I'm trying to learn the basics of homotopy theory by studying from Strom's Modern Classical Homotopy Theory, a book in which the proof of every proposition presented is given as a guided exercise to the reader.

Right now I'm working with homotopy colimits, and I've been stuck with Project 6.26 at page 154 since this morning.

In this problem it is asked to study the set $\mathcal{S}(f)$ of homotopy classes of maps $\Sigma X\rightarrow\Sigma Y$ induced on homotopy colimits by the natural transformation $$ \require{AMScd} \begin{CD} \ast @<<< X @>>> \ast\\ @VVV @V{f}VV @VVV \\ \ast @<<< Y @>>>\ast \end{CD} $$ As suggested by the author, the two maps $\Sigma f$ and $-\Sigma f$ can be obviously found just by taking the cofibrant replacements $$ \require{AMScd} \begin{CD} C_{-}X @<<< X @>>> C_{+}X\\ @V{Cf}VV @V{f}VV @VV{Cf}V \\ C_{-}Y @<<< Y @>>> C_{+}Y \end{CD} $$ and $$ \require{AMScd} \begin{CD} C_{+}X @<<< X @>>> C_{-}X\\ @V{Cf}VV @V{f}VV @VV{Cf}V \\ C_{-}Y @<<< Y @>>> C_{+}Y \end{CD} $$ which may even be in the same homotopy class.

My question is: is there any other element in $\mathcal{S}(f)$? If not, how can I prove that?

  • 1
    $\begingroup$ In the simplest case take $X$, $Y$ as wedges $X\simeq X_1\vee X_2$, $Y\simeq Y_1\vee Y_2$ and do the same with the map $f=f_1\vee f_2:X\rightarrow Y$. From the example you give, its easy to construct four potentially different maps, and not any harder to take $X_1=X_2=Y_1=Y_2=S^n$ and $f_1,f_2$ non-trivial to show that it is possible for all four maps to be different. If I come up with a more exciting example I'll write you a full answer later. $\endgroup$ – Tyrone Oct 3 '18 at 12:31

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