5/6/05: Extra class meeting at 3:35 pm, Friday, VinH 213. Come,
if you can, to support Chris! The Adams spectral sequence. [Chris's
presentation, McCleary's Guide, Chapter 9]
5/6/05: Extra class meeting at 3:35 pm, Friday, VinH 213. Come,
if you can, to support Chris! The Adams spectral sequence. [Chris's
presentation, McCleary's Guide, Chapter 9]
5/6/05: Simplicial approximation. [Joao's presentation, Hatcher:
Section 2.C]
5/4/05: The Atiyah-Singer Index Theorem. [Antoine's presentation,
Booss and Bleecker's book]
5/2/05: Extra class meeting at 3:35 pm, Monday, VinH, second
floor. Come, if you can, to support Dan! Classifying spaces of
posets and categories. [Dan's presentation, Quillen, LNM 341,
Gabriel-Zisman, or Goerss-Jardine, Simplicial homotopy theory]
5/2/05: Curvature and characteristic classes. [Josef's presentation,
Appendix to Milnor-Stasheff's book]
4/29/05: The classical Riemann-Roch theorem. [Wenliang's presentation,
Ahlfors-Sario's book]
4/27/05: Vector bundles. [Sato: Section 8.2]
4/25/05: The Hurewicz theorem (relative). Another theorem of
Whitehead. [Hatcher: pp. 366-368, 369-371]
4/22/05: The existence of CW approximation (absolute and
relative). The Hurewicz theorem (absolute). [Hatcher: pp. 366-367,
369-370, 486, 364]
4/20/05: The Freudenthal suspension theorem. The Hurewicz
theorem. [Hatcher: pp. 360-364, 366]
4/18/05: Finishing the proof of the Dold-Thom Theorem. The Freudenthal
suspension theorem. [Hatcher: pp. 483-486, 360]
4/15/05: Further properties of infinite symmetric products. The
Dold-Thom Theorem. [Hatcher: pp. 483-486]
4/13/05: Example: SP (S^2). Properties of infinite symmetric products
(functroiality, homotopy invariance). [Hatcher: pp. 481-482]
4/11/05: Cellular approximation. Infinite symmetric
products. [Hatcher: pp. 348-351, 282, 481; Selick: pp. 72-73]
4/08/05: "Functorial" properties of CW approximation. [Hatcher:
pp. 355-356; Selick: p. 73]
4/06/05: CW approximation. [Hatcher: pp. 353-354; Selick: pp. 72-73]
4/04/05: ). Corollaries from Whitehead's theorem. [Hatcher:
pp. 346-348, 352, 357; Selick: p. 73]
4/01/05: HELP (proof). Proof of Whitehead's theorem. [Hatcher:
pp. 346-348, 15; Selick: pp. 72, 73]
3/30/05: HELP. HEP for (relative) CW complexes. Whitehead's
theorem. [Hatcher: pp. 346-348, 15; Selick: pp. 72, 68, 73]
3/28/05: Class canceled - family emergency.
3/25/05: n-equivalences and n-connected spaces and
pairs. Examples. [Hatcher: p. 346; Selick: pp. 68, 71]
3/23/05: Change of basepoint (continued). Weak equivalences. [Hatcher:
pp. 421-423, 352; Selick: p. 65]
3/21/05: The homotopy groups of a product and a union. Examples of the
n-torus T^n, the complex projective spaces CP^n and CP^infty. The
generalized Hopf fibrations for quaternions and Cayley numbers and
consequences for the homotopy groups of spheres S^4 and S^8. Change of
basepoint. [Hatcher: pp. 343, 378-379, 421-423; Sato: Section 3.3;
Selick: p. 65]
3/14-20/05: The Spring Break.
