Relative Cohomology and Projective Twistor Diagrams
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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. -
Topics in Geometric Group Theory. Part I
Topics in Geometric Group Theory. Part I Daniele Ettore Otera∗ and Valentin Po´enaru† April 17, 2018 Abstract This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (qsf), geometric simple connec- tivity (gsc), topological inverse-representations, and the notion of easy groups. As we will explain, these properties are central in the theory of discrete groups seen from a topo- logical viewpoint at infinity. Also, we shall outline the main steps of the achievements obtained in the last years around the very general question whether or not all finitely presented groups satisfy these conditions. Keywords: Geometric simple connectivity (gsc), quasi-simple filtration (qsf), inverse- representations, finitely presented groups, universal covers, singularities, Whitehead manifold, handlebody decomposition, almost-convex groups, combings, easy groups. MSC Subject: 57M05; 57M10; 57N35; 57R65; 57S25; 20F65; 20F69. Contents 1 Introduction 2 1.1 Acknowledgments................................... 2 arXiv:1804.05216v1 [math.GT] 14 Apr 2018 2 Topology at infinity, universal covers and discrete groups 2 2.1 The Φ/Ψ-theoryandzippings ............................ 3 2.2 Geometric simple connectivity . 4 2.3 TheWhiteheadmanifold............................... 6 2.4 Dehn-exhaustibility and the QSF property . .... 8 2.5 OnDehn’slemma................................... 10 2.6 Presentations and REPRESENTATIONS of groups . ..... 13 2.7 Good-presentations of finitely presented groups . ........ 16 2.8 Somehistoricalremarks .............................. 17 2.9 Universalcovers................................... 18 2.10 EasyREPRESENTATIONS . 21 ∗Vilnius University, Institute of Data Science and Digital Technologies, Akademijos st. 4, LT-08663, Vilnius, Lithuania. e-mail: [email protected] - [email protected] †Professor Emeritus at Universit´eParis-Sud 11, D´epartement de Math´ematiques, Bˆatiment 425, 91405 Orsay, France. -
Experience Implementing a Performant Category-Theory Library in Coq
Experience Implementing a Performant Category-Theory Library in Coq Jason Gross, Adam Chlipala, and David I. Spivak Massachusetts Institute of Technology, Cambridge, MA, USA [email protected], [email protected], [email protected] Abstract. We describe our experience implementing a broad category- theory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed sub- stantial engineering effort to teach Coq to cope with large categorical constructions without slowing proof script processing unacceptably. In this paper, we share the lessons we have learned about how to repre- sent very abstract mathematical objects and arguments in Coq and how future proof assistants might be designed to better support such rea- soning. One particular encoding trick to which we draw attention al- lows category-theoretic arguments involving duality to be internalized in Coq's logic with definitional equality. Ours may be the largest Coq development to date that uses the relatively new Coq version developed by homotopy type theorists, and we reflect on which new features were especially helpful. Keywords: Coq · category theory · homotopy type theory · duality · performance 1 Introduction Category theory [36] is a popular all-encompassing mathematical formalism that casts familiar mathematical ideas from many domains in terms of a few unifying concepts. A category can be described as a directed graph plus algebraic laws stating equivalences between paths through the graph. Because of this spar- tan philosophical grounding, category theory is sometimes referred to in good humor as \formal abstract nonsense." Certainly the popular perception of cat- egory theory is quite far from pragmatic issues of implementation. -
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. -
Motivating Abstract Nonsense
Lecture 1: Motivating abstract nonsense Nilay Kumar May 28, 2014 This summer, the UMS lectures will focus on the basics of category theory. Rather dry and substanceless on its own (at least at first), category theory is difficult to grok without proper motivation. With the proper motivation, however, category theory becomes a powerful language that can concisely express and abstract away the relationships between various mathematical ideas and idioms. This is not to say that it is purely a tool { there are many who study category theory as a field of mathematics with intrinsic interest, but this is beyond the scope of our lectures. Functoriality Let us start with some well-known mathematical examples that are unrelated in content but similar in \form". We will then make this rather vague similarity more precise using the language of categories. Example 1 (Galois theory). Recall from algebra the notion of a finite, Galois field extension K=F and its associated Galois group G = Gal(K=F ). The fundamental theorem of Galois theory tells us that there is a bijection between subfields E ⊂ K containing F and the subgroups H of G. The correspondence sends E to all the elements of G fixing E, and conversely sends H to the fixed field of H. Hence we draw the following diagram: K 0 E H F G I denote the Galois correspondence by squiggly arrows instead of the usual arrows. Unlike most bijections we usually see between say elements of two sets (or groups, vector spaces, etc.), the fundamental theorem gives us a one-to-one correspondence between two very different types of objects: fields and groups. -
