MA3H6 Algebraic topology
Term II 2013
||Date of Monday
||Pages in Hatcher
||Introduction, simplices, barycentric coordinates.
Δ-complexes, free Abelian groups, chains. Boundaries, chain
complexes, simplical homology, computation for the circle.
||97 - 106
||Prof. Mond prepared exercises on Abelian groups for last year's class.
||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
||106 - 112
||Jeff Erickson gives a very readable discussion
of how to use Smith normal form to compute simplicial homology.
||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
||112 - 117
||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 \(A\).
||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
|| 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
||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
||Excision for Δ-complexes, wedge sum and reduced homology,
the five lemma. Naturality. Skeleta of Δ-complexes,
quotients, equivalence of simplicial and singular homology.
128 - 130
||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.
||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
||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 \(X\)?
|| 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
||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
||A student asks: Are there non-homeomorphic manifolds with
isomorphic homology groups?
||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