DOCSLIB.ORG

Geometry and Topology of Cohomology Jump Loci [6Pt

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometry and Topology of Cohomology Jump Loci [6Pt GEOMETRY AND TOPOLOGY OF COHOMOLOGY JUMP LOCI LECTURE 1: CHARACTERISTIC VARIETIES Alex Suciu Northeastern University MIMS Summer School: New Trends in Topology and Geometry Mediterranean Institute for the Mathematical Sciences Tunis, Tunisia July 9–12, 2018 ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 1 / 25 OUTLINE 1 CAST OF CHARACTERS The character group The equivariant chain complex Characteristic varieties Degree1 characteristic varieties 2 EXAMPLES AND COMPUTATIONS Warm-up examples Toric complexes and RAAGs Quasi-projective manifolds 3 APPLICATIONS Homology of finite abelian covers Dwyer–Fried sets Duality and propagation ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 2 / 25 CAST OF CHARACTERS THE CHARACTER GROUP THE CHARACTER GROUP Throughout, X will be a connected CW-complex with finite q-skeleton, for some q ¥ 1. We may assume X has a single 0-cell, call it e0. 0 Let G = p1(X, e ) be the fundamental group of X: a finitely 1 1 generated group, with generators x1 = [e1], ... , xm = [em]. The character group, G = Hom(G, Cˆ) Ă (Cˆ)m is a (commutative) algebraicp group, with multiplication r ¨ r1(g) = r(g)r1(g), and identity G Ñ Cˆ, g ÞÑ 1. 1 Let Gab = G/G – H1(X, Z) be the abelianization of G. The » projection ab: G Ñ Gab induces an isomorphism Gab ÝÑ G. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMERp SCHOOLp2018 3 / 25 CAST OF CHARACTERS THE CHARACTER GROUP The identity component, G0, is isomorphic to a complex algebraic torus of dimension n = rank Gab. p The other connected components are all isomorphic to G0 = (Cˆ)n, and are indexed by the finite abelian group Tors(Gab). p Char(X ) = G is the moduli space of rank1 local systems on X: ˆ p r : G Ñ C Cr the complex vector space C, viewed as a right module over the group ring ZG via a ¨ g = r(g)a, for g P G and a P C. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 4 / 25 CAST OF CHARACTERS THE EQUIVARIANT CHAIN COMPLEX THE EQUIVARIANT CHAIN COMPLEX Let p : X Ñ X be the universal cover. The cell structure on X lifts to a cell structure on X. 0 ´1 0 0 Fixing ar lift e˜ P p (e ) identifies G = p1(X, e ) with the group of deck transformationsr of X. Thus, we may view the cellular chain complex of X as a chain complex of left ZG-modules,r r B˜i+1 B˜i ¨ ¨ ¨ / Ci+1(X, Z) / Ci (X, Z) / Ci´1(X, Z) / ¨ ¨ ¨ . ˜ 1 0 B1(e˜ i ) = (xi ´ 1)er˜ . r r ˜ 2 m f 1 B2(e˜ ) = i=1 Br/Bxi ¨ e˜ i , where r P F = xx , , x y is the word traced by the attaching map of e2; mř 1 ... m Br/Bxi P ZFm are the Fox derivatives of r; f : ZFm Ñ ZG is the linear extension of the projection Fm G. