Algebraic Topology Simplicial Homology Wayne Lawton Department of Mathematics National University of Singapore S14-04-04, matwml@nus.edu.sg http://math.nus.edu.sg/~matwml 1

Homology Groups A differential group is an abelian group and an endomorphism that satisfies is the differential or boundary operator subgroup of cycles subgroup of boundaries Remark Clearly Definition is the homology group and its elements are homology classes. 2

Oriented Simplices Definition An orientation of a d-dimensional simplex is one of two equivalence classes of orderings (permutations) of its vertices where two permutations are equivalent if they differ by an even permutations. denotes the equivalence class for permutation An oriented simplex is a simplex with an orientation. Examples Result Two permutations are equivalent iff one can be obtained from the other using an even number of transpositions. 3

Group of q-Chains Definition Let be a simplicial complex and let denote the abelian group generated by the oriented simplices in whenever together with the relations and are oriented simplices having the same simplices but inequivalent orientations. Example is a free abelian group whose rank equals the number of simplices in and is called the group of chains in 4

Boundary Operator Definition The group of chains of a simplicial complex is Lemma The homomorphisms given by where we define are well defined, satisfy and define a boundary operator by Proof 1st claim: a direct tedious computation gives whence the claim follows since every permutation is a product of transpositions (permutation switching 2 things). 5

Boundary Operator Lets do this tedious work since it is necessary ! 6

Boundary Operator Proof 2nd claim: it suffices to show that We compute The result now follows since 7

Boundary Operator Proof 3rd claim: follows since Remark (the trivial group) and is a graded group and is a graded homomorphism of degree 8

Computation of Homology Groups Definition is the th homology group of Result Example 1. Example 2. 9

Computation of Homology Groups Example 3. is represented by the following 6 x 4 matrix Linear Algebra  where and hence The sphere has one 3-dim hole 10

Computation of Homology Groups Example 4. represented 6 x 4 matrix M on preceding page and hence The ball does NOT have a 3-dim hole 11

Smith Normal Form http://en.wikipedia.org/wiki/Smith_normal_form Definition A commutative ring R is principle if every ideal in R has the form Rg for some g in R Examples Z = ring of (rational) integers, F[z] = ring of polynomials over a field F Theorem [1] Every m x n matrix A over a PID R equals where Proof and Algorithm: see the very informative article http://en.wikipedia.org/wiki/Smith_normal_form which also discusses applications to homology and the structure theorem for finitely generated modules over pid 1] Smith, H. M. S., On systems of indeterminate equations and congruences, Philos. Trans., 151(1861), 293--326. http://en.wikipedia.org/wiki/Henry_John_Stephen_Smith 12

MATLAB Code MATLAB Code available at http://mathforum.org/kb/thread.jspa?forumID=80&threadID=257763&messageID=835342#835342 >> help smith Smith normal form of an integer matrix. [U,S,V] = smith(A) returns integer matrices U, S, and V such that A = U*S*V', S is diagonal and nonnegative, S(i,i) divides S(i+1,i+1) for all i, det U =+-1, and det V =+-1. s = smith(A) just returns diag(S). Uses function ehermite. [U,S,V] = smith(A); This function is in some ways analogous to SVD. Remark The matrix V is the transpose of the matrix appearing in the previous page! 13

Example >> A = [2 4 4;-6 6 12;10 -4 -16] A = 2 4 4 -6 6 12 >> U U = 1 0 0 -12 -3 -2 17 4 3 >> V' ans = 1 2 2 -1 -3 -2 0 0 -1 >> U*S*V' 2 4 4 -6 6 12 10 -4 -16 >> A = [2 4 4;-6 6 12;10 -4 -16] A = 2 4 4 -6 6 12 10 -4 -16 >> [U,S,V] = smith(A); >> S S = 2 0 0 0 6 0 0 0 12 14

