DOCSLIB.ORG

Problems on Simplicial Complexes and Their Face Rings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Problems on Simplicial Complexes and Their Face Rings Problems on Simplicial Complexes and their Face Rings In the problems below, A always denotes a finite simplicial complex of dimension d — 1 with vertex set V and face ring fc[A]. Properties like "acyclic," "Cohen-Macaulay," etc., are always taken with respect to a fixed ground field k. 1. Let A be a simplicial complex with 159 3-dimensional faces. For i < 2, find the smallest possible number of i~faces of A. For i > 4, find the largest possible number of i-faces of A. 2. A {d — l)-dimensional simplicial complex A is pure if all its facets (= maximal faces) have dimension d — 1. Let (/o, /ij • • •, fd-i) be the /-vector of a pure {d — l)-dimensional simplicial complex. (a) Show that the vector (/d_2, /d-a,..., /o, 1) is the /-vector of a simplicial complex. (b) Show that fi < fd-2-i for -1 < i < [{d - 3)/2j, and that fo < fl < '" < f[id-l)/2i' (c) (unsolved) Characterize (or at least obtain significant new con­ ditions on) the /-vector of a pure simplicial complex. (A com­ plete characterization is probably hopeless, as it would give, for instance, all orders of finite projective planes.) 3. Give an example of a pure simplicial complex A which fails for some i to satisfy the "Upper Bound Inequality" hi < (^~^^*~M. Here n = /o(A), the number of vertices of A. (There exists such an example with /i(A) = (1,2,3,4,5, -26,23, -8,1).) 4. Give an example of two simplicial complexes Ai and A2 such that I All ^ IA2I, but such that /i(Ai) > 0 and /^(Aa) ^ 0. What is the smallest possible dimension of Ai and A2? (The symbol ^ denotes homeomorphism of topological spaces.) 5. Let A be a (d—l)-dimensional simplicial complex. For 0 < j < d—1, define the j-skeleton Aj of A by A^ = {F G A : dimF < j}. Express the /i-vector /i(Ad-2) in terms of /i(A). 6. Given integers n > d > 1, let A(n, d) be the simplicial complex with vertex set [n] whose facets consist of all d-subsets F of [n] with the 136 Problems on Simplicial Complexes the following property: If {i, i -h 1,..., j} C F, i — 1 0 F, j +1 ^ F, 2 > 1, and j < n, then j — i is odd. Find the /i-vector /i(A(n,d)) of A(n,rf). 7. Let r and A be simplicial complexes on disjoint vertex sets V and VK, respectively. Define the join F * A to be the simplicial complex on the vertex setVUW with faces FuG, where F G F and G € A. (If F consists of a single point, then F * A is the cone over A. If F consists of two disjoint points, then F * A is the suspension of A.) (a) Compute the /i-vector h{r * A) in terms of /i(F) and /i(A). (b) The boundary A(d) of the {d ~ I)-dimensional cross-polytope (as an abstract simplicial complex) has vertex set V = {xi,..., ^diVii" ",yd}^ with F C V a face of A(rf) if and only if {xuVi} 2 F for all i. Find h{A{d)). 8. Let A be a triangulation of a (d ~ l)-sphere, d> 3. (a) Show that the link of every vertex is simply-connected (i.e., the fundamental group of \\k{v)\ is trivial for every vertex v). (b) (very difficult) Show that for d > 6, the link of an edge need not be simply-connected. 9. Let A be a (d - l)-dimensional simplicial complex. (a) Let a : A —^ A be an automorphism of A of prime order p. Suppose that for all nonempty faces F of A we have a{F) ^ F. Let (/lo,..., /id) be the /i-vector of A. Show that /ii(A) = (-l)»(^^(modp). Deduce that x(^) = — 1 (mod p), (b) More generally, suppose G is a group of automorphisms of A, and let #G = g. Suppose that for all 1 ^^ a € G and 0 7?^ F G A we have (T(F) ^ F. Show that the congruences in (a) are still valid, with p replaced by g. 10. Let A be a (d — l)-dimensional simplicial complex with /-vector (/o? • • •) id-\) and /i-vector (fto? • • • ? ^d)* We say that A satisfies the Problems on Simplicial Complexes 137 Dehn'Sommerville equations if hi = hd-i for all i. Show that if d is even and A satisfies the Dehn-Sommerville equations, then /o - /i + /3 - /4 H- /e - A + /9 - /lo + • •' = 0. 11. Show that there does not exist a triangulation A of a ball B**""^ for d > 3 such that every facet intersects the boundary in a (rf — 2)-face together with the opposite vertex. 