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 5 / 25 CAST OF CHARACTERS THE EQUIVARIANT CHAIN COMPLEX H˚(X, Cr) is the homology of the chain complex of C-vector spaces Cr bZG C‚(X, Z): rB˜i+1(r) B˜i (r) ¨ ¨ ¨ / Ci+1(X, C) / Ci (X, C) / Ci´1(X, C) / ¨ ¨ ¨ , where the evaluation of B˜i at r is obtained by applying the ring homomorphism ZG Ñ C, g ÞÑ r(g) to each entry of B˜i . Alternatively, consider the universal abelian cover, X ab, and its ab equivariant chain complex, C‚(X , Z) = ZGab bZG C‚(X, Z), ab with differentials Bi = id b Bi . r Then H˚(X, Cr) is computed from the resulting C-chain complex, ab ˜r with differentials Bi (r) = Bi (r). The identity1 P Char(X ) yields the trivial local system, C1 = C, and H˚(X, C) is the usual homology of X with C-coefficients. Denote by bi (X ) = dimC Hi (X, C) the ith Betti number of X. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 6 / 25 CAST OF CHARACTERS CHARACTERISTIC VARIETIES CHARACTERISTIC VARIETIES DEFINITION The characteristic varieties of X are the sets i Vk (X ) = tr P Char(X ) | dimC Hi (X, Cr) ¥ ku. i i i For each i, get stratification Char(X ) = V0 Ě V1 Ě V2 Ě ¨ ¨ ¨ i 1 P Vk (X ) ðñ bi (X ) ¥ k. 0 0 V1 (X ) = t1u and Vk (X ) = H, for k ¡ 1. i ˆ Define analogously Vk (X, |) Ă Hom(G, | ), for arbitrary field |. i i ˆ Then Vk (X, |) = Vk (X, K) X Hom(G, | ), for any | Ď K. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 7 / 25 CAST OF CHARACTERS CHARACTERISTIC VARIETIES LEMMA i For each 0 ¤ i ¤ q andk ¥ 0, the set Vk (X ) is a Zariski closed subset of the algebraic group G = Char(X ). PROOF (FOR i ă q). p Let R = C[Gab] be the coordinate ring of G = Gab. By definition, a i character r belongs to Vk (X ) if and only if p p ab ab rank Bi+1(r) + rank Bi (r) ¤ ci ´ k, where ci = ci (X ) is the number of i-cells of X. Hence, i ab ab Vk (X ) = tr P G | rank Bi+1(r) ¤ r ´ 1 or rank Bi (r) ¤ s ´ 1u r+s=c ´k+1; r,s¥0 i £ p ab ab = V Ir (Bi ) ¨ Is(Bi+1) , r+s=c ´k+1; r,s¥0 i ¸ where Ir (j) = ideal of r ˆ r minors of j. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 8 / 25 CAST OF CHARACTERS CHARACTERISTIC VARIETIES The characteristic varieties are homotopy-type invariants of a space: LEMMA SupposeX » X 1. There is then an isomorphism G 1 – G, which i 1 i restricts to isomorphisms Vk (X ) – Vk (X ), for alli ¤ q andk ¥ 0. p p PROOF. Let f : X Ñ X 1 be a (cellular) homotopy equivalence. 