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

    For XX and YY pointed topological spaces, the monoidalness of the suspension spectrum functor induces a canonical morphism of spectra

    Σ Maps(X,Y)Maps(Σ X,Σ Y) \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 XX and YY are finite complexes with a connectivity bound on YY but no dimension bound on XX? (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,)Maps(X,-)

    Σ Maps(X,Y)p 1(P 1Maps(X,))(Y)=(D 1Maps(X,))(Y)=Maps(Σ X,Σ Y) \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(π 0(p 1))coker(\pi_0(p_1)).

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 6th 2018

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

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 6th 2018

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