MA4J7 Cohomology and Poincaré duality
Term II 2019-2020
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Schedule
Week |
Date of Monday |
Topics |
Pages in Hatcher |
Example sheet |
Comments |
1 |
Jan. 6 |
Introduction and examples. Categories, (covarient) functors,
natural transformations. Examples. Rings, modules, graded
modules, sequences, and chain complexes. The homology functor
factors. \(\Hom_R(P, \ast)\) is covariant. \(\Hom_R(\ast, Q)\)
is contravariant.
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162 - 165, 190 - 191 |
Example sheet 1 |
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2 |
Jan. 13 |
Fundamental theorem of \(R\)-modules (for \(R\) a PID). Chains,
cycles, boundaries, homology. Exact, short, split sequences.
Sections. Cochains, cocycles, coboundararies, cohomology. The
universal coefficient theorem (UCT). Definition of, properties of,
naturality of \(h\). \(\Hom(\cdot, G)\) is right exact.
\(\Ext\) measures the failure of \(\Hom(\cdot, G)\) to be left
exact. Free resolutions, definition of \(\Ext\), examples.
Functoriality of \(\Ext\) (Lemma 3.1).
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191 - 195 |
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3 |
Jan. 20 |
Finish the proof of UCT. UCT is a functor. Hands on cohomology
of \(\RR^n\), of \(S^1\), of \(\TT^2\). Reduced cohomology.
Triples of spaces, long exact sequence for relative cohomology.
Degree shift. Relative cochains are "simpler" than relative
chains. The map \(h\) is a natural transformation between
connecting homomorphisms. Review of the five lemma. Long exact
sequences, induced homomorphisms, homotopy invariance, excision
for cohomology.
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129, 195 - 202 |
Example sheet 2 |
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4 |
Jan. 27 |
Axioms for cohomology - homotopy invariance, long exact
sequences, excision, and disjoint unions. (We follow Hatcher in
omitting the dimension axiom.) Singular cohomology is a
cohomology theory. Simplical cohomology is isomorphic to
singular. Statement of Proposition 3.17. Review of CW
complexes. Two definitions of cellular cohomology; natural
isomorphism between them. Cellular cohomology is isomorphic to
singular. Dualising split short exact sequences. Review of
relative Mayer-Vietoris for homology: excisive covers,
subordinate chain complexes, chain homotopy equivalences.
Dualise to obtain relative Mayer-Vietoris for cohomology.
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202 - 204 |
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Answers to questions asked by students on 2020-01-23. |
5 |
Feb. 3 |
Cup product on cochains. Example computations. Graded Leibniz
rule. Induced cup product on cohomology, induced homomorphisms.
\(R\)-algebras, graded \(R\)-algebras, cohomology rings.
Example of the torus. Graded (skew) commutivity of cup product
at level of cohomology classes. Cup products in two-complexes,
for real projective plane. Hatcher's introduction to
cohomology. Cohomology of real, complex, and quaternionic
projective spaces (statements).
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186 - 189, 206 - 208, 210 - 211 |
Example sheet 3 |
Answers to questions asked by students on 2020-02-04. |
6 |
Feb. 10 |
Relative cup product. Tensor products of modules, of
\(R\)-algebras. Exterior algebras. Cross product. Statement
of Künneth formula, applications, overview of proof.
Natural transformation of theories giving isomorphism of
theories. Mayer–Vietoris for cohomology theories. Mapping
telescopes, transferring the proof from finite to infinite
dimensional complexes.
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209, 212 - 217 |
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7 |
Feb. 17 |
Checking that \(h, k\) are cohomology theories via the axioms:
homotopy invariance, excision, long exact sequences, disjoint
unions. Degree shifts for long exact sequences. Tensor
products and direct products. Naturality of the cross product.
The completely relative Künneth formula, cross product of
generators for spheres. Cohomology rings for real, complex, and
quaternionic projective spaces (statement).
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217 - 220 |
Example sheet 4 |
Thursday lecture missed due to strike. |
8 |
Feb. 24 |
Proof of cohomology of (real) projective spaces, using
naturality of cross product.
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220 - 222 |
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Monday and Tuesday lectures missed due to strike. |
9 |
Mar. 2 |
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All lectures missed due to strike. |
10 |
Mar. 9 |
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All lectures missed due to strike. |
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