MA5209 Algebraic Topology Wayne Lawton Department of Mathematics National University of Singapore S , Lecture 5. Simplicial Homology (18, 29 September, 2, 13, 16, 20, 23 October 2009)

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

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

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

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

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

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

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

Computation of Homology Groups Definition is theth homology group of Example 1. Result Example 2.

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

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

Smith Normal Form Theorem [1] Every m x n matrix A over a PID R equals 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 where 1] Smith, H. M. S., On systems of indeterminate equations and congruences, Philos. Trans., 151(1861), Proof and Algorithm: see the very informative article which also discusses applications to homology and the structure theorem for finitely generated modules over pid

MATLAB Code MATLAB Code available at >> 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!

Example >> A = [2 4 4; ; ] A = >> [U,S,V] = smith(A); >> S S = >> U U = >> V' ans = >> U*S*V' ans =

Finitely Generated Abelian Groups Theorem Every finitely generated abelian group Proof If where is generated by elementsthen there exists a unique homomorphism such thatSinceis surjective the 1 st Clearly for groups gives where is any matrix whose columns generate Letbe the Smith form ofwhere The result follows by

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

Homology of the Torus all 27 edges oriented all 18 faces oriented counterclockwise Question Compute the matrices for 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 and use

Homology of the Torus

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

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

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

Commutator Subgroups Definition Letbe a group. The commutator subgroup (also called the first derived subgroup and ordenoted by by the set ) Proof Left to the reader. Lemma For every homomorphism of is the subgroup generated is a normal subgroup of constructProof For each Corollary by Clearlyis a homomorphism and hence the lemma implies that for every sois normal.

The Abelianization Functor Lemma If define is abelian. is a group then Proof It suffices to show that Then compute Given Question For groups and such that choose and a homomorphismand by Show thatis a functor from the category of groups to the category of abelian groups. Question Prove is the left adjoint of the inclusion functor. See left adjoint

The Abelianization Functor Theorem If is orientable and is a closed surface with genus when then Proof Theorem Corollary whenis nonorientable. whenis orientable and Proof To be given later. whenis nonorientable. Proof The first assertion is left to the reader. Clearly if is nonorientable thenwhere whence the 2 nd assertion.

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

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

Chain Complexes and Chain Maps Chain Complex: differential graded abelian group 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 Chain Maps: deg 0 homomorphism Proof Left to the reader.

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

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

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

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

Chain Homotopy that there exists a chain map Definition A chain contraction of a chain complex such Lemma Chain equivalence Definition A chain equivalence between two chain complexes is a chain map withchain homotopic torespectively. same homology. is a chain homotopy betweenand Lemma A contractible chain complex is acyclic Theorem A free acyclic chain comp.is contractible. Prooffree satisfies (all its homology groups equal 0).

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

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

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

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

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

Contiguity Classes Lemma Let is the Lebesgue number for the open cover be maps such that where of thenis an open cover ofand there existsand a simplicial approximationto both Proof The first assertion is obvious. Choose such that hence Then constructso that

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

Contiguity Classes Theorem If are simplicial maps that are in the same contiguity class then their For induced chain maps andare contiguous. are chain homotopic. Proof We may assume that defineand observe thatis a subset of a simplex (called the carrier of) inand that Assume we have homomorphisms forso that andis a chain in the the carrier ofFor an oriented n-simplexlet Thenso let Corollary

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

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

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

Invariance Theorem 3. Ifare homotopic Proof Homework due Friday 30 October then

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