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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 3rd 2018
• (edited Nov 3rd 2018)

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?)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 6th 2018
• (edited Nov 6th 2018)

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 $coker(\pi_0(p_1))$.

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeNov 6th 2018

Perhaps I’m being dense, but what is the $S^4$ doing there in #2?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 6th 2018

Thanks for catching. That’s just the choice for $Y$ that I have in mind, of course. Fixed now.