Here's a naive question: Whats the relationship between the connectivity of a map and connectivity of its cofiber? In the case where all fundamental groups are zero I know how to prove (using hurevitz and serre ss) that if the cofiber is $n$-connective then the map is $n$-connective (maybe upto a constant minus/plus 1). I'm wondering about the general case, is it true without restriction on $\pi_1$?

Just in case somebody knows and is willing to help. (I have to admit that I know almost nothing about 2-categories, bicategories and similar stuff.)

If I may come back to my issue with equivariant cohomotopy of the point in non-trivial RO(G)-degree, on the off chance some equivariant homotopy theorist here could offer some help:

I have most of the proof that the G-equivariant cohomotopy of the points in some RO(G)-degree V is an extension of the Burnside ring A(G/N) whenever N is a normal subgroup of G such that V^N = 0.

I have written this out here:

But currently I use one step in this proof which looked trivial on first sight, but really requires an argument.

So there remains a gap.

ncatlab.org/nlab/show/…

@UrsSchreiber The G-set S in A(G) is the trace of the strongly dualizable spectrum Sigma^infty S_+; this is intrinsically defined and preserved by any symmetric monoidal functor (as you probably know better than I do). Consequently the map A(G) -> A(G/N) induced by the symmetric monoidal partial geometric fixed points functor Phi^N must send S to the trace of Phi^N(Sigma^infty S_+). Since Phi commutes with Sigma^infty, your claim follows.

@TomBachmann thanks for offering help! Sorry for being slow, but I need to think about this: 1)How do we see that the comparison map to the partial fixed point spectrum is monoidal? and 2) Why is the G-set that trace?

@UrsSchreiber (1) That the geometric fixed point functor is symmetric monoidal is a classical fact (proven, e.g., in Hill-Hopkins-Ravenel). One way of seeing this is that it is the localization induced by the idempotent object $Σ^∞\widetilde{EP_N}$ where $P_N$ is the family of subgroups not containing N

(2) follows from the identification of the unit and counit morphisms for the self dual spectrum S (plus the tom Dieck splitting, ofc).

A very high-powered way of proving (1) is also Remark 9.9 here: arxiv.org/pdf/1711.03061.pdf

Thanks. I am aware that the geometric fixed point functor is symmetric monoidal, but what I'd seem to need here is that the comparison map from E-cohomology to Phi^N E-cohomology to be symmetric monoidal, no?

I'm not sure I follow. The two spectra don't even live in the same category

Here by "trace" Tom means the thing that associates to a (strictly) dualizable object X of a sm category the composition $tr(X):=[1→X^\vee\wedge X→1]∊[1,1]$

So every time you have a symmetric monoidal functor F you get the identity $F(tr(X))=tr(F(X))$

@DenisNardin the cohomologies though are both spectra or abelian groups. Explicitly, I am concerned with this map right here: ncatlab.org/nlab/show/geometric+fixed+point+spectrum#Hmm

I still do not follow. The map A(G)→A(G/N) is literally the map $[1,1]_{Sp^G}→[1,1]_{Sp^{G/N}}$ induced by $\Phi^N$

Tom's description of [S] as the trace of Σ^∞_+S shows that this map sends [S] to [S^N] (as we would expect)

Thanks. In that proof I was trying to complete, the map A(G) --> A(G/N), does not manifestly arise this way, instead it arises as a component map of the colimit in Prop. II 9.13 IN Lewis-May-Steinberger.

Maybe I should abandon that whole proof strategy, but this is in any case what I was asking about: Why is that component projection behind the link I gave equivalent to that evident/canonical map?

Oh, but that's pretty much the same thing

You can identify the two things if you identify $\Phi^N$ with the smashing localization associated to the idempotent $Σ^∞\widetilde{EP_n}$

Since you can show that $EP_n$ is the colimit of all those S^V's

(the argument for that is in the proof of II.9.13 in LMS, where they call what I call $\widetilde{EP_N}$ $\tilde E\mathcal{F}_N$

Thanks! That helps. The statement about smashing localization is secretly what they have around Prop. Ii 9.4, I suppose.

I expect it would all be pretty trival if there were a 2018-analog of Lewis-May-Steinberger, and probably that's exactly what exists internal to you guys.

My task is to write out a proof that can be recognized by readers who are less than homotopy experts...

I will have to call it quits for tonight, thanks a lot for the chat.

But while we are at it, allow me to ask: I just meant to prove here that S_G^V(pt) --> S^0_G/N(pt) is surjective when V^N = 0. But ideally I'd like to know S^V_G(pt) completely. What can one say?

So, thanks to the tom Dieck splitting $(Σ^∞S^V)^G=\bigoplus_{(H)} (Σ^∞S^{V^H})_{hWH}$, so the group you're asking for is $\bigoplus_{(H)}π_0(S^{V^H})_{hWH}

(I'm assuming the group is finite for the moment 'cause I can never remember how the shift go in the general case)

So, I guess the question is, for which H is (S^{V^H})_{hWH} connected?

Note that the homotopy quotient is taken in pointed spaces