12. Let A be a triangulation of a (d— l)-dimensional ball, with h(A) = (/io,/ii,...,/irf)- (a) Show that h{dA) — {ho — hd, ho-\- hi — hd — hd-i, ho-i- hi -\- h2 — hd — hd-i — hd-2, ..., /lo -f h hd-i — hd — ' " — hi). (b) (unsolved) Show that hi > hd-i for 0 < i < [d/2j. (c) Show that hi = hd-i for 1 < z < d — 1 if and only if the boundary of A is the boundary of a simplex (i.e., has exactly d facets). (d) (unsolved) Characterize /i-vectors of balls. 13. Let A be a pure {d ~ l)-dimensional simplicial complex. A face of A of dimension < d — 1 is called free if it lies on exactly one facet. (a) If A is acyclic, must there exist a free face? What if A is contractible? (b) What if also A is shellable? 14. Show that if A is a shellable simplicial complex, then so is the barycentric subdivision sd(A). (We can define sd(A) to be the order complex of the poset of nonempty faces of A.) 15. Let A be a pure (d — l)-dimensional simplicial complex, with h- vector /i(A) = (/IQ, /ii,..., /i^). Let T be the order complex of the poset of nonempty faces of A. Thus F is a balanced complex with colors 1,2,..., d, where an i-dimensional face of A (regarded as a 138 Problems on Simplicial Complexes vertex of T) is colored i+1. Let 5 C [d]. Show that the flag /i-vector of r is given by t=0 where Dd^i{Sji + 1) denotes the number of permutations w = W1W2 ' • • Wd^i of [d+1] such that S := [rf+1] — 5 = {j : u/j > Wj^i] (the descent set of ti;) and K;d^_i = z + 1. 16. Find explicitly every simplicial complex A with the property that every ordering of its facets is a shelling. 17. Let P be a finite graded poset with ranks 1,2,...,n. Let S C [n]. Show that if P is shellable (i.e., the order complex 0{P) is shellable), then so is the rank-selected subposet P5. 18. A d'pyramid V is the convex hull of a (d— l)-polytope Q, called the basis of Vy and a point x ^ aff Q (= the affine span of Q), called the apex of P. Show that /i(P) = /i{Q) + /i-i(Q), with the convention /-i(Q) = /d-i(Q) = 1- 19. Let Q be a (d — l)-polytope and / a closed line segment such that relint(Q) n relint(/) is a single point. (Here relint denotes relative interior.) Then the d-polytope P = conv(Q U /) is called a d- bipyramid with basis Q, Show that /i(^) = /i(Q) + 2/i-i(Q), 0<z<d~2 /rf,l(P) = 2/,^2(Q). 20. Show that the Euler relation /o-/i+- • -f-(-l)''~7d~i = l+(-l)''"^ is the most general linear relation holding among the components of the /-vector of a d-polytope. Equivalently, the affine span in R*' of all /-vectors of rf-polytopes has codimension 1 (or dimension d~ 1). HINT: Use the previous two problems. 21. Show that the Dehn-Sommerville equations are the most general linear relations holding among the components of the /-vector (or /i-vector) of a simplicial d-polytope. Problems on Simplicial Complexes 139 22. A convex d-polytope V is cubical if every fax^et (or equivalently every proper face) is combinatorially equivalent to a cube of the appropriate dimension. Let Cd denote the set of all /-vectors of cubical d-polytopes. (a) Show that the afiine span aff Q of C has dimension at most [d/2\. (b) By constructing suitable examples, show that in fact aff Cd has dimension exactly [d/2j. (c) Show that a cubical polytope of dimension at least four has an even number of vertices. (The conclusion is false for three- dimensional cubical poly topes.) (d) (surprisingly difficult) Show that a cubical d-polytope has at least 2^ vertices. (e) Find a cubical 4-polytope for which /a > /o- (f) (unsolved) Given d and n, what is the maximum number of z-dimensional faces of a cubical cf-polytope with n vertices? 23. (generalizing (d) above) Suppose that the d-polytope V has no tri­ angular 2-face. Show that for every facet F of P, there is a facet G disjoint from F. Deduce that V has at least 2^ vertices. 24. Here we give two unsolved problems due to Kalai which illustrate our abysmal ignorance of certain aspects of convex poly topes. (a) Show that if P is a centrally-symmetric d-polytope, then /_i -h /o + • • • + /d-i > 3^. (b) Let j be a positive integer.