3/11/05: More computations of homotopy groups. [Hatcher: pp. 377-378,
380; Sato: Section 3.3]
3/9/05: The homotopy sequence of a fibration: relation to the homotopy
sequence of the pair (E,F). First computations of homotopy
groups. [Hatcher: pp. 376, 342; Selick: p. 65; Sato: Section 3.3]
3/7/05: The homotopy sequence of a pair: the descrption of the
maps. The homotopy sequence of a fibration. [Hatcher: pp. 343-345,
376; Selick: p. 65]
3/4/05: Homotopy groups. The homotopy sequence of a pair. [Hatcher:
pp. 339-341, 343-345; Selick: pp. 64-65]
3/2/05: Cofiber sequences, finished: the proof of the last
lemma. Relation to fiber sequences. [Hatcher: pp. 397-399, 462, 339-341,
344-345; Selick: p. 66]
2/28/05: Cofiber sequences, continued: the end of proving that the
long sequence of based spaces is exact. [Hatcher: p. 399]
2/25/05: HW collected. Cofiber sequences. [Hatcher: pp. 397-398, 461;
Selick: pp. 61-62]
2/23/05: Fiber sequences. [Hatcher: p. 409; Selick: pp. 59-60, 65]
2/21/05: Based fibrations. The homotopy fiber. [Hatcher: pp. 407-408;
Selick: p. 59]
2/18/05: Based loop spaces. [Hatcher: pp. 395-396; Selick: p. 59]
2/16/05: Fibrations: change of fiber, local triviality up to
homotopy. [Hatcher: pp. 406-407; Selick: pp. 54-55]
2/14/05: Fibrations: fiber homotopy equivalence, change of
fiber. [Hatcher: pp. 405-406; Selick: pp. 53, 55]
2/11/05: Fibrations: a local fibration is a fibration (proving the
local finiteness lemmas from Step 1). [Hatcher: pp. 379-380; Selick:
p. 56]
2/9/05: Fibrations: a local fibration is a fibration (Steps 2 and
3). [Hatcher: pp. 379-380; Selick: p. 56]
2/7/05: Fibrations: a local fibration is a fibration (Plan of proof
and Step 1 of 2). [Hatcher: pp. 379-380; Selick: p. 56]
2/4/05: Fibrations: local triviality of covering spaces, fiber
bundles, examples, a local fibration is a fibration, in particular, a
fiber bundle is a fibration. [Hatcher: pp. 69, 376-378; Selick:
pp. 99, 56]
2/2/05: Fibrations: the mapping path space, replacing maps by
fibrations. [Hatcher: p. 407; Selick: p. 59]
1/31/05: Fibrations: the homotopy lifting property (HLP), the pullback
of a fibration is a fibration, a covering space is a fibration with a
unique homotopy lifting property (HLP), the dual of a cofibration is a
fibration. [Hatcher: pp. 375-376, 60; Selick: pp. 53-54]
1/28/05: Cofiber homotopy equivalences. Homotopy equivalence of
pairs. The homology of an NDR-pair. NDR-pairs and CW
complexes. [Hatcher: pp. 16-17; Selick: pp. 56-58; Bredon: pp. 430-434
(especially Corollary 1.4); Spanier: Sections 1.4 and 7.6 (especially
Corollary 2 and Theorem 12)]
1/26/05: Cofibrations and NDR-pairs. [Hatcher: p. 15; Selick: pp. 56-58]
1/24/05: Cofibrations and mapping cylinders. Replacing maps by
cofibrations. [Selick: pp. 56-57; Hatcher: pp. 15-17, 461]
1/21/05: Cofibrations, the homotopy extension property (HEP). The
pushout of a cofibration is a cofibration. [Hatcher: pp. 14, 460-461;
Selick: pp. 56]
1/19/05: A review of compactly generated spaces. [Hatcher:
pp. 523-525; Selick: Section 2.6]
12/15/04: Ingredients of Poincaré duality: the definition of
the cap product, the notions of R-orientation (local and global) and
an R-fundamental class. The idea behind proving Poincaré
duality. The only important result I forgot to mention in class is
that every manifold is Z/2Z-orientable and in a unique way. This is
because there is a unique choice of a local Z/2Z-orientation at each
point. These local orientations will automatically be compatible,
because H_n (M, M\U; Z/2Z) --> H_n (M, M\x; Z/2Z) ~ Z/2Z for x in U is
an isomorphism. [Hatcher: pp. 233-238, 239-241, 249, and, if you want
to see how Poincaré duality is proven, 242-248]
12/13/04: Poincaré duality and its consequences: the
nonsingularity of the cup product pairing mod torsion and over a field
(proof), the cohomology ring of the complex projective
space. [Hatcher: pp. 250-251]
12/10/04: Poincaré duality and its consequences: the idea of a cap
product, capping with the fundamental class, the top homology and
cohomology of a connected compact oriented manifold, the
nonsingularity of the cup product pairing mod torsion and over a
field. [Hatcher: pp. 249-250]
12/08/04: The Borsuk-Ulam theorem: proof. Pairing (and duality)
