MATH 663-ALGEBRAIC TOPOLOGY I-FALL 2025-A. FREIRE
Syllabus
(includes a list of references)
Course log:
8/19 Simplicial homology: main definitions, simplicial
approximation theorem (statement), finite generation
Ref: [Rotman], p.131--p.145; [HiltonWylie], sect. 1.1, 1.2, 1.3, 1.7
(to p.35), 1.8, 2.2
8/21 Simplicial homology: subcomplexes, star of a face, cone
over a complex (its homology), barycentric
subdivision, chain homomorphism from C(K) to C(K'). Ref: [Rotman]
Simplicial
homology notes
(in progress--version date 8/26, 3:20PM.)
Simplicial
homology: problems
8/26: diameter reduction under subdivision/proof of simplicial
approximation/induced map in simplicial homology,
contiguous simplicial maps. Examples: homology of the boundary of a
simplex/ euler characteristic
Ref: [Rotman], p. 139, p. 146/ notes.
8/28 Singular homology: Eilenberg-Steenrod `axioms', zeroth homology
group, homology of path components, long exact sequence.
Long
exact sequence in homology (geometric proof)
(includes one problem.)
9/2 Singular homology: geometric meaning of relative homology (esp,
the case of H_n(X,x_0))/Reduced homology/The Hurewicz homomorphism
from the fundamental group to first homology.
9/4 Poincare's theorem (fundamental group and first
homology)/homotopy invariance of homology
Ref: Rotman, Hatcher
9/9 Excision in simplicial and singular homology/Mayer-Vietoris.
Notes
on the excision theorem.
(final version: 9/13, 12:30 AM) Includes three problems.
9/11 Proof of excision theorem (ref. [Rotman], ch.6)
9/16 Excision and Mayer-Vietoris: examples
relative homology of `good pairs'/local homology/equivalence of
singular and simplicial homology
9/18 Class cancelled
9/23: Excision
and Mayer-Vietoris: examples
homology of sphere minus disk, sphere minus surface/invariance of
domain/degree and local degree
9/25: homology sequence of cell attachment/examples: real and
complex projective spaces
Ref: Rotman, p. 180--193.
additional problems on homology: [Hatcher] p. 155: 2, 3, 4,
7, 31/ [Rotman] 6.17, 6.18, 6.20, 6. 23, 8.17
9/30: DeRham cohomology on manifolds: definitions, map induced in
cohomology, homotopy invariance
Ref: Bott-Tu, ch. 1.
10/2 DeRham cohomology: Mayer-Vietoris sequence/examples/de Rham
cohomology with compact supports
10/7 FALL BREAK (No lecture, first five problems due)
10/9 proper maps, compact support examples
10/14 compact support: homotopy invariance, Mayer-Vietoris sequence
10/16: oriented manifolds, integration, Stokes' theorem/Poincare
duality (start)
10/21: class canceled
10/23: Poincare duality in de Rham cohomology (proof)
de
Rham cohomology: problems
10/28 Degree formula (integral of the pullback)/ singular cohomology
(basic def'ns)
10/30 Universal coefficient theorem in cohomology; Ext(H; G).
makeup class: review of manifolds, transversality, differential
forms
11/2 ELECTION DAY--no classes at UTK
11/4 Sufficiency of smooth chains and cochains
Smooth
chains suffice
11/9 Proof of deRham's theorem