Load more

Recommended publications
  • K-THEORY of FUNCTION RINGS Theorem 7.3. If X Is A
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 69 (1990) 33-50 33 North-Holland K-THEORY OF FUNCTION RINGS Thomas FISCHER Johannes Gutenberg-Universitit Mainz, Saarstrage 21, D-6500 Mainz, FRG Communicated by C.A. Weibel Received 15 December 1987 Revised 9 November 1989 The ring R of continuous functions on a compact topological space X with values in IR or 0Z is considered. It is shown that the algebraic K-theory of such rings with coefficients in iZ/kH, k any positive integer, agrees with the topological K-theory of the underlying space X with the same coefficient rings. The proof is based on the result that the map from R6 (R with discrete topology) to R (R with compact-open topology) induces a natural isomorphism between the homologies with coefficients in Z/kh of the classifying spaces of the respective infinite general linear groups. Some remarks on the situation with X not compact are added. 0. Introduction For a topological space X, let C(X, C) be the ring of continuous complex valued functions on X, C(X, IR) the ring of continuous real valued functions on X. KU*(X) is the topological complex K-theory of X, KO*(X) the topological real K-theory of X. The main goal of this paper is the following result: Theorem 7.3. If X is a compact space, k any positive integer, then there are natural isomorphisms: K,(C(X, C), Z/kZ) = KU-‘(X, Z’/kZ), K;(C(X, lR), Z/kZ) = KO -‘(X, UkZ) for all i 2 0.
    [Show full text]
  • 1 CW Complex, Cellular Homology/Cohomology
    Our goal is to develop a method to compute cohomology algebra and rational homotopy group of fiber bundles. 1 CW complex, cellular homology/cohomology Definition 1. (Attaching space with maps) Given topological spaces X; Y , closed subset A ⊂ X, and continuous map f : A ! y. We define X [f Y , X t Y / ∼ n n n−1 n where x ∼ y if x 2 A and f(x) = y. In the case X = D , A = @D = S , D [f X is said to be obtained by attaching to X the cell (Dn; f). n−1 n n Proposition 1. If f; g : S ! X are homotopic, then D [f X and D [g X are homotopic. Proof. Let F : Sn−1 × I ! X be the homotopy between f; g. Then in fact n n n D [f X ∼ (D × I) [F X ∼ D [g X Definition 2. (Cell space, cell complex, cellular map) 1. A cell space is a topological space obtained from a finite set of points by iterating the procedure of attaching cells of arbitrary dimension, with the condition that only finitely many cells of each dimension are attached. 2. If each cell is attached to cells of lower dimension, then the cell space X is called a cell complex. Define the n−skeleton of X to be the subcomplex consisting of cells of dimension less than n, denoted by Xn. 3. A continuous map f between cell complexes X; Y is called cellular if it sends Xk to Yk for all k. Proposition 2. 1. Every cell space is homotopic to a cell complex.
