MA 751: Topics in Topology

Kate Ponto
Fall 2019

The course description and syllabus.


My office hours this semester are MWTh 10:30-11:30.

Copies of the references for this class are available on the course canvas page.

Vocabulary responsibilties

Vocabulary Source Finished!
SC exact sequence of pointed set, coproduct, replete M, p. 4 8/26
CS natural homeomorphism, small and weak generator, natural equivalence M, p. 5 8/30
TW shift suspension, wedge axiom M, p. 5 8/30
JB Eckman-Hilton duality, loop space functor, M, p. 6 9/4
OM colimit, additive category M p. 7,8 9/6
SC sum, zero object, symmetric monoidal category M p. 8, 11 9/9
CS Hurewicz theorem, Whitehead theorem M. p. 11 9/9
TW graded category, additive functor M p. 21, 22 9/23
JB small skeleton, mapping telescope M p. 25, 30 9/30
OM H-space, adjoint A 126, 131 10/7
SC generalized cohomology theory, representability A 132 10/7
CS Eilenberg-Mac Lane space, BU, BO, BSp A 134 10/9
TW direct system, direct limit, Bott periodicity A 136 10/9
JB pairing, hopf map A 139, 141 10/11
OM J-homomorphism , "usual categorical things about sums and products", trivial object, split short exact A 142, 151, 152, 156 11/11
SC Brown Representablity, "addition in the sets [X,Y]", functor of two variables A 156, 157 11/13
CS telescope, Thom complex, 2-plane bundle A 171,175 11/20
TW SO(2) bundle, element that classifies a bundle, \pi_1(SO) A 175, 180 11/25
JB Whitney sum, pull back a bundle, pi_3(BSO(3)) A 180, 182, 184 12/2
OM Compactly generated, weak Hausdorff space, enriched category, tensor and cotensor in an enriched category L 234, 243 12/9
SC Yoneda Lemma, tensors and continuous functors L 243, 244 12/9
CS Internal/categorical hom, complete and cocomplete with limits and colimits constructed levelwise MMSS 447 12/9
TW equalizer, comparison of represented functors, MMSS 447, 448 12/9

Completed Reading

Monday Wednesday Friday
8/26-8/30 M p. 3,4 M p. 4,5 M p. 5,6
9/2-9/6 M p. 6,7 M p. 7,8
9/9-9/13 M p. 8-11 M p. 11
9/16-9/20 M p. 11-14 M p. 14-17
9/23-9/27 M p. 17-21 M p. 22-28 M p. 28
9/30-10/4 M p. 28-31 M p. 31-32, A p. 123-124 A p. 124-128
10/7-10/11 A p. 128-134 A p. 134-138 A p. 138-143
10/14-10/18 A p. 143-147 A p. 147-148 A p. 148-149
10/21-10/25 A p. 149-151 A p. 151
10/28-11/1 A p. 151-152
11/4-11/8 A p. 152-153 A p. 153 A p. 153-154
11/11-11/15 A p. 154-156 A p. 156-161 A p. 161-169
11/18-11/22 A p. 169-174 A p. 174-178 A p. 178-180
11/25-11/29 A p. 180-184
12/2-12/6 A p. 184-190 L p. 233-240 L p. 240-243
12/9-12/13 L p. 243-254, MMSS p. 446-448 MMSS p. 448-450 MMSS p. 450-453

Things we got stuck on

10/16Where should the primes be the proof of 3.2?A p. 148
11/8 "usual cateorical things about sums and products"A p. 153
11/11 cone on a cofinal spectrum cofinal in the cone on the original spectrum? A p. 154

Things to know about the sources

8/23 Triangulated categories will not be a big deal for us (all we need is in prop 1 of M) so look at appendix 2 if you are really interested, but you can also just ignore it.
8/23We will not be covering the proofs of properties of the homotopy category of CW complexes that are outsourced from M.
9/13 In M chapter 2 there are several technical terms that are used more informally: "derived" means that we've taken a category and forced some collection of maps to be isomorphisms. "coherence" is asking about what sort of comptibility is reasonable to ask for between some morphisms. Reasonable usually means what is given by the examples. "completion" means that we have a category that doesn't have enough objects, morphisms, or both and we want to find a category it sits inside that is just big enough to contain the things we are missing. "handcrafted smash products" is just the techical term for this particular functor.
9/13 When we get to it, read the introduction (chapter 1) of Adams for motivation - becoming really comfortable with all the ideas he references would take many semesters.