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MA3H6 Algebraic topology
Term II 2013


Week Date of Monday Topics Pages in Hatcher Example sheet Comments
1 Jan. 7 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 last year's class.
2 Jan. 14 Homology of the torus, real projective plane, singular simplices, chains, and homology. Path components, homeomorphism invariance. Chain complexes, chain maps, functoriality. Funtoriality again, examples of chain maps, chain homotopies, the prism operator. 106 - 112 Two Jeff Erickson gives a very readable discussion of how to use Smith normal form to compute simplicial homology.
3 Jan. 21 Prism operator again, 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 We fix a mistake made in Lecture 8: the pair \((X,A)\) is a good pair if \(X\) is a topological space and \(A\) is a non-empty closed subset having a neighborhood in \(X\) that deformation retracts to \(A\).
4 Jan. 28 Topological interpretation of the connecting homomorphism, two versions of excision, open covers. Linear simplices, linear chains, coning, subdivision, diameter decreases under subdivision. Subdivision is chain homotopic to the identity, retraction of the complex of singular chains to the complex of singular chains subordinate to a cover. 118 - 124 Four
5 Feb. 4 Review of the retraction proof, excision from retraction, equivalence of relative homology of a pair and homology of the quotient space. Commutivity of maps of spaces implies commutivity of maps of homologies, the exact triangle for quotients, reduced homology of spheres. Invariance of domain, the Brouwer fixed point theorem, explicit generators of homology of spheres. 124 - 126 Five Cantor proved that all unit cubes (in any positive dimension) are in bijection with each other. Peano proved that any unit cube is a continuous image of any other.

However, Brouwer's invarance of domain shows that cubes of different dimensions are not homeomorphic.

6 Feb. 11 Excision for Δ-complexes, wedge sum and reduced homology, the five lemma. Naturality. Skeleta of Δ-complexes, quotients, equivalence of simplicial and singular homology. 126,
128 - 130
Six Show that the homology of the Hawaiian earring is an uncounable group. Show that it is not a free Abelian group. For a discussion of the fundamental group see Cannon and Conner and the references therein.
7 Feb. 18 Homology of the infinite wedge of circles, of the infinite dimensional dunce cap \(R^\infty\), of connected orientable surfaces without boundary. Degrees of self-maps of spheres, maps of the circle, the antipodal map. Hairy-ball theorem. 134 - 135 Seven We fix a mistake made in Lecture 20: one must pay careful attention to orientations when proving that topological boundaries are algebraic boundaries.

A student asks: is every endomorphism of \(H_*(X)\) induced by a self-map of \(X\)?

8 Feb. 25 One-point compactification, local degrees, the fundamental theorem of algebra. All degrees are realized, CW-complexes, \(k\)-skeleta, examples. Lemmas on CW-complexes, cellular homology is "homology squared". 136 - 139 Eight
9 Mar. 4 Review of CW-complexes, axioms following Whitehead, computation of the cellular boundary map. Cellular homology of surfaces, of two-complexes, of spheres including \(S^\infty\). Homology of real projective space, definition of Euler characteristic and examples. 140 - 142, 144, 146 Nine A student asks: Are there non-homeomorphic manifolds with isomorphic homology groups?
10 Mar. 11 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, Eleven