    [Show full text]
  • Local Homology of Abstract Simplicial Complexes
    Local homology of abstract simplicial complexes Michael Robinson Chris Capraro Cliff Joslyn Emilie Purvine Brenda Praggastis Stephen Ranshous Arun Sathanur May 30, 2018 Abstract This survey describes some useful properties of the local homology of abstract simplicial complexes. Although the existing literature on local homology is somewhat dispersed, it is largely dedicated to the study of manifolds, submanifolds, or samplings thereof. While this is a vital per- spective, the focus of this survey is squarely on the local homology of abstract simplicial complexes. Our motivation comes from the needs of the analysis of hypergraphs and graphs. In addition to presenting many classical facts in a unified way, this survey presents a few new results about how local homology generalizes useful tools from graph theory. The survey ends with a statistical comparison of graph invariants with local homology. Contents 1 Introduction 2 2 Historical discussion 4 3 Theoretical groundwork 6 3.1 Representing data with simplicial complexes . .7 3.2 Relative simplicial homology . .8 3.3 Local homology . .9 arXiv:1805.11547v1 [math.AT] 29 May 2018 4 Local homology of graphs 12 4.1 Basic graph definitions . 14 4.2 Local homology at a vertex . 15 5 Local homology of general complexes 18 5.1 Stratification detection . 19 5.2 Neighborhood filtration . 24 6 Computational considerations 25 1 7 Statistical comparison with graph invariants 26 7.1 Graph invariants used in our comparison . 27 7.2 Comparison methodology . 27 7.3 Dataset description . 28 7.4 The Karate graph . 28 7.5 The Erd}os-R´enyi graph . 30 7.6 The Barabasi-Albert graph .
    [Show full text]
  • Homology and Homological Algebra, D. Chan
    HOMOLOGY AND HOMOLOGICAL ALGEBRA, D. CHAN 1. Simplicial complexes Motivating question for algebraic topology: how to tell apart two topological spaces? One possible solution is to find distinguishing features, or invariants. These will be homology groups. How do we build topological spaces and record on computer (that is, finite set of data)? N Definition 1.1. Let a0, . , an ∈ R . The span of a0, . , an is ( n ) X a0 . an := λiai | λi > 0, λ1 + ... + λn = 1 i=0 = convex hull of {a0, . , an}. The points a0, . , an are geometrically independent if a1 − a0, . , an − a0 is a linearly independent set over R. Note that this is independent of the order of a0, . , an. In this case, we say that the simplex Pn a0 . an is n -dimensional, or an n -simplex. Given a point i=1 λiai belonging to an n-simplex, we say it has barycentric coordinates (λ0, . , λn). One can use geometric independence to show that this is well defined. A (proper) face of a simplex σ = a0 . an is a simplex spanned by a (proper) subset of {a0, . , an}. Example 1.2. (1) A 1-simplex, a0a1, is a line segment, a 2-simplex, a0a1a2, is a triangle, a 3-simplex, a0a1a2a3 is a tetrahedron, etc. (2) The points a0, a1, a2 are geometrically independent if they are distinct and not collinear. (3) Midpoint of a0a1 has barycentric coordinates (1/2, 1/2). (4) Let a0 . a3 be a 3-simplex, then the proper faces are the simplexes ai1 ai2 ai3 , ai4 ai5 , ai6 where 0 6 i1, .
    [Show full text]
  • Algebraic Topology
    algebraic topology Lectures delivered by Michael Hopkins Notes by Akhil Mathew Fall 2010, Harvard Last updated 11/29/2010 Contents Lecture 1 9/1 Lecture 2 9/3 x1 A basic construction 7 x2 Definition 8 x3 Simplicial complexes 9 x4 ∆-complex 10 Lecture 3 9/8 x1 Chain complexes from ∆-complexes 12 Lecture 4 9-10 Lecture 5 9/13 x1 Completion of the proof of homotopy invariance 17 x2 Excision 18 Lecture 6 [Section] 9/13 x1 A discussion of naturality 20 Lecture 7 9/15 x1 Excision 22 x2 Some algebra 24 Lecture 8 9/17 Lecture 9 9/20 x1 Some algebra 29 x2 Relative homology 31 x3 A substantial theorem 32 Lecture 10 9/22 x1 Finishing up last week 35 x2 Triples 36 x3 Another variant; homology of the sphere 36 x4 Equivalence of simplicial and singular homology 38 1 Lecture 11 9/24 x1 Degree of a map 39 x2 Computing the degree 42 Lecture 12 9/27 x1 Recap 42 x2 Degree can be calculated locally 43 x3 Cellular homology 44 Lecture 13 [Section] 9/27 x1 Jordan curve theorem 46 x2 Suspensions 47 x3 Example of cellular homology 47 Lecture 14 9/29 x1 An application of degree 47 x2 Cellular homology 48 Lecture 15 10/1 x1 The cellular boundary formula 50 x2 Examples 51 Lecture 16 10/4 x1 Lefschetz fixed point formula 53 x2 Simplicial approximation 55 x3 Proof of the theorem 55 Lecture 17 [Section] 10/4 Lecture 18 10/6 x1 Simplicial approximation theorem 57 x2 Stars 58 x3 Proof of the simplicial approximation theorem 58 x4 Lefschetz fixed point theorem 59 Lecture 19 10/8 x1 Tensor products 60 x2 Torsion products 62 x3 Homology with coefficients 63 x4 A loose end: the trace on a f.g.
