MATH 663--ALGEBRAIC TOPOLOGY--FALL 2021--A. FREIRE
Syllabus
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)
10/1 F FALL BREAK
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.)