Subject: Re: three postings
From: Tom Goodwillie <tomg@math.brown.edu>
Date: Tue, 18 Jul 2006 10:42:57 -0400

The proof that I sketched the other day for Victor Turchin 
(who must be the inquiring party here) assumed a connectivity hypothesis, 
which I will now explain.

For a cosimplicial space X consider, for each n>0, the n-dimensional 
cubical diagram which I will call X[n]. It has X^n as "first" space 
and X0 as "last". Each map in it is a codegeneracy map. 
(To be more precise, it is induced by the inclusion of the poset of 
subsets of an n-element set into the category of all nonempty finite 
ordered sets.) 
The hypothesis is that this diagram is (an+b)-cartesian, meaning that the 
canonical 
map from the first space to the holim of the others is (at least) 
(an+b)-connected. 
(This basically means that the nondegenerate part of X^n is 
(an+b-1)-connected. 
Modulo the usual need to fuss about pi_0 and pi_1, it is saying that the 
homotopy group 
spectral sequence vanishes above a certain line.)

If a>1 and b is nonnegative, then it's not at all hard to show (using what 
I call the 
"higher Blakers-Massey theorem and its dual" in my "Calculus 2") that -

(1) the tower of partial totalizations {Tot^s} converges in the sense that 
the connectivity 
of the map Tot-->Tot^s tends to infinity with s,

(2) the same is true for the (levelwise) suspension SigmaX of the 
cosimplicial space X,

(3) the canonical map from Sigma(Tot^s X) to Tot^s(Sigma X) also has a 
connnectivity tending 
to infinity,

and therefore

(4) in the limit you get a weak equivalence SigmaTotX --> TotSigmaX.


It seems to me that the condition b\geq 0 can be relaxed somewhat, 
something like b>-a.

Of course you need some hypothesis to get (4). But it may be that there is 
some sneaky 
approach I am unaware of that makes (4) true even in some cases when (1) 
is not. I'd be curious.


I have not thought about what this same argument gives for more general 
hocolim instead of suspension.

    Tom Goodwillie

>
>
> From: "vitia-t" <vitia-t@yandex.ru>
> Date: Tue, 18 Jul 2006 11:27:33 +0400 (MSD)
>
> Is it well known that the homotopy totalization of a cosimplicial space 
commutes with the suspension
> (under some mild conditions)? I have a sketch of the proof from Tom 
Goodwillie, but I wonder whether
> it was already known.
> Probably in general the homotopy totalization commutes with homotopy 
colimits?
> ____________________________________________________________