    [Show full text]
  • 07-20-2015 Contents 1. Axioms for a Homology Theory and Excision 1 2
    07-20-2015 Contents 1. Axioms for a homology theory and Excision 1 2. Axioms for a Reduced homology theory 5 3. Simplicial Homology 5 4. Equivalence of singular and simplicial homology 6 5. The simplicial complex of a poset 8 P 6. Deriving more properties for H∗ 9 1. Axioms for a homology theory and Excision We summarize what we've shown: Proposition 1.1. Let H∗ : pairs in Top ! Ab∗ be the singular homology functor. Then (1) (Homotopy Invariance) If f ' g :(X; A) ! (Y; B), then H∗(f) = H∗(g): H∗(X; A) ! H∗(Y; B) (2) (Dimension Axiom) H0(∗) = Z and Hn(∗) = 0 if n > 0. ` ∼ L (3) (Additivity) If (X; A) = (Xi;Ai), then H∗(X; A) = i H∗(Xi;Ai). (4) (Exactness) If A ⊂ X, there is a long exact sequence ::: ! Hn(A) ! Hn(X) ! Hn(X; A) ! Hn−1(A) ! ::: This exact sequence is natural. Given a map of pairs : (X; A) ! (Y; B), we get an induced map of long exact sequences: ::: / Hn(A) / Hn(X) / Hn(X; A) / Hn−1(A) / ::: ::: / Hn(B) / Hn(Y ) / Hn(Y; B) / Hn−1(B) / ::: Our next goal is to explain why singular homology also satisfies the following axioms: (5) (Excision) Suppose that A ⊂ V ⊂ X is such that A ⊂ int(V ). Then the inclusion i :(X − A; V − A) ! (X; V ) induces an isomorphism ∼= H∗(i): H∗(X − A; V − A) −! H∗(X; V ): (6) (Weak Equivalence) If f : X ! Y is a weak equivalence and induces a weak equivalence on subspaces f : A ! B, then H∗(f): H∗(X; A) ! H∗(Y; B) is an isomorphism.
    [Show full text]
  • CW-Complexes
    CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a col- n n lection of characteristic maps φα : D −!X. Here n ranges over the non- negative integers, Dn is the unit n-disc, and α ranges over some index set (or one index set for each n, if you prefer) that is customarily suppressed from the notation. These maps are subject to the following axioms: n n A1. The restriction of each φα to the interior of D is an embedding. n n n n A2. Let eα = φα(Int D ). Then as a set, X is the disjoint union of the eα. n (The eα are called the cells of X.) n n−1 A3. φα(S ) lies in a finite union of cells of lower dimension (i.e. of dimen- sion < n). n −1 n A4. A subset W of X is closed if and only if (φα) W is closed in D for all n; α. The term CW-complex comes from “closure-finite with the weak topol- ogy", where “closure-finite” refers to A3 and \weak topology" refers to A4. A finite complex is a CW-complex with only finitely many cells. Observe that if X is a finite complex, A4 is redundant, since W is the union of the n n −1 compact sets φα((φα) W ), and these are closed since X is Hausdorff. If X has cells of dimension n but no cells of higher dimension, we say that X is n-dimensional.
