12/15/21: Homework is due by the beginning of the class. The Hurewicz
and Whitehead theorems (no proofs, just some indication
thereof). Example of homotopy inequivalent simply connected CW
complexes with isomorphic integral homology: ℂℙ2
\/ ℂℙ2 and S4 \/ (S2 x
S2). [Class notes. Text: Chapter 4 (pp. 366-371, 346-348)]
12/13/21: Relative homotopy groups and the long exact sequence of
pair. The action of the fundamental group on homotopy groups. [Class
notes. Text: Chapter 4 (pp. 343-345, 341-342)]
12/10/21: Stiefel and Grassmann manifolds. Bott periodicity. [Class
notes. Text: Chapter 4 (pp. 381-382, 383-384)]
12/8/21: Hopf fibrations and homotopy groups of low-dimensional
spheres. Projective spaces and associated fiber bundles. [Class
notes. Text: Chapter 4 (pp. 377-379)]
12/6/21: The (long) Puppe sequences of pointed homotopy classes of
maps associated with fibrations and cofibrations. Homotopy groups. The
long exact sequence of homotopy groups of a fibration. Example: the
universal cover of S1. The homotopy groups of S1
and the low-dimensional homotopy groups of Sn. [Class
notes. Text: Chapter 4 (pp. 398-399, 339-342, 376, 380, 349)]
12/3/21: Exact sequences of pointed homotopy classes of maps
associated with fibrations and cofibrations. Pointed mapping
spaces. Adjunction between reduced suspension and based loops. The
group structures on the sets of homotopy classes of maps from the
(iterated) reduced suspension and maps to the (iterated) loop
space. [Class notes. Text: Chapter 4 (pp. 398-399, 395-396)]
12/1/21: Homework is due by the beginning of the class. The iterated
(long) fibration and cofibration sequences. Exact sequences of
homotopy classes of maps to a fibration sequence and from a
cofibration sequence. [Class notes. Text: Chapter 4 (pp. 398, 399, 409)]
11/29/21: Fibration and cofibration sequences. [Class notes. Text:
Chapter 4 (p. 409)]
11/26/21: Thanksgiving Break: no classes!
11/24/21: Cofibrations and neighborhood deformation retracts
(NDRs). Replacing a map by a cofibration. [Class notes. Text: Section
4.H (pp. 460-461)]
11/22/21: Replacing a map by a fibration. [Class notes. Text: Chapter
4 (pp. 407-408)]
11/19/21: The free and based path space, the (based) loop space. The
path-space fibrations and their fibers. [Class notes.]
11/17/21: The homotopy lifting property. Fibrations in the sense of
Hurewicz. Examples: covering spaces, trivial fibrations, local
fibrations, locally trivial fibrations, i.e., fiber bundles. Maps of
fibrations. The pullback of a fibration. Homotopy equivalence of
fibers of a fibration. The monodromy action of the fundamental group
of the base. [Class notes. Text: Chapter 4 (pp. 375, 376-377,
379-380)]
11/15/21: Homework is due by the beginning of the class. Orientation
on manifolds over rings R other than ℤ. Poincar&eaigu; duality
over R. Manifolds are canonically oriented over ℤ/2. Perfectness
of the intersection pairing over ℤ/2. The cohomology ring of the
real projective space ℝℙn with ℤ/2
coefficients. Compactly generated spaces and mapping spaces. The
compact-open topology. [Class notes. Text: Chapter 3 (p. 250),
Appendix A (pp. 523 and 529-531)]
11/12/21: Intersection pairing and the cohomology ring of the complex
projective space ℂℙn. [Class notes. Text: Chapter
3 (pp. 249-250)]
11/10/21: Poincaré, Poincaré-Lefschetz, and Alexander
duality. Cohomology with compact support and the sketch of proof of
Poincaré duality. Intersection pairing. [Class notes. Text:
Chapter 3 (pp. 241, 252-255, 249)]
11/8/21: Orientation on manifolds: a consistent choice of local
orientations and the degree-n (relative) homology group of the
n-manifold. [Class notes. Text: Chapter 3 (pp. 234, 236)]
11/5/21: The relative cap products. Review of manifolds (topological,
smooth, with or with no boundary) and orientation. [Class notes. Text:
Chapter 3 (pp. 240-241, 233-234)]
11/3/21: The relative cup products. [Class notes. Text (Hatcher uses
concrete maps for the Eilenberg-Zilber and Alexander-Whitney maps,
while we use their existence from the acyclic model theorem in class):
Chapter 3 (pp. 209-210)]
11/1/21: The Mayer-Vietoris sequence. [Class notes. Text: Chapter 3
(pp. 203-204)]
10/29/21: Homework is due by the beginning of the class. Class meeting
is canceled.
