAL GROUP AND SEIFERT-VAN KAMPEN'S THEOREM KATHERINE GALLAGHER Abstract. The fundamental group is an essential tool for studying a topo- logical space since it provides us with information about the basic shape of the space. In this paper, we will introduce the notion of free products and free groups in order to understand Seifert-van Kampen's Theorem, which will prove to be a useful tool in computing fundamental groups. Contents 1. Introduction 1 2. Background Definitions and Facts 2 3. Free Groups and Free Products 4 4. Seifert-van Kampen Theorem 6 Acknowledgments 12 References 12 1. Introduction One of the fundamental questions in topology is whether two topological spaces are homeomorphic or not. To show that two topological spaces are homeomorphic, one must construct a continuous function from one space to the other having a continuous inverse. To show that two topological spaces are not homeomorphic, one must show there does not exist a continuous function with a continuous inverse. Both of these tasks can be quite difficult as the recently proved Poincar´econjecture suggests. The conjecture is about the existence of a homeomorphism between two spaces, and it took over 100 years to prove. Since the task of showing whether or not two spaces are homeomorphic can be difficult, mathematicians have developed other ways to solve this problem. One way to solve this problem is to find a topological property that holds for one space but not the other, e.g. the first space is metrizable but the second is not. Since many spaces are similar in many ways but not homeomorphic, mathematicians use a weaker notion of equivalence between spaces { that of homotopy equivalence. -
Lefschetz Section Theorems for Tropical Hypersurfaces
LEFSCHETZ SECTION THEOREMS FOR TROPICAL HYPERSURFACES CHARLES ARNAL, ARTHUR RENAUDINEAU, AND KRISTIN SHAW Abstract. We establish variants of the Lefschetz hyperplane section theorem for the integral tropical homology groups of tropical hypersur- faces of toric varieties. It follows from these theorems that the integral tropical homology groups of non-singular tropical hypersurfaces which n are compact or contained in R are torsion free. We prove a relation- ship between the coefficients of the χy genera of complex hypersurfaces in toric varieties and Euler characteristics of the integral tropical cellu- lar chain complexes of their tropical counterparts. It follows that the integral tropical homology groups give the Hodge numbers of compact non-singular hypersurfaces of complex toric varieties. Finally for tropi- cal hypersurfaces in certain affine toric varieties, we relate the ranks of their tropical homology groups to the Hodge-Deligne numbers of their complex counterparts. Contents 1. Introduction 1 Acknowledgement 5 2. Preliminaries 6 3. Tropical Lefschetz hyperplane section theorem 16 4. The tropical homology of hypersurfaces is torsion free 24 5. Betti numbers of tropical homology and Hodge numbers 27 References 32 1. Introduction Tropical homology is a homology theory with non-constant coefficients for polyhedral spaces. Itenberg, Katzarkov, Mikhalkin, and Zharkov, show that under suitable conditions, the Q-tropical Betti numbers of the tropical limit arXiv:1907.06420v1 [math.AG] 15 Jul 2019 of a family of complex projective varieties are equal to the corresponding Hodge numbers of a generic member of the family [IKMZ16]. This explains the particular interest of these homology groups in tropical and complex algebraic geometry. -
Algebraic Topology
Algebraic Topology Vanessa Robins Department of Applied Mathematics Research School of Physics and Engineering The Australian National University Canberra ACT 0200, Australia. email: [email protected] September 11, 2013 Abstract This manuscript will be published as Chapter 5 in Wiley's textbook Mathe- matical Tools for Physicists, 2nd edition, edited by Michael Grinfeld from the University of Strathclyde. The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology. arXiv:1304.7846v2 [math-ph] 10 Sep 2013 1 Contents 1 Introduction 3 2 Homotopy Theory 4 2.1 Homotopy of paths . 4 2.2 The fundamental group . 5 2.3 Homotopy of spaces . 7 2.4 Examples . 7 2.5 Covering spaces . 9 2.6 Extensions and applications . 9 3 Homology 11 3.1 Simplicial complexes . 12 3.2 Simplicial homology groups . 12 3.3 Basic properties of homology groups . 14 3.4 Homological algebra . 16 3.5 Other homology theories . 18 4 Cohomology 18 4.1 De Rham cohomology . 20 5 Morse theory 21 5.1 Basic results . 21 5.2 Extensions and applications . 23 5.3 Forman's discrete Morse theory . 24 6 Computational topology 25 6.1 The fundamental group of a simplicial complex . 26 6.2 Smith normal form for homology . 27 6.3 Persistent homology . 28 6.4 Cell complexes from data . 29 2 1 Introduction Topology is the study of those aspects of shape and structure that do not de- pend on precise knowledge of an object's geometry.