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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 3rd 2018
   • (edited Nov 3rd 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexFor $X$ 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?)

   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?)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 6th 2018
   • (edited Nov 6th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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 $coker(\pi_0(p_1))$.

   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 $coker(\pi_0(p_1))$.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 6th 2018
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   Author: DavidRoberts
   Format: MarkdownItexPerhaps I'm being dense, but what is the $S^4$ doing there in #2?

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 6th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for catching. That's just the choice for $Y$ that I have in mind, of course. Fixed now.

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