between cohomology and cohomology. Poincaré duality as a pairing
between cohomology and homology. [Hatcher: pp. 176, 229, 191-192, 195,
198, 230-233, 241]
12/06/04: Discussion of the homework: Problems 1 and 3. Odd maps
between spheres: proof. [Hatcher: pp. 122-123, 140, 229]
12/03/04: Maps between real projective spaces inducing nonzero maps on
the fundamental groups. Antipodal (odd) maps between spheres. The
Borsuk-Ulam theorem. [Hatcher: pp. 176, 229]
12/01/04: The naturality and homotopy invariance of the cup
product. The cohomology of the real projective space with integral and
mod 2 coefficients. The ring structure of that cohomology mod
2. [Hatcher: pp. 212-214]
11/29/04: Checking the remaining axioms (exactness, additivity, and
excision) for cellular cohomology. The cross and cup
products. [Hatcher: pp. 199-202, 278-280, 206-212, 215]
11/23/04: Discussion of the Kuenneth formula for cohomology. Axioms
for a cohomology theory. Checking some axioms (dimension,
functoriality, and homotopy) for cellular cohomology. [Hatcher:
pp. 277, 201-203; Sato: pp. 55-59]
11/21/04: Singular cohomology of spaces. Functors Ext and Hom. The
Universal Coefficients Formula in cohomology. Proof. [Hatcher:
pp. 197-198, 191-195]
11/19/04: The cellular chain complex of the product of CW
complexes. Step 4 of the proof: combining the computation of the
boundary of the product of the top cells in two cubes (Step 2) with
the naturality of the cell product map alpha (Step 3) to prove the
product formula for arbitrary cells. Dualization of a chain
complex. Cochain complexes and their cohomology. The singular,
cellular, and simplicial cohomology of spaces, CW complexes, and
Delta-complexes, resp. [Hatcher: pp. 271 and 185-191]
11/17/04: The cellular chain complex of the product of CW
complexes. Step 1: proving the formula d(e_1 x ... e_n) = \sum_i
(-1)^i e_1 x ... x de_i x ... e_n for the n-cube I^n (finished). Step
2: d(e^p x e^q) = de^p x e^q + (-1)^p e^p x de^q for I^n. Step 3: the
naturality of the cross-product map alpha: C^CW (X) \tensor C^CW (Y)
--> C^CW (X x Y). Lemma: the degree of the reduced
suspension. [Hatcher: pp. 270-271 and 137]
11/15/04: The topological Kuenneth Formula. The cellular chain complex
of the product of CW complexes. Step 1: proving the formula d(e_1 x
... e_n) = \sum_i (-1)^i e_1 x ... x de_i x ... e_n for the
n-cube. [Hatcher: pp. 268-270]
11/12/04: The proof of the algebraic Kuenneth Formula and its
corollary for a field. Step 2: The case of a general complex
C. [Hatcher: pp. 274-275]
11/10/04: The algebraic Kuenneth Formula. The Universal Coefficients
Formula as a particular case of the Kuenneth Formula. The topological
Kuenneth Formula. Corollary: the case of coefficients in a field. Step
1 of a proof of the corollary: when the complex C is concentrated at
one place. [Hatcher: pp. 273-274, 275-276]
11/08/04: The functors Tor for a general commutative ring. The
Universal Coefficients Formula in algebra and topology. Example: the
cellular homology of RP^n with coefficients in Z_2 from Universal
Coefficients. [Hatcher: pp. 263-266]
11/05/04: The cellular homology of RP^n with coefficients in Z_2. The
Brouwer Fixed Point theorem. More homological algebra: towards the
homology of the tensor product of (chain and CW) complexes and the
universal coefficients formula. The tensor product of a (chain)
complex by a module. Functor Tor_*. [Hatcher: pp. 154, 114-115,
261-263, 195]
11/03/04: Computations of degrees. The cellular homology of
RP^n. [Hatcher: pp. 136-137, 144]
11/01/04: The local degree and its relation to the global
one. [Hatcher: pp. 135-136]
10/29/04: The degree of a self-map of a sphere. Properties of
degree. The cellular homology of CP^n and CP^infty. [Hatcher:
pp. 134-135, 140; Selick: p. 41; Sato: p. 52]
10/27/04: Cellular homology equals singular. [Hatcher: pp. 137-139;
Selick: pp. 39-40; Sato: pp. 40-44]
10/25/04: Cellular homology: identification of the cellular chain
groups and the cellular differential. [Hatcher: pp. 137, 139, 140-141;
Selick: p. 39]
10/22/04: Simplicial homology equals singular homology. [Hatcher:
pp. 128-130]
10/20/04: Computation of the simplicial homology of the real
