Course outline

8/18  W  homotopy equivalence, mapping cylinder, CW complexes (Hatcher ch. 0)

8/20  F Subcomplexes, cell structures on spheres, real and complex projective spaces

8/23 M genl. topology of CW complexes (Hatcher, apppendix)-canonical neighborhoods

attaching by a map [Rotman, p. 184--188]

HW problems: Rotman, 8.12, 8.13 (presentation)
Hatcher ch. 0: 16, 19, 21, 29 (presentation)

Presentation topic: cell structure for the grassmanian (Milnor Characteristic Classes, ch.6)
Presentation topic: Morse theory

8/25 W CW complexes are locally contractible/ cell structure given by a Morse function (torus example)
Quotient spaces, cone over  space/ two theorems on homotpy equivalence (statement)

8/27 F Two criteria for homotopy equivalence (Hatcher ch. 0)

8/30 M homotopy equivalence criteria/simplicial complexes

9/1 W simplicial homology,  first properties

9/3 F singular homology: chain maps, induced map on  homology, homeomorphism invariance

9/6 M Labor Day (no classes)

9/8 W singular homology:  first properties:  dimension axiom, H_0(X), reduced homology, homotopy invariance

9/10 F presentations
Problems 16 and 29 (Hatcher ch.0) (Sathya)
Problem 16 notes
Problems on attachment spaces [Rotman] (Liam)

9/13 M presentation: cell structure for the Grassmannian (Ben)

9/15 W homotopy invariance

9/17 F relative homology of pairs, long exact sequence, connection with homology of quotient spaces

9/20  M long exact sequence, triples, applications (homology of spheres)

9/22  W local homology groups, degree, excision (statement); relative homology and reduced homology of the quotient
Presentation topic: local computation of degree (GV)

9/24  F Hurewicz theorem: H_1 is the abelianized fundamental group. (ref: [Bredon])

9/27 M no lecture  (to be replaced)

9/29 W no lecture (to be replaced)


10/4 M Proof of excision, I: barycentric subdivision

10/6 W Proof of excision, II:  subdivision operators in singular homology

10/8 F Proof of excision,  III: conclusion--chain homotopy equivalence

10/11 M Applications:  Mayer-Vietoris sequence

10/13 W Student presentations
Local computation of the degree (Guillermo)
Simplicial Approximation Theorem (Tariq)

10/15 F Simplicial homology and singular homology

10/18 M Simplicial vs. singular homology (end)/Euler charactristic (simplicial)-invariance

10/20 W Manifolds homotopy equiv to CW complexes, via Morse theory (Patrick)

10/22 F Morse Theory (conclusion)- Patrick

10/25 M Cellular homology groups, isomorphism with singular homology

10/27 W The cellular boundary map/ first examples

10/29 F Homology computations via cellular homology (examples)

11/1 M Moore spaces and lens spaces: homology

11/3 W Cohomology (singular, de Rham); homology with  coefficients, tensor products and exact sequences

11/5 FLefschetz fixed-point theorem (Bryan)

11/8 M Mayer-Vietoris sequences for de Rham cohomology, and dR cohomology with compact supports

11/10 W de Rham cohomology: integration, Poincare lemma, homotopy invariance, cup product

11/12 F Poincare duality in de Rham cohomology

11/15 M Cellular approximation theorem (Jeffrey)
Group actions on spheres/Commutative division algebras  (Betsy)

11/17 W Singular cohomology
afternoon lecture: cup product,cohomology ring (examples)
Ref: [Hatcher], p. 206-208 and 215-216.

11/19 F Universal coefficient theorems: Tor and Ext (Nathan)

11/22 M  Homology of pseudomanifolds (Ivy, Bernardo)

11/24 W  no class (Thanksgiving break)

11/26 F  no class (Thanksgiving break)

11/29 M de Rham's theorem (ref. [Bredon])

12/ 1  W orientation class of oriented topological manifolds: existence/uniqueness.
(ref: [Hatcher] p. 236-238;  [Massey]  p. 351-356.)