    [Show full text]
  • Introduction to Simplicial Homology (Work in Progress)
    Introduction to simplicial homology (work in progress) Jakob Jonsson February 3, 2011 2 Contents 0 Algebraic preliminaries 5 0.1 Groups,rings,andfields . 5 0.1.1 Groups ............................ 5 0.1.2 Rings ............................. 6 0.1.3 Fields ............................. 6 0.2 Modules................................ 7 0.2.1 Definitionofvectorspacesandmodules . 7 0.2.2 Homomorphisms and isomorphisms . 7 0.2.3 Sumsanddirectsums .. .. .. .. .. .. .. .. 8 0.2.4 Freemodules ......................... 9 0.2.5 Finitelygeneratedabeliangroups. 9 0.2.6 Quotients ........................... 10 0.2.7 Isomorphismtheorems . 11 1 Simplicial homology 15 1.1 Simplicialcomplexes . 15 1.1.1 Abstractsimplicialcomplexes . 15 1.1.2 Geometric realizations of simplicial complexes . ... 16 1.2 Orientedsimplices .......................... 18 1.3 Boundarymaps............................ 20 1.4 Orientationsoforientedsimplices . 22 1.5 Productsofchaingroupelements . 24 1.6 Definitionofsimplicialhomology . 25 1.7 Explicithomologycomputations . 26 1.7.1 Runningexample1. 26 1.7.2 Runningexample2. 27 1.7.3 Runningexample3. 29 1.8 Homologyinlowdegrees. 31 1.8.1 Homology in degree −1 ................... 31 1.8.2 Homologyindegree0 . 32 1.8.3 Homology in degree 1 for 1-dimensional complexes . 32 2 Combinatorial techniques 35 2.1 Generalchaincomplexes. 35 2.2 Splittingchaincomplexes . 36 2.3 Collapses ............................... 38 2.4 Collapsinginpractice . 41 2.5 BasicsofdiscreteMorsetheory . 46 3 4 CONTENTS 2.6 Joins,deletions,andlinks . 49 2.6.1 Joins.............................. 49 2.6.2 Linksanddeletions. 51 2.7 Independencecomplexes . 53 2.7.1 Example............................ 54 3 Algebraic techniques 57 3.1 Exactsequences............................ 57 3.2 Chainmapsandchainisomorphisms . 58 3.3 Relativehomology .......................... 59 3.4 Mayer-Vietorissequences . 63 3.5 Independencecomplexesrevisited. 65 Chapter 0 Algebraic preliminaries The study of simplicial homology requires basic knowledge of some fundamental concepts from abstract algebra.
    [Show full text]
  • Duality Theorems for Simplicial Complexes
    FACULTY OF SCIENCE UNIVERSITY OF COPENHAGEN Department of Mathematical Sciences, University of Copenhagen Bachelor thesis in mathematics Jens Jakob Kjær Duality theorems for simplicial complexes Advisor: Jesper Michael Møller June 4th, 2011 Abstract This thesis deals with the homology and cohomology groups of simplicial complexes, and especially with the two duality theorems which willdemon- strate links between the two notions. For this we will need some results, and we will therefore prove the long exact sequence for the reduced relative homology group; and to link the homology and cohomolgy to our geometric notion of a simplicial complex we will see that these are connected to the Euler characteristic of the complex. Then by rewriting the relative homol- ogy group we will be able to prove the Alexander duality which is the major theorem of this thesis. We will use the result that if realizations of two different simplicial complexes are homotopic then their homology and coho- mology groups are isomorphic in order to prove another duality, by showing that there is a different complex constructed by the nonfaces of our complex which is homotopic to the Alexander dual of the complex. For this we shall need some homotopy theorems which will be duly proven. Resum´e Dette projekt beskæftiger sig med homologi- og kohomologigrupperne af simplicielle komplekser, og specielt med to dualitets sætninger som vil vise forbindelsen mellem disse to strukturer. For at gøre dette vil vi bevise den lange eksakte sekvens for den reducerede relative homologigruppe, og for at forbinde homologi med kohomologi til vores geometriske intuition for simplicielle komplekser vil vi vise, at begge begreber er forbundet til Euler- karakteristikken af et kompleks.
