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