MA 751: Topics in Topology

Kate Ponto
Fall 2019

The course description and syllabus.

Announcements

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
JB small skeleton, mapping telescope M p. 25, 30
OM H-space, adjoint A 126, 131
SC generalized cohomology theory, representability A 132
CS Eilenberg-Mac Lane space, BU, BO, BSp A 134
TW direct system, direct limit, Bott periodicity A 136
JB pairing, hopf map A 139, 141
OM J-homomorphism , representable functor A 142, 151
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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
9/23-9/27
9/30-10/4
10/7-10/11

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.