Finitely Generated Abelian Groups Theorem Every finitely generated abelian group where Proof If is generated by elements then there exists a unique homomorphism such that Since is surjective the 1st for groups gives http://en.wikipedia.org/wiki/Isomorphism_theorem where Clearly is any matrix whose columns generate Let be the Smith form of where The result follows by 15

Smith Form in Homology Consider and choose a fixed bases for then and is represented by and is represented with Smith form where so we let be its be the lower submatrix of and Smith form with Then 16

Homology of the Torus all 27 edges oriented all 18 faces oriented counterclockwise Question Compute the matrices for and use them with MATLAB to compute the homology groups of Suggestion : I attempted to compute the matrix for on the next page but did not check it for errors, the matrix should have rank = 18 - dim ker = 17 17

Homology of the Torus 18

Computation of Theorem Proof are in the same connected component of the polyhedron iff there exist a path such that The simplicial approximation theorem implies that this occurs iff there exist such that Then the chain satisfies and hence We leave it to the reader to prove the converse. 19

Why Elements in are Called Cycles We consider the case Definition A loop in K is a chain where such that Clearly and hence so every loop is a cycle. Theorem Every element in is a lin. comb. of loops. Proof Express with then construct a loop and such that has less coefficients, then use induction. We suggest that the reader work out a detailed proof. 20

Normal Subgroups Definition Let be a group. A subgroup is normal if for every we have Lemma If is a normal subgroup of then the set of left cosets of is a group (called the quotient ) under the binary operation defined by of by Moreover, if are normal subgroups then every homomorphism that satisfies induces a homomorphism defined by Proof Left to the reader. 21

Commutator Subgroups Definition Let be a group. The commutator subgroup (also called the first derived subgroup and is the subgroup generated denoted by or ) of by the set Lemma For every homomorphism Proof Left to the reader. Corollary is a normal subgroup of Proof For each construct by Clearly is a homomorphism and hence the lemma implies that for every so is normal. 22

The Abelianization Functor is abelian. Lemma If is a group then Proof It suffices to show that such that Given choose and Then compute Question For groups and and a homomorphism define by Show that is a functor from the category of groups to the category of abelian groups. Question Prove is the left adjoint of the inclusion functor. See http://en.wikipedia.org/wiki/Commutator_subgroup 23

The Abelianization Functor Theorem If is a closed surface with genus then when is orientable and when is nonorientable. Proof http://wapedia.mobi/en/Fundamental_polygon Theorem Proof To be given later. Corollary when is orientable and when is nonorientable. Proof The first assertion is left to the reader. Clearly if is nonorientable then where whence the 2nd assertion. 24

Homology from Homotopy Theorem If is a simplicial complex whose polyhdron is connected and then Proof Let be the edge group defined on page 132 in Armstrong’s Basic Topology (BT). Theorem 6.1 in BT proves (using the simplicial approximation theorem) that and hence it suffices to prove that Define by Clearly equivalent edge loops e give homologous z(e), so is well defined. is surjective since every cycle is a linear combination of ‘loops’ of the form z(g). Moreover 25

Homology from Homotopy For each choose an edge path joining to and construct the edge loop L is a max tree Since and hence Moreover implies that Therefore for every oriented edge (a,b) that occurs k-times in the oriented edge (b,a) also occurs k-times. Since by Theorem 6.12 in BT we have 26

Chain Complexes and Chain Maps Chain Complex: differential graded abelian group Chain Maps: deg 0 homomorphism that gives a commutative diagram: Theorem Chain complexes & chain maps are a category. Homology is a functor from it to the cat. of graded abelian groups&deg-0 homomorphisms Proof Left to the reader. 27

Example: Simplicial Homology Theorem If are simplicial complexes and is a simplicial map then the maps defined by: ( = 0 if are not distinct) give a chain map Proof Left to the reader. Corollary is a functor from the category of simplicial complexes & simplicial maps to the category of chain complexes and chain maps. Corollary is a functor from SIMP to GAB. 28