    [Show full text]
  • Simplicial Complexes
    Simplicial complexes 1. First definitions. An (abstract) simplicial complex ∆ on a finite set S is a collection of subsets of S, closed under the operation of taking subsets. The elements of a simplicial complex ∆ are called faces. An element σ 2 ∆ of cardinality i + 1 is called an i-dimensional face or an i-face of ∆. The empty set, ;, is the unique face of dimension −1. Faces of dimension 0, i.e., elements of S, are vertices and faces of dimension 1 are edges. The maximal faces under inclusion are called facets. To describe a simplicial complex, it is often convenient to simply list its facets|the other faces are exactly determined as subsets. The dimension of ∆, denoted dim(∆), is defined to be the maximum of the dimensions of its faces. A simplicial complex is pure if each of its facets has dimension dim(∆). Example 1. If G = (V; E) is a simple connected graph (undirected with no multiple edges or loops), then G is the pure one-dimensional simplicial complex on V with E as its set of facets. Example 2. Figure1 pictures a simplicial complex ∆ on the set [5] := f1; 2; 3; 4; 5g: ∆ := f;; 1; 2; 3; 4; 5; 12; 13; 23; 24; 34; 123g; writing, for instance, 23 to represent the set f2; 3g. 3 1 4 5 2 Figure 1. A 2-dimensional simplicial complex, ∆. The sets of faces of each dimension are: F−1 = f;g F0 = f1; 2; 3; 4; 5g F1 = f12; 13; 23; 24; 34g F2 = f123g: Its facets are 5, 24, 34, and 123.
    [Show full text]
  • Algebraic Topology I
    Lectures on Algebraic Topology I Lectures by Haynes Miller Notes based on a liveTEXed record made by Sanath Devalapurkar Images created by Xianglong Ni Fall 2016 Revised August 2019 i ii Preface Over the 2016–2017 academic year, I ran the graduate algebraic topology sequence at MIT. The first semester traditionally deals with singular homology and cohomology and Poincaré duality; the second builds up basic homotopy theory, spectral sequences, and characteristic classes. My goal was to give a pretty standard classical approach to these subjects. In the first semester, I had various more specific objectives as well. I wanted to introduce students to the basic language of category theory and simplicial sets, so useful throughout mathematics and finding their first real manifestations in algebraic topology. I wanted to stress the methods of homological algebra, for similar reasons. And I especially wanted to give an honest account of the machinery – relative cap product and Čech cohomology – needed in the proof of Poincaré duality. The present document contains a bit more detail on these last matters than was presented in the course itself. On the other hand I barely touched on some important subjects. I did not talk about simplicial complexes at all, nor about the Lefschetz fixed point theorem. I gave only a brief summary of the theory of covering spaces and the fundamental group, in preparation for a proper understanding of orientations. I avoided some point set topology by working with only compact subspaces rather than general closed subspaces in the development of Poincaré duality. I was lucky enough to have in the audience a student, Sanath Devalapurkar, who spontaneously decided to liveTEX the entire course.
    [Show full text]
  • Algebraic Topology
    Algebraic Topology DRAFT 2013 - U.T. Contents 0 Introduction 2 1 Chain complexes 2 2 Simplicial homology 4 3 Singular homology 6 4 Chain homotopy and homotopy invariance 8 4.1 Chain homotopy . .8 4.2 Homotopy invariance . .9 5 The Snake Lemma and relative homology 10 6 The Five Lemma and excision 12 6.1 Atool ............................................. 12 6.2 Excision . 13 6.3 Locality . 14 6.4 Mayer-Vietoris sequence . 15 7 The degree of a map of spheres 15 8 Cellular homology 17 9 Cohomology and the Universal Coefficient Theorem 20 9.1 Cohomology . 20 9.2 Universal Coefficient Theorem . 21 1 0 Introduction We study topological spaces by associating algebraic objects which are invariant under homotopy (deformation). Sets: number of path components π0 Numbers: Euler characteristic χ winding number w γ Groups: fundamental group π1 ( ) Graded abelian groups: Homology H∗ Graded rings: Cohomology H∗ This course will introduce and study homology and cohomology. Why? • Nonabelian groups are difficult to work with • π1 only tells us something about the 2-skeleton of a space n • higher dimensional analogues, πn fhomotopy classes of based maps from S to the space k of interestg, are very hard to compute. Even πn S are not all know for n k. = Homology, H∗, is a refinement of the Euler characteristic and still relatively easy> to compute. References Everything contained in this course is covered in • chapters 2 and 3 of `Algebraic Topology' by Allen Hatcher, available online at http://www.math.cornell.edu/∼hatcher/ 1 Chain complexes Definition 1.1. Let C0;C1;::: be abelian groups (or vector spaces) and let @n Cn Cn−1 be homomorphisms (or linear maps).
    [Show full text]