Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
1 to 4 of 4
For $X$ and $Y$ pointed topological spaces, the monoidalness of the suspension spectrum functor induces a canonical morphism of spectra
$\Sigma^\infty Maps(X,Y) \longrightarrow Maps(\Sigma^\infty X, \Sigma^\infty Y)$Can we say anything useful about this transformation? For instance, may we get control over its cokernel, say when both $X$ and $Y$ are finite complexes with a connectivity bound on $Y$ but no dimension bound on $X$? (So Freudenthal suspension theorem fails, but can we quantify how much it fails, stably?)
I failed to notice the obvious, namely that the map in #1 is right away the projection to the first stage in the Goodwillie-Taylor tower of the functor $Maps(X,-)$
$\Sigma^\infty Maps(X, Y) \overset{p_1}{\longrightarrow} (P_1 Maps(X,-))(Y) = (D_1 Maps(X,-))(Y) = Maps( \Sigma^\infty X, \Sigma^\infty Y )$Thanks to Charles Rezk over on the MO homotopy-chat for patiently pointing this out.
I’d still like get a better idea of bounding $coker(\pi_0(p_1))$.
Perhaps I’m being dense, but what is the $S^4$ doing there in #2?
Thanks for catching. That’s just the choice for $Y$ that I have in mind, of course. Fixed now.
1 to 4 of 4