1. MA5209 Algebraic Topology Wayne Lawton Department of Mathematics National University of Singapore S14-04-04, matwml@nus.edu.sgmatwml@nus.edu.sg http://math.nus.edu.sg/~matwml Lecture 7. Relative Homology Pairs and Triangulable Pairs Relative Chain Groups and Homology (Homework) Exact Sequences of Chain Complexes The Exact Homology Sequence (Homework) The Mayer-Vietoris Sequence Axiomatic Homology Theory (3, 6 November 2009)

2. Pairs and Triangulable Pairs Theorem If to the pair Definition The category of pairs has objects and for objectsand where We observe that is a topological space and We will call these morphisms simply maps from the pair Definition A pair is triangulable if there exists a simplicial complex and a subcomplexsuch that the pair (category) isomorphic to This means is

3. Homework Due Friday 30 October Theorem If be a subcomplex. The group of q-chains of be a simplicial complex and let modulois the quotient group Definition Let Question 1 Prove thatis free and describe a basis. Question 2 Explain how to construct a chain complex by defining appropriate boundary operators. Question 3. Letand letthe collection of proper faces ofCompute the homology groupsof this chain complex.

4. Exact Sequence of Chain Complexes Theorem If monomorphism Definition A short exact sequence of groups such that A short exact sequence of chain maps of chain complexes such that for every integer is a short exact sequence of (abelian) groups. the sequence and epimorphism consists of a consists Example

5. Exact Sequence of Chain Complexes then for each integer Therefore, if SES of chain complexes commutes and each row is exact is a the following diagram

6. Exact Sequence of Chain Complexes Lemma Proof exactness of last row 

7. Homework Due Tuesday 3 November Theorem If Question 1 Prove that determines so so gives a map calls this map the connecting homomorphism. defines a mapSpanier is exact (called the Exact Homology Sequence). Question 2 Prove that QUESTIONS refer to diagram&Lemma on preceding page Question 3 Prove that Question 4 Prove that the (infinite) sequence

8. The Mayer-Vietoris Short Exact Sequence such that Let be a simplicial complex and let be subcomplexes of and Let Theorem This gives a short exact sequence be chain maps induced by inclusions and

9. The Mayer-Vietoris Sequence Proof Follows from the previous theorem Theorem The MV sequence below is exact. together with the Zig-Zag Lemma assigned for the last homework (2 slides before). A detailed proof of the ZZ lemma is given in the recent handouts (from Munkre’s Elements of Algebraic Topology).

10. The MV Sequence for Topological Spaces is a topological space andIfare subspaces such that the pairs are triangulable and and then (it can be proved) that we have the exact MV sequence for the pair

11. Application of the Mayer-Vietoris Sequence Theorem Ifis a triangulable top. space then suspension Where the is the quotient space of Proof andwhere so cones  and since is an isomorphism.

12. Homework Due Friday 6 November Theorem If Question 1 Prove the Theorem on slide 8 that the Question 2 Consider the 3 dimensional torus MV short exact sequence is an exact sequence. This space is triangulable but you don’t need to actually triangulate it to answer this question. i. Considerwhere and compute the MV sequence for the pair ii. Use this MV seq. to compute

13. Homework Due Tuesday 10 November Theorem If Question 1 Consider a map where i. Construct a linear map such that and ii. Derive a formula for the sequence induces a map that satisfies in terms of the eigenvalues of the matrix that represents the linear map iii. Prove that the Lefschetz zeta function - see http://en.wikipedia.org/wiki/Lefschetz_zeta_function ofis a rational function. iv. Discuss implications for periodic points of

14. Admissible Categories of Pairs Theorem If Definition A class P of pairs of topological spaces is admissible if 1. 2. 3. Examples Polyhedral pairs, pairs of topological spaces, CW complex pairs.

15. Axiomatic Homology Theorem If Definition A homology theoryand i. a covariant functor consists of from an admissible category of topological pairs to the category of graded abelian groups and homomorphisms of ii. a natural transformationof degree -1 from to (so) that satisfies the following four axioms: degree 0.

16. Four Axioms of Homology Theorem If1. Homotopy Ifare homotopic then 2. Exactness For any pair with inclusions there is an exact sequence 3. Excision For any pairand open withthe excision map induces an isomorphism 4. Dimension Reference p.199-200 Spanier Algebraic Topology

17. Examples of Homology Theories Theorem If Simplicial Theory Admissible Pairs Singular Polydedral Topological Cech ? Poly. or CW CellularCW Cech: see handouts from Hocking and Young CW: see handouts from Munkres exactness fails on some comact pairs, see* * Eilenberg and Steenrod, Foundations Alg. Top.