0 1 10 The induced homomorphism f7 : p1(X, e ) Ñ p1(X , e ), yields an 1 isomorphism of algebraic groups, fˆ7 : G Ñ G. Lifting f to a cellular homotopy equivalence, f˜ X X 1, defines x p : Ñ 1 1 isomorphisms Hi (X, Cr˝f7 ) Ñ Hi (X , Cr), for each r P G . r r ˆ i 1 i Hence, f7 restricts to isomorphisms Vk (X ) Ñ Vk (X ). p ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 9 / 25 CAST OF CHARACTERS DEGREE 1 CHARACTERISTIC VARIETIES DEGREE 1 CHARACTERISTIC VARIETIES 1 2 Vk (X ) depends only on G = p1(X ) (in fact, only on G/G ), so we 1 may write these sets as Vk (G). Suppose G = xx1, ... , xm | r1, ... , rpy is finitely presented 1 ab Away from1 P G, we have that Vk (G) = V (Ek (B1 )), the zero-set of the ideal of codimension k minors of the Alexander matrix p ab ab p m B1 = Bri /Bxj : ZGab Ñ ZGab. If j : G Q is an epimorphism, then, for each k ¥ 1, the induced monomorphism between character groups, j˚ : Q ãÑ G, restricts 1 1 to an embedding Vk (Q) ãÑ Vk (G). p p Given any subvariety W Ă (Cˆ)n defined over Z, there is a finitely n 1 presented group G such that Gab = Z and V1 (G) = W . ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 10 / 25 EXAMPLES AND COMPUTATIONS WARM-UP EXAMPLES WARM-UP EXAMPLES EXAMPLE (THE CIRCLE) We have S1 = R. 1 ˘1 Identify p1(S , ˚) = Z = xty and ZZ = Z[t ]. Then: Ă 1 ˘1 t´1 ˘1 C‚(S ) : 0 / Z[t ] / Z[t ] / 0 For r P Hom(Z, CĂˆ) = Cˆ, we get r´1 1 Cr bZZ C‚(S ) : 0 / C / C / 0 1 1 which is exact, except for rĂ= 1, when H0(S , C) = H1(S , C) = C. Hence: 0 1 1 1 V1 (S ) = V1 (S ) = t1u i 1 Vk (S ) = H, otherwise. ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 11 / 25 EXAMPLES AND COMPUTATIONS WARM-UP EXAMPLES EXAMPLE (THE n-TORUS) n n n ˆ ˆ n Identify p1(T ) = Z , and Hom(Z , C ) = (C ) . Using the Koszul n resolution C‚(T ) as above, we get n Ă i n t1u if k ¤ ( i ), Vk (T ) = #H otherwise. EXAMPLE (NILMANIFOLDS) More generally, let M be a nilmanifold. An inductive argument on the nilpotency class of p1(M), based on the Hochschild-Serre spectral sequence, yields i t1u if k ¤ bi (M), Vk (M) = #H otherwise ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 12 / 25 EXAMPLES AND COMPUTATIONS WARM-UP EXAMPLES EXAMPLE (WEDGE OF CIRCLES) n 1 ˆ ˆ n Identify p1( S ) = Fn, and Hom(Fn, C ) = (C ) . Then: Ž ˆ n n (C ) if k ă n, 1 1 Vk S = $t1u if k = n, ' ª &H if k ¡ n. %' EXAMPLE (ORIENTABLE SURFACE OF GENUS g ¡ 1) Write p1(Sg ) = xx1, ... , xg, y1, ... , yg | [x1, y1] ¨ ¨ ¨ [xg, yg ] = 1y, and ˆ ˆ 2g identify Hom(p1(Sg ), C ) = (C ) . Then: (Cˆ)2g if i = 1, k ă 2g ´ 1, i Vk (Sg ) = $t1u if i = 1, k = 2g ´ 1, 2g; or i = 2, k = 1, &'H otherwise. %' ALEX SUCIU (NORTHEASTERN) COHOMOLOGY JUMP LOCI MIMS SUMMER SCHOOL 2018 13 / 25 EXAMPLES AND COMPUTATIONS TORIC COMPLEXES TORIC COMPLEXES AND RAAGS Given L simplicial complex on n vertices, define the toric complex n TL as the subcomplex of T obtained by deleting the cells corresponding to the missing simplices of L: s s n TL = T , where T = tx P T | xi = ˚ if i R su.