10/27/21: The cap product. [Class notes. Text (Hatcher uses concrete
maps for the Eilenberg-Zilber and Alexander-Whitney maps, while we use
the acyclic model theorem in class): Chapter 3 (pp. 239-240, 240-241
(i.e., skip the relative cap products for the time being), 249)]
10/25/21: The cohomology ring (the proof). [Class
notes. Text (Hatcher uses concrete maps for the Eilenberg-Zilber and
Alexander-Whitney maps, while we use the acyclic model theorem in
class): Chapter 3 (pp. 207, 210-212)]
10/22/21: The cohomology cross product and cup product. The cohomology
ring (only the wording of the theorem). [Class notes. Text (Hatcher
uses concrete maps for the Eilenberg-Zilber and Alexander-Whitney maps,
while we use the acyclic model theorem in class): Chapter 3 (pp. 186,
206-210, 212-213, 214-215), Section 3.B (pages 278-280)]
10/20/21: Remarks on the Eilenberg-Zilber map and homology cross
product. The shuffle product as a way to chop the product of simplices
into simplices in order to construct the Eilenberg-Zilber map. The
topological Künneth formula over a field. Example: the integral
homology of RP2 x RP2; the
geometric interpretation of the generator of
H3(RP2 x RP2; ℤ) =
ℤ/2. [Class notes. Text: Chapter 3 (pp. 185), Section 3.B (pages
276, 277-278, Example 3B.4 on p. 272 for X = Y
= RP2, the G= ℤ, m = n = 2, homology part of
Exercise 1 on p. 280)]
10/18/21: The algebraic Künneth theorem. The Eilenberg-Zilber
maps, homology cross product, and topological Künneth
formula. [Class notes. Text: Chapter 3 (p. 185), Section 3.B (pages
268, 273--276)]
10/15/21: Homework is due by the beginning of the class. Identifying
cohomology for R = ℤ via the torsion and free parts of
homology. The UCT for homology. Betti numbers from integral and
ℚ- and ℝ-homology and cohomology. [Class notes. Text (covers
mostly the case R = ℤ): Chapter 3 (p. 196) Section 3.A (pages
264-265, 267)]
10/13/21: The proof of the algebraic universal coefficient theorem for
cohomology. Corollary for complexes of vector spaces: the
nondegeneracy of the Kronecker pairing. [Class notes. Text (covers
mostly the case R = ℤ): Chapter 3 (pp. 191-195, 198-199)]
10/11/21: Back to topology: the universal coefficient theorem (UCT)
for cohomology, algebraic and topological. [Class notes. Text (covers
mostly the case R = ℤ): Chapter 3 (pp. 195-196, 196-198, 201)]
10/8/21: Comparison of resolutions (the idea of proof). The Horseshoe
lemma and the properties T2 and E2: the long exact sequences for
Tor(M,N) and Ext(M,N) coming from a short exact sequence in the first
argument, M. [Class notes. Text (covers mostly the case R = ℤ):
Chapter 3 (pp. 197), Section 3.A (pages 265, 267), Section 3.F
(pp. 317-318). Rotman's An Introduction to Homological Algebra:
pp. 340-343, 349-350]
10/6/21: Existence of Tor and Ext: proofs. Comparison of resolutions
(no proof). Independence of Tor(M,N) and Ext(M,N) of the choice of a
projective resolution of M. Functoriality of Tor(M,N) and Ext(M,N) in
M. [Class notes. Text (covers mostly the case R = ℤ): Chapter 3
(pp. 193-195), Section 3.A (pages 263-264, 265-266). Rotman's An