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. -
Oka Manifolds: from Oka to Stein and Back
ANNALES DE LA FACULTÉ DES SCIENCES Mathématiques FRANC FORSTNERICˇ Oka manifolds: From Oka to Stein and back Tome XXII, no 4 (2013), p. 747-809. <http://afst.cedram.org/item?id=AFST_2013_6_22_4_747_0> © Université Paul Sabatier, Toulouse, 2013, tous droits réservés. L’accès aux articles de la revue « Annales de la faculté des sci- ences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://afst.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Annales de la Facult´e des Sciences de Toulouse Vol. XXII, n◦ 4, 2013 pp. 747–809 Oka manifolds: From Oka to Stein and back Franc Forstnericˇ(1) ABSTRACT. — Oka theory has its roots in the classical Oka-Grauert prin- ciple whose main result is Grauert’s classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomor- phic maps from Stein manifolds and Stein spaces to Oka manifolds. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. In this expository paper we discuss Oka manifolds and Oka maps. We de- scribe equivalent characterizations of Oka manifolds, the functorial prop- erties of this class, and geometric sufficient conditions for being Oka, the most important of which is Gromov’s ellipticity. -
Arxiv:Math/9703207V1 [Math.GT] 20 Mar 1997
ON INVARIANTS AND HOMOLOGY OF SPACES OF KNOTS IN ARBITRARY MANIFOLDS V. A. VASSILIEV Abstract. The construction of finite-order knot invariants in R3, based on reso- lutions of discriminant sets, can be carried out immediately to the case of knots in an arbitrary three-dimensional manifold M (may be non-orientable) and, moreover, to the calculation of cohomology groups of spaces of knots in arbitrary manifolds of dimension ≥ 3. Obstructions to the integrability of admissible weight systems to well-defined knot invariants in M are identified as 1-dimensional cohomology classes of certain generalized loop spaces of M. Unlike the case M = R3, these obstructions can be non-trivial and provide invariants of the manifold M itself. The corresponding algebraic machinery allows us to obtain on the level of the “abstract nonsense” some of results and problems of the theory, and to extract from others the essential topological (in particular, low-dimensional) part. Introduction Finite-order invariants of knots in R3 appeared in [V2] from a topological study of the discriminant subset of the space of curves in R3 (i.e., the set of all singular curves). In [L], [K], finite-order invariants of knots in 3-manifolds satisfying some conditions were considered: in [L] it was done for manifolds with π1 = π2 = 0, and in [K] for closed irreducible orientable manifolds. In particular, in [L] it is shown that the theory of finite-order invariants of knots in two-connected manifolds is isomorphic to that for R3; the main theorem of [K] asserts that for every closed oriented irreducible 3- manifold M and any connected component of the space C∞(S1, M) the group of finite- arXiv:math/9703207v1 [math.GT] 20 Mar 1997 order invariants of knots from this component contains a subgroup isomorphic to the group of finite-order invariants in R3. -
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. -
Perspective the PNAS Way Back Then
Proc. Natl. Acad. Sci. USA Vol. 94, pp. 5983–5985, June 1997 Perspective The PNAS way back then Saunders Mac Lane, Chairman Proceedings Editorial Board, 1960–1968 Department of Mathematics, University of Chicago, Chicago, IL 60637 Contributed by Saunders Mac Lane, March 27, 1997 This essay is to describe my experience with the Proceedings of ings. Bronk knew that the Proceedings then carried lots of math. the National Academy of Sciences up until 1970. Mathemati- The Council met. Detlev was not a man to put off till to cians and other scientists who were National Academy of tomorrow what might be done today. So he looked about the Science (NAS) members encouraged younger colleagues by table in that splendid Board Room, spotted the only mathe- communicating their research announcements to the Proceed- matician there, and proposed to the Council that I be made ings. For example, I can perhaps cite my own experiences. S. Chairman of the Editorial Board. Then and now the Council Eilenberg and I benefited as follows (communicated, I think, did not often disagree with the President. Probably nobody by M. H. Stone): ‘‘Natural isomorphisms in group theory’’ then knew that I had been on the editorial board of the (1942) Proc. Natl. Acad. Sci. USA 28, 537–543; and “Relations Transactions, then the flagship journal of the American Math- between homology and homotopy groups” (1943) Proc. Natl. ematical Society. Acad. Sci. USA 29, 155–158. Soon after this, the Treasurer of the Academy, concerned The second of these papers was the first introduction of a about costs, requested the introduction of page charges for geometrical object topologists now call an “Eilenberg–Mac papers published in the Proceedings. -
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.