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

MA3H6 Algebraic topology
Term II 2014


Week Date of Monday Topics Pages in Hatcher Example sheet Comments
1 Jan. 6 Introduction, simplices, barycentric coordinates. Δ-complexes, free Abelian groups, chains. Boundaries, chain complexes, simplical homology, computation for the circle. 97 - 106 One Prof. Mond prepared exercises on Abelian groups for his class.

Fixed a typo (missing 2's) in Exercise 1.1.

2 Jan. 13 Computations of homology of circle, torus, and the projective plane. Singular simplices, chains, and homology. Computation of singular homology of a point and of disjoint unions. Computation of \(H_0\) of nonempty, path-connected space, reduced homology, chain maps, induced maps on homology. Homtopies. 106 - 111 Two Questions asked on Monday.

For 2c: Dan Moskovich discusses the failure of the triangulation conjecture. There are compact connected orientable manifolds without boundary in dimension six (and above) that admit no \(\Delta\)-complex structure.

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

3 Jan. 20 Homotopy equivalent spaces, chain homotopies, prism operator, homology is a homotopy invariant (Theorem 2.10). Exact sequences, relative homology. The connecting homomorphism. The long exact sequence of homologies (of a short exact sequence of chain complexes), applied to a pair or a triple. 112 - 117 Three Questions asked on Monday.
4 Jan. 27 Topological interpretation of the connecting homomorphism, two versions of excision, open covers. Linear simplices, linear chains, coning, subdivision, diameter decreases under subdivision. Subdivision, iterated subdivision is chain homotopic to the identity. 118 - 123 Four Questions asked on Monday.
5 Feb. 3 Retraction of the complex of singular chains to the complex of singular chains subordinate to a cover. Deduce excision. Review quotient spaces. Isomorphism of relative homology of a good pair and reduced homology of the quotient. Exact triangle of reduced homologies for a pair. Functoriality, proving the isomorphism, (reduced) homology of spheres. Invariance of domain, the Brouwer fixed point theorem. 123 - 124 Five Questions asked on Monday.

Fixed a typo (missing ι) in Exercise 5.8.

6 Feb. 10 Explicit generators for homology of spheres, excision for Δ-complexes. Wedge sum, the five lemma. Naturality, skeleta of Δ-complexes, start Theorem 2.27 showing the equivalence of simplicial and singular homology. 125 - 126,
128 - 130
Six Questions asked on Monday.
7 Feb. 17 Finish Theorem 2.27. Homology of graphs, of surfaces. Final details of proof of Theorem 2.27, statement of the classification of surfaces. Degrees of maps of spheres, examples. Degree and homotopy, the hairy ball theorem, covers of the circle. 134 - 135 Seven Questions asked on Monday.
8 Feb. 24 Odd versus even, degree, Borsuk-Ulam theorem, explicit generators for local homology, manifolds. Local and global orientations, local degree. All degrees are realized, CW-complexes, \(k\)-skeleta. 136 - 137, 174, 176, 233 - 234, 519 Eight
9 Mar. 3 Cellular homology, computing the cellular boundary map. Cellular homology made to order, of two-complexes, of surfaces, of real projective space, and of other examples. 137 - 142, 144 Nine
10 Mar. 10 Cellular structure on \(\CP^n\). Euler characteristic of CW-complexes. Mayer-Vietoris theorem versions one and two. \(H_1\) and the fundamental group, discussion of topics in topology. 146 - 148, 149 - 151, 166 - 168 Ten Questions asked in weeks nine and ten.