MA3H6 Algebraic topology
Term II 20162017

Schedule
Week 
Date of Monday 
Topics 
Pages in Hatcher 
Example sheet 
Comments 
1 
Jan. 9 
Introduction, basic examples, quotient topology. Simplices,
Δcomplex 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, pathconnected
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 Δcomplexes. Wedge sum, the five
lemma. Skeleta of Δcomplexes, 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. Onepoint
compactifications, proper maps. Review CWcomplexes,
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 CWcomplexes. Euler characteristic, examples.
Classification of surfaces. MayerVietoris theorem versions one
and two, applications. The various meanings of "boundary". An
overview of the module via category theory. 
146  148, 149  151 
Exercises




