MA3H6 Algebraic topology
Term II 2014

Schedule
Week 
Date of Monday 
Topics 
Pages in Hatcher 
Example sheet 
Comments 
1 
Jan. 6 
Introduction, simplices, barycentric coordinates.
Δcomplexes, 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, pathconnected 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
Δcomplexes. Wedge sum, the five lemma. Naturality,
skeleta of Δcomplexes, 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, BorsukUlam theorem, explicit
generators for local homology, manifolds. Local and global
orientations, local degree. All degrees are realized,
CWcomplexes, \(k\)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 twocomplexes, 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
CWcomplexes. MayerVietoris 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. 