Load more

Recommended publications
  • Kernelization of the Subset General Position Problem in Geometry Jean-Daniel Boissonnat, Kunal Dutta, Arijit Ghosh, Sudeshna Kolay
    Kernelization of the Subset General Position problem in Geometry Jean-Daniel Boissonnat, Kunal Dutta, Arijit Ghosh, Sudeshna Kolay To cite this version: Jean-Daniel Boissonnat, Kunal Dutta, Arijit Ghosh, Sudeshna Kolay. Kernelization of the Subset General Position problem in Geometry. MFCS 2017 - 42nd International Symposium on Mathematical Foundations of Computer Science, Aug 2017, Alborg, Denmark. 10.4230/LIPIcs.MFCS.2017.25. hal-01583101 HAL Id: hal-01583101 https://hal.inria.fr/hal-01583101 Submitted on 6 Sep 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Kernelization of the Subset General Position problem in Geometry Jean-Daniel Boissonnat1, Kunal Dutta1, Arijit Ghosh2, and Sudeshna Kolay3 1 INRIA Sophia Antipolis - Méditerranée, France 2 Indian Statistical Institute, Kolkata, India 3 Eindhoven University of Technology, Netherlands. Abstract In this paper, we consider variants of the Geometric Subset General Position problem. In defining this problem, a geometric subsystem is specified, like a subsystem of lines, hyperplanes or spheres. The input of the problem is a set of n points in Rd and a positive integer k. The objective is to find a subset of at least k input points such that this subset is in general position with respect to the specified subsystem.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Interpolation
    Current Developments in Algebraic Geometry MSRI Publications Volume 59, 2011 Interpolation JOE HARRIS This is an overview of interpolation problems: when, and how, do zero- dimensional schemes in projective space fail to impose independent conditions on hypersurfaces? 1. The interpolation problem 165 2. Reduced schemes 167 3. Fat points 170 4. Recasting the problem 174 References 175 1. The interpolation problem We give an overview of the exciting class of problems in algebraic geometry known as interpolation problems: basically, when points (or more generally zero-dimensional schemes) in projective space may fail to impose independent conditions on polynomials of a given degree, and by how much. We work over an arbitrary field K . Our starting point is this elementary theorem: Theorem 1.1. Given any z1;::: zdC1 2 K and a1;::: adC1 2 K , there is a unique f 2 K TzU of degree at most d such that f .zi / D ai ; i D 1;:::; d C 1: More generally: Theorem 1.2. Given any z1;:::; zk 2 K , natural numbers m1;:::; mk 2 N with P mi D d C 1, and ai; j 2 K; 1 ≤ i ≤ kI 0 ≤ j ≤ mi − 1; there is a unique f 2 K TzU of degree at most d such that . j/ f .zi / D ai; j for all i; j: The problem we’ll address here is simple: What can we say along the same lines for polynomials in several variables? 165 166 JOE HARRIS First, introduce some language/notation. The “starting point” statement Theorem 1.1 says that the evaluation map 0 ! L H .ᏻP1 .d// K pi is surjective; or, equivalently, 1 D h .Ᏽfp1;:::;peg.d// 0 1 for any distinct points p1;:::; pe 2 P whenever e ≤ d C 1.
    [Show full text]
  • 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.
    [Show full text]
  • Algebraic Topology
    Algebraic Topology John W. Morgan P. J. Lamberson August 21, 2003 Contents 1 Homology 5 1.1 The Simplest Homological Invariants . 5 1.1.1 Zeroth Singular Homology . 5 1.1.2 Zeroth deRham Cohomology . 6 1.1.3 Zeroth Cecˇ h Cohomology . 7 1.1.4 Zeroth Group Cohomology . 9 1.2 First Elements of Homological Algebra . 9 1.2.1 The Homology of a Chain Complex . 10 1.2.2 Variants . 11 1.2.3 The Cohomology of a Chain Complex . 11 1.2.4 The Universal Coefficient Theorem . 11 1.3 Basics of Singular Homology . 13 1.3.1 The Standard n-simplex . 13 1.3.2 First Computations . 16 1.3.3 The Homology of a Point . 17 1.3.4 The Homology of a Contractible Space . 17 1.3.5 Nice Representative One-cycles . 18 1.3.6 The First Homology of S1 . 20 1.4 An Application: The Brouwer Fixed Point Theorem . 23 2 The Axioms for Singular Homology and Some Consequences 24 2.1 The Homotopy Axiom for Singular Homology . 24 2.2 The Mayer-Vietoris Theorem for Singular Homology . 29 2.3 Relative Homology and the Long Exact Sequence of a Pair . 36 2.4 The Excision Axiom for Singular Homology . 37 2.5 The Dimension Axiom . 38 2.6 Reduced Homology . 39 1 3 Applications of Singular Homology 39 3.1 Invariance of Domain . 39 3.2 The Jordan Curve Theorem and its Generalizations . 40 3.3 Cellular (CW) Homology . 43 4 Other Homologies and Cohomologies 44 4.1 Singular Cohomology .