projective plane. The relative simplicial homology. [Hatcher:
pp. 106-107, 128; compare with Sato: pp. 50-51]
10/18/04: Delta-complexes and simplicial homology: examples and
computations. [Hatcher: pp. 102 and 106; compare with Sato: p. 51,
where the simplicial complex structure (rather than semi-) on the real
projective plane is way more complicated than the Delta-complex
structure of Hatcher on the same space]
10/15/04: The excision axiom for singular homology. Delta-complexes
(i.e., Eilenberg-Zilber's semisimplicial complexes) and simplicial
homology. [Hatcher: pp. 119, 124, 102-106; Selick: p. 37; compare with
Sato: pp. 45-50, where he considers simplicial complexes (rather than
semi-) and their simplicial homology]
10/13/04: The subcomplex of singular chains subordinate to an open
covering is homotopy equivalent to the singular chain
complex. [Hatcher: pp. 123-124; Selick: pp. 36-37]
10/11/04: The subcomplex of singular chains subordinate to an open
covering. Attempting to use the barycentric subdivision to show the
subcomplex is homotopy equivalent to the singular chain
complex. [Hatcher: p. 123; Selick: p. 37]
10/08/04: Discussion of Problem 1 from Section 2.3 of Hatcher. The
barycentric subdivision and its properties. [Hatcher: p. 119-123, 165;
Selick: pp. 35-37 through the proof of Theorem 5.2.4]
10/06/04: Completing the proof of the Homotopy Axiom. Excision for
topological spaces and how it implies one for CW complexes, via the
double mapping cylinder construction. [Hatcher: p. 110-113, 119;
Selick: pp. 35-36]
10/04/04: Discussion of Problem 2 about the contractible path
space. Brush-up on using the algebraic homotopy to see that the
singular homology of a convex set is trivial in all degrees but zero,
in which the homology is G. [Hatcher: pp. 529-530 regarding the
compact-open topology, which we introduce on the path spaces; p. 113]
10/01/04: Step 2 in detail: computing the singular homology of a
convex set in a Euclidean space. Algebraic homotopy games. [Hatcher:
p. 113; Selick: pp. 21-24, 35]
09/29/04: Singular homology: checking the axioms. Additivity and
homotopy (1st way: using the Eilenberg-Zilber homotopy equivalence;
2nd way: Steps 1 and 2, to be continued). [Hatcher: pp. 109, 110-111;
Selick: p. 35]
09/27/04: Singular homology: checking the axioms (functoriality,
naturality of the connecting homomorphism "del", exactness,
dimension). [Hatcher: pp. 108-111, 113-118, 127-128; Selick:
pp. 34-35]
09/24/04: Computations from the axioms: the homology of the n-sphere,
showing that the constant self-map of the sphere is not homotopic to
the identity map. Singular chains and singular homology. [Hatcher:
pp. 114, 108; Sato: Section 4.3(d)]
09/22/04: Reduced and unreduced homology: the suspension isomorphism,
expressing the connecting homomorphism in the long exact sequence
through the suspension homomorphism, equivalence of reduced and
unreduced homology theories. [Hatcher: pp. 160-162]
09/20/04: Reduced and unreduced homology. Relative homology and
reduced homology of the quotient. [Hatcher: pp. 160-161]
09/17/04: More of homological algebra: chain maps, homotopies, tensor
products, Koszul rule of signs. Generalized homology
theories. [Hatcher: notions (in boldface) and statements of
homological algebra introduced on pp. 111, 113, 218, 273; remark on
uniqueness on p. 161]
09/15/04: Axiomatic homology theory, continued. Basics of category
theory and homological algebra. [Hatcher: pp. 160-165, notions (in
boldface) and statements of homological algebra introduced on pp. 106,
111, 113-114, 116-117]
09/13/04: More on CW complexes: RP^n and CP^n. Operations on CW
complexes. CW pairs, cellular maps, cellular homotopy, the homotopy
categories of CW complexes and CW pairs. Axiomatic homology theory:
axioms for (absolute) homology of CW pairs. [Hatcher: pp. 6-7, 8-10,
160-165]
09/10/04: CW complexes, cellular maps. Examples. Pushouts (or
attaching or gluing spaces). [Hatcher: pp. 5-8, 11, 12-14]
09/08/04: What is Algebraic Topology? The fundamental theorem of
algebra: a topological proof. [Hatcher: pp. 1-4 (Homotopy and Homotopy
Type), pp. 29-31 (The Fundamental Group of the Circle through Theorem
1.8), and pp. 97-101 (Introduction to Chapter 2)]