Thread by @dzackgarza

1/ Finally had some time to review my conference notes, so here's a thread on the talks!

The actual abstracts are here: math.virginia.edu/2019/09/mid-at…

2/ The first talk was about how certain H-spaces (a generalization of topological groups) -- those with no "v_n torsion" -- can be expressed as product of certain irreducible H-spaces Y_k that 1) also have no v_n torsion, and 2) are somewhat understood.

3/ This parallels how spaces can be characterized (up to homotopy equivalence) as "twisted products" of Eilenberg-MacLane spaces (the "atoms" of homotopy theory) using Postnikov towers.

4/ You can build these towers by attaching cells to a space via maps that annihilate homotopy, and you can analogously annihilate v_n torsion in a similar tower (I think).

5/ It seems that these Y_k are realized as certain spectra (in a range), and their homology is computable using the Atiyah–Hirzebruch spectral sequence.

6/ Some of these Y_k have been identified (S^1 and CP∞), another has something to do with Morava K-theory, but an open problem is sorting out what they are!

7/ The 2nd talk was related to the h-cobordism theorem, which generalizes the Poincaré conjecture (theorem!).

8/ Two n-manifolds X, Y are cobordant when there's an n+1 manifold M whose boundary is X ∐ Y; an h-cobordism just requires that the inclusions of X,Y into M are homotopy equivalences.

9/ The main question here is: if two manifolds are homotopy equivalent, when are they also diffeomorphic?

10/ A strategy here is to build a moduli/classifying space H(M) of "h-cobordisms starting at M".

Then the homotopy groups of H(M) tell you about obstructions to h-cobordisms being trivial.

11/

(A trivial h-cobordism over M just looks like MxI, which is diffeomorphic to M, so I *think* the idea here is that if you can find an h-cobordism H between M and N, then M~N, and if H is trivial, this forces M≅N.)

12/ It seems that H(M) is tough to understand in general, so we try to reduce the problem by stabilizing and look at something called H∞(M).

13/ It seems that Goodwillie believe that this should decompose as a product, but the argument involves "smoothing manifolds with corners", which introduces some choice/coherence issues.

14/ Towards this end, a "multicategory" is introduced, which generalizes a category but allows morphisms to take a k-tuple of inputs. These simultaneously generalize both operads and monoidal categories.

15/ A priori, it seems like there are 2 k-theories around the produce the necessary spectra here, one by Waldhausen and one by Segal. In the multicategory setting, through, it turns out that there is a "multi-natural transformation" between the two.

16/ The third talk was related to the Eilenberg-Maclane spectrum HF2 (where here F2 ≅ Z/2Z as fields).

17/ Spectra represent homology theories, i.e. H^*(X; F2) = [Σ∞X, HF2], where you take a suspension spectrum over X and then extract n'th degree cohomology as the (homotopy classes of) degree -n maps.

18/

Given a group G, you can consider representations of G into vector spaces and get something called the "representation ring". If you replace vector spaces with sets, you get the "Burnside ring", which is supposed to capture how G can act on sets.

19/

Then Segal's conjecture (theorem!) essentially says that you can recover the Burnside ring of G from the stable homotopy groups of the classifying space BG (modulo many many details)

20/ If you specialize to G = C2 (cyclic group of order 2) it seems like this conjecture is equivalent to checking if F2 ≃ (F2∧F2)^tC2.

21/ The inside is a smash product, and the exponent denotes the "Tate construction" (which is the cofiber of a certain trace map where you sum over homotopy orbits to get fixed points, I think).

22/ The role this plays is that if you take the trace map (F2)_hC2 ->(F2)^hC2, there is a lift of this map (the "transfer" map) F_hC2 -> F^C2, where this new thing is the *actual* (not just homotopy) fixed points.

23/ So the idea seems to be that we can sometimes produce maps that relate homotopy fixed points to actual fixed points, where the former is perhaps more amenable to homotopy theory?

24/ There are some open questions here concerning how classical objects fit into this framework. For example, Hill-Hopkins-Ravenel work with a "norm" for spectra X, denoted NX, to resolve Kervaire Invariant One.

25/ There is a pullback diagram that similarly relates (NX)^C2 (actual fixed points) to (X∧X)^hC2 (homotopy fixed points).

26/ The statement of the Segal conjecture here is that the map (NX)^C2 -> (X∧X)^hC2 is in fact 2-completion. Letting X = F2, we can produce a seemingly "classical" object (F2∧F2)^hC2, but it's not known what its homotopy groups are.

27/ It seems like various spectral sequences might be an approach (Homotopy Fixed Point, Adams, Slice).

28/ The last talk was on operads, which I admittedly know less about! The idea of an operad seems to be to model operations with many inputs and a single output (i.e. "k-ary" functions).

29/ There is some natural way to get an operad out of any monoid, so there is some equivalence between operads and symmetric monoidal categories.

The idea is to produce a type of localization that "inverts" these operations.

30/ If you are in a category with weak equivalences (e.g. Top with homotopy-equivalences), there is the usual "localize at weak equivalences" functor.

31/ It turns out we can build an equivalent category out of simplicial sets and taking its homotopy category, called the "hammock localization".

In operads, we can do a similar construction: given an operad O and a submonoid W, we can "localize at W" to get L_W(O).

32/ The construction involves writing operations as directed trees, and then we glue leaves together based on the structure of W.

So there is some natural sequence W -> O -> L_W(O), and we can consider the pairs of categories arising from W and O, say C(W) and C(O).

33/ So we could compare the usual hammock localization of (C(O), C(W)) on one hand, or the hammock localization of the categories (C(L_W(O)), C(W)).

34/

An open question here is whether or not there are any equivalences here, i.e. does the operad localization commute with the categorical one?

35/ And that's it! I can't claim I understand much about the topics, but it's really fun trying to wrap my head around new ideas. Hopefully I'll get around to some threads about older conferences sometime!

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