Introduction to Homological Algebra: pp. 340-343]
10/4/21: Resolutions. Existence of Tor(M,N) and Ext(M,N): construction
using a projective resolution of M. Tor0 (M,N) = M ⊗
N and Ext0 (M,N) = Hom (M, N). [Class notes. Text (covers
only R = ℤ): Chapter 3 (p. 193), Section 3.A
(p. 263). Rotman's An Introduction to Homological Algebra
(available online through the library): pp. 325-327, 346, 353-354,
370, 372]
10/1/21: Homework is due at the start of the class. Projective and
injective modules. [Class notes. Your favorite Homological Algebra
text, e.g., Weibel, Rotman, or Mac Lane.]
9/29/21: Computation of Tor and Ext. [Class notes. Text: Chapter 3
(bottom of p. 195 and the top of p. 196), Section 3.A (the bottom of
p. 263 and the top of p. 264, pp. 265, Section 3.F (pp. 317-318
(skipping the proof of Proposition 3F.11 for the time being))]
9/27/21: Left and right exact functors. Functors Tor and Ext: defining
properties, examples, uniqueness. [Class notes. Text: Chapter 3 (Ext
at the bottom of p. 195 and top of p. 196), Section 3.A (Tor on page
265)]
9/24/21: The Eilenberg-Steenrod axioms. Existence and uniqueness
theorems for (co)homology theories. [Class notes. Text: Chapter 2
(pages 160-162), Chapter 3 (pages 201-202). Note that Hatcher does the
reduced (co)homology version of the axioms; see Wikipedia or another
textbook for the list of unreduced axioms, if you have not taken them
down in class]
9/22/21: Functoriality of cohomology and long exact sequence for
cohomology of a pair. [Class notes. Text: Chapter 2 (pages 162-165),
Chapter 3 (pages 199-201)]
9/20/21: Homework is due at the beginning of the class. Review of
cochain complexes and cohomology. Pairing between cohomology and
homology. Cohomology computation example: RP2. [Class
notes. Text: Chapter 2 (page 144), Chapter 3 (pages 190-191)]
9/17/21: Tensor products and Hom. Homology with coefficients in a
module. Cohomology. [Class notes. Text: Chapter 3 (pages 215, 190-193,
197-199, simplicial and cellular cohomology on pages 202-203)]
9/15/16: The Hurewicz theorem for the fundamental group. Simplicial
complexes and simplicial homology. Example: RP2. [Class
notes. Text: Chapter 2 (pp. 104-107, 128). Note the difference between
the notion of a simplicial complex and that of a Δ-complex,
discussed on p. 107.]
9/13/16: Reduced singular homology. Identification of the differential
in the cellular complex via the degree of a map between two
spheres. The (reduced) singular homology of Sn. Cellular
homology. The cellular homology of S1. Isomorphism between
cellular and singular homology. [Class notes. Text: Chapter 2 (pages
139-141).]
9/10/16: CW complexes and the cellular chain complex. [Class
notes. Text: Chapter 2 (pages 5, 124, 137-141)]
9/8/16: Introduction. Syllabus handed out. Singular homology: quick
review.
[Syllabus.
Class notes. Text: Chapter 2 (pages 97-103, skipping the notion of a
Δ-complex, which we will not be using in the course, top of page
106, page 108, top paragraph of page 109, pages 111, 115-117)]
Last modified: (2021-12-15 11:29:01 CST)