Example: Singular Homology let Definition For be a standard ordered n-simplex and define simplicial maps by (this maps onto ) the face of For any topological space let be the free abelian group generated by maps let the boundary operator be Theorem is a functor from category TOP to category CHAIN and = sing. hom. functor. 29

Chain Homotopy between two chain maps is a degree 1 homomorphism such that Lemma Theorem Proof 30

Chain Homotopy of chain Lemma Composites homotopic maps are chain homotopic. This means that a chain homotopy such that Proof Let Then is a degree 1 homomorphism and 31

Chain Homotopy Definition A chain equivalence between two chain complexes is a chain map such that there exists a chain map with chain homotopic to respectively. Lemma Chain equivalence same homology. Definition A chain contraction of a chain complex is a chain homotopy between and Lemma A contractible chain complex is acyclic (all its homology groups equal 0). Theorem A free acyclic chain comp.is contractible. Proof free satisfies 32

Chain Homotopy Definition An augmentation of a chain complex with is a chain complex such that Lemma If is a simplicial complex then has a unique augmentation given by Theorem If is a cone (Lecture 2, slide 8) then is acyclic. Proof Define a hom. by and Homework due 20 Oct: compute 33

Stellar Subdivision Lemma Barycentric subdivision can be obtained by a sequence of stellar subdivisions of the form where is the barycenter of Step 1 Lemma Stellar subdivision defines a chain map (the subdivision operator) by Proof (Step 1) 34

Stellar Subdivision Theorem If is a stellar subdivision of is a simplicial approximation of that induces the chain map defined by then Corollary Proof (Step 1) Theorem Proof Subcomplex of simplices containing is a cone and Finish it! 35

Contiguity Classes are Definition Simplicial maps contiguous if (delete repeated vertices) and in the same contiguity class if there exist simplicial maps with and contiguous. Lemma If are simplicial approximations to a map then and are contiguous. (review simplicial approx. on slide 17, lecture 3) Proof 36

Contiguity Classes that are Lemma Simplicial maps in the same contiguity class are homotopic. Proof Without loss of generality we may assume that and are contiguous. Proof Define a homotopy by Remark Let be a simplicial complex and realize its polyhedron as a subset of Euclidean space with norm Let be the Lebesque number for the open cover of of Hence and 37

Contiguity Classes Lemma Let be maps such that where is the Lebesgue number for the open cover of then is an open cover of and there exists and a simplicial approximation to both Proof The first assertion is obvious. Choose such that hence Then construct so that 38

Contiguity Classes Corollary If are homotopic then there exist and simplicial approximations to to and such that and are in the same contiguity class. Proof Homework Due Tuesday 27 October. 39

Contiguity Classes are simplicial maps Theorem If that are in the same contiguity class then their induced chain maps are chain homotopic. Proof We may assume that and are contiguous. For define and observe that is a subset of a simplex (called the carrier of ) in and that Assume we have homomorphisms for so that and is a chain in the the carrier of For an oriented n-simplex let Then so let Corollary 40

Invariance Lemma If and are simplicial complexes, is a map with simplicial approx. with and are subdivision chain maps (by composing stellar subdiv. maps), is a simplicial approx. of and chain maps induced by the simp. maps, then and Proof. Since both they are are simplicial approx. to contiguous so Also so 41

Invariance Theorem 1. Every map induces a homomorphism where by the rule: is a simplicial approxroximation to and is the subdivision chain map. Proof Preceding lemma  the rule is well defined. 42

Invariance satisfies Theorem 2. The identity and maps satisfy be a simplicial approx. Proof Let then let be to a simplicial approx. to be Let be subdivision chain maps and be a simplicial approx. to (a standard simp. map) and its induced chain map. Since are simp. approx. to resp. 43

Invariance Theorem 3. If are homotopic then Proof Homework due Friday 30 October 44

Applications Theorem Proof Corollary Proof Theorem (Brouwer-Fixed Point) Every map has a fixed point. Proof Otherwise there exists a retraction so that Since is contractible However and this ccontradiction concludes the proof. 45