\( \newcommand{\EE}{\mathbb{E}} \newcommand{\RR}{\mathbb{R}} \newcommand{\CP}{\mathbb{CP}} \newcommand{\orb}{\mbox{orb}} \newcommand{\Isom}{\operatorname{Isom}} \)

MA3H6 Algebraic topology
Term II 2016-2017


Week Date of Monday Topics Pages in Hatcher Example sheet Comments
1 Jan. 9 Introduction, basic examples, quotient topology. Simplices, Δ-complex structures. Free Abelian groups, chains. 97 - 105 Exercises Model solutions Prof. Mond prepared exercises on Abelian groups for his class.
2 Jan. 16 Boundary operator, chain complexes, simplical homology, computation for the circle, torus, and the projective plane. Smith normal form. Singular simplices, chains, and homology. Computation of singular homology of a point and of disjoint unions. Computation of \(H_0\) of a nonempty, path-connected space. 105 - 110 Exercises Model solutions Questions asked on Monday.

For 7: Jeff Erickson gives a very readable discussion of how to use Smith normal form to compute simplicial homology.

3 Jan. 23 Reduced homology, chain maps, induced maps on homology. Homtopies, homotopy equivalent spaces, chain homotopies, prism operator, singular homology is a homotopy invariant (Theorem 2.10). Exact sequences. 110 - 114 Exercises Model solutions Questions asked on Monday.
4 Jan. 30 Pairs, good pairs. Relative homology, the connecting homomorphism. The exact triangle: the long exact sequence of homologies (of a short exact sequence of chain complexes), applied to a pair or a triple. Topological interpretation of the connecting homomorphism, two versions of excision, covers, subordinate chains. The complex of singular chains retracts to the complex of subordinate chains. Deducing excision. 114 - 119 Exercises
5 Feb. 6 Convexity, linear simplices, linear chains, coning, subdivision, the Fine lemmas. Subdivision, iterated subdivision is chain homotopic to the identity. Transport of structure, the refinement operator. Finish the proof of excision. 119 - 124 Exercises Model solutions Questions asked on Tuesday.
6 Feb. 13 Isomorphism of relative homology of a good pair and reduced homology of the quotient. Exact triangle of reduced homologies for a pair, (reduced) homology of spheres. The Brouwer fixed point theorem, local homology, invariance of domain. Review, explicit generators for \(H_*(S^n)\) I. 124 - 126, 114 - 115 Exercises Model solutions
7 Feb. 20 Explicit generators for \(H_*(S^n)\) II, pointed spaces. Hawaiian earring. Excision for Δ-complexes. Wedge sum, the five lemma. Skeleta of Δ-complexes, the equivalence of simplicial and singular homology. 125, 128 - 130 Exercises
8 Feb. 27 Manifolds, manifolds with boundary, examples. Surfaces, projective spaces. Fundamental problems in topology: classification, homeomorphsim. Cone, suspension. Reflection, antipodal map. Explicit generators for local homology of manifolds, local and global orientations. Degrees of maps of spheres, examples (hairy ball theorem). 134 - 136, 233 - 235 Exercises
9 Mar. 6 Local degrees, all degrees realized. One-point compactifications, proper maps. Review CW-complexes, subcomplexes, comforting facts. Cellular homology is homology squared, computing the cellular boundary map, examples. 136, 519 - 523, 137 - 142, 144 Exercises
10 Mar. 13 Exercises on CW-complexes. Euler characteristic, examples. Classification of surfaces. Mayer-Vietoris theorem versions one and two, applications. The various meanings of "boundary". An overview of the module via category theory. 146 - 148, 149 - 151 Exercises