MATH 663--ALGEBRAIC TOPOLOGY I--FALL 2023

Syllabus

Course outline

Student presentations


Th 8/24  Singular homology: basic definitions, H_*(point), H_0(X)

Tu 8/29 class canceled

Th 8/31 class canceled

Tu 9/5 (Zoom) Induced homomorphism/ homeo. invariance/ reduced homology/ homotopy invariance (start)

Th 9/7 (Zoom) Homotopy invariance (end)/ Ex: def retracts and retracts/ Relative homology, long exact seq.

Scanned notes, part A (6pp)
Scanned notes, part B (4pp)

Tu 9/12 (Zoom) Split exact sequence/ H_n(X,x_0) equals reduced homology of X/case of retracts/Def retracts have vanishing rel. homology/
              Homotopy invariance of rel. homology/ long exact seq for triples/excision thm (statement) Thm: relative homology equals
              reduced homology of the quotient (for "good pairs")/Five Lemma

Scanned notes (7 pp.)


Th. 9/14: rel. homeo invariance/ locality principle (statement), excision (proof)
Scanned notes


Tu  9/19 Mayer-Vietoris proof of locality-barycentric subdivision

Th 9/21 Examples: Proof of locality (barycentric subdiv): conclusion/Hurewicz theorem (start)

Tu 9/26  Hurewicz proof (conclusion)/ examples: figure-eight, circle, spheres

Th 9/28  Generators for the homology of circle and spheres/Brouwer degree/oriented surfaces/projective plane/wedge of spheres

Tu 10/3  Attachment spaces: characterization via preimages, examples (spheres, real and complex projective spaces); homology pf RP(2).

Th 10/5 Attachment spaces: homology of n-cell attachment, interpretation, examples (product of spheres).
[Ref: Vick, ch. 2]

Tu 10/10 FALL BREAK

Th 10/12 Finite CW complexes: definition, subcomplexes/ cellular homology (start)

Tu 10/17 cellular homology: isomorphic to singular homology

Th 10/19 cellular homology, examples: projective space, euler characteristic

Tu 10/24 Morse theory (start)

Th 10/26 homology of finite graphs (Cannon)/ Morse Theory (cont.)

afternoon (4:05-5:20): cell structure of Grassmannians (Sam)/ Morse inequalities

Tu 10/31 Simplicial homology (outline)

Th 11/2 simplicial homology: isomorphism with singular/ Cell structure of SO(n) (George)

Tu  11/7 homology w/ general coefficient groups, Tor(G,H), univ. coeff theorem

Th 11/9 Eilenberg-Steenrod axioms, smooth manifolds are triangulable (Ivy)

Tu 11/14 singular cohomology and Ext (H,G)/ univ. coeff. thm. for cohomology

Th 11/16 cup product, external products in cohomology, Kuenneth formula/Cohomology rings of n-torus, projective space

Tu 11/21 manifolds and orientation (local homology)

Th 11/23 Thanksgiving

Tu 11/28    Discrete Morse theory (Sagnik)/Cellular approximation theorem (Amer)

Afternoon: homology of pseudomanifolds (Tariq)/ Hopf's theorem on maps to spheres (Dalen)

Th 11/30 deRham cohomology, Hodge theorem/

Afternoon: proof of DeRham's theorem (Fotis) /A discrete Hodge theorem (Jared)

Tu 12/5 Lefschetz fiexed-point theorem (Andrew)/Intro to the Hopf Invariant (Chris)

Hopf invariant: equivalence of two definitions