    [Show full text]
  • Classical Algebraic Geometry
    CLASSICAL ALGEBRAIC GEOMETRY Daniel Plaumann Universität Konstanz Summer A brief inaccurate history of algebraic geometry - Projective geometry. Emergence of ’analytic’geometry with cartesian coordinates, as opposed to ’synthetic’(axiomatic) geometry in the style of Euclid. (Celebrities: Plücker, Hesse, Cayley) - Complex analytic geometry. Powerful new tools for the study of geo- metric problems over C.(Celebrities: Abel, Jacobi, Riemann) - Classical school. Perfected the use of existing tools without any ’dog- matic’approach. (Celebrities: Castelnuovo, Segre, Severi, M. Noether) - Algebraization. Development of modern algebraic foundations (’com- mutative ring theory’) for algebraic geometry. (Celebrities: Hilbert, E. Noether, Zariski) from Modern algebraic geometry. All-encompassing abstract frameworks (schemes, stacks), greatly widening the scope of algebraic geometry. (Celebrities: Weil, Serre, Grothendieck, Deligne, Mumford) from Computational algebraic geometry Symbolic computation and dis- crete methods, many new applications. (Celebrities: Buchberger) Literature Primary source [Ha] J. Harris, Algebraic Geometry: A first course. Springer GTM () Classical algebraic geometry [BCGB] M. C. Beltrametti, E. Carletti, D. Gallarati, G. Monti Bragadin. Lectures on Curves, Sur- faces and Projective Varieties. A classical view of algebraic geometry. EMS Textbooks (translated from Italian) () [Do] I. Dolgachev. Classical Algebraic Geometry. A modern view. Cambridge UP () Algorithmic algebraic geometry [CLO] D. Cox, J. Little, D.
    [Show full text]
  • 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.
    [Show full text]
  • Bordism and Cobordism
    [ 200 ] BORDISM AND COBORDISM BY M. F. ATIYAH Received 28 March 1960 Introduction. In (10), (11) Wall determined the structure of the cobordism ring introduced by Thorn in (9). Among Wall's results is a certain exact sequence relating the oriented and unoriented cobordism groups. There is also another exact sequence, due to Rohlin(5), (6) and Dold(3) which is closely connected with that of Wall. These exact sequences are established by ad hoc methods. The purpose of this paper is to show that both these sequences are ' cohomology-type' exact sequences arising in the well-known way from mappings into a universal space. The appropriate ' cohomology' theory is constructed by taking as universal space the Thom complex MS0(n), for n large. This gives rise to (oriented) cobordism groups MS0*(X) of a space X. We consider also a 'singular homology' theory based on differentiable manifolds, and we define in this way the (oriented) bordism groups MS0*{X) of a space X. The justification for these definitions is that the main theorem of Thom (9) then gives a 'Poincare duality' isomorphism MS0*(X) ~ MSO^(X) for a compact oriented differentiable manifold X. The usual generalizations to manifolds which are open and not necessarily orientable also hold (Theorem (3-6)). The unoriented corbordism group jV'k of Thom enters the picture via the iso- k morphism (4-1) jrk ~ MS0*n- (P2n) (n large), where P2n is real projective 2%-space. The exact sequence of Wall is then just the cobordism sequence for the triad P2n, P2»-i> ^2n-2> while the exact sequence of Rohlin- Dold is essentially the corbordism sequence of the pair P2n, P2n_2.
    [Show full text]