# The homotopy cofiber of the smash product of two maps of spectra

It is a standard fact that smashing with a fixed spectrum $Z$ preserves cofiber sequences. So if I have a cofiber sequence $$X \xrightarrow{f} Y \rightarrow C_f$$ then there is also a cofiber sequence $$Z \wedge X \rightarrow Z \wedge Y \rightarrow Z \wedge C_f$$

If more generally I have a map $Z \xrightarrow{g} W$, is there any formula for the cofiber of the map $$Z \wedge X \xrightarrow{g \wedge f} W \wedge Y$$ in terms of $C_f$ and $C_g$? (The above discussion corresponding to $g = \mathrm{id}_Z$).

By factoring $g\wedge f$ as $X\wedge f$ followed by $g\wedge Y$ you see that it's in the middle of a cofiber sequence $Z\wedge C_f\to \ ?\to C_g\wedge Y$. Similarly it's in the middle of a cofiber sequence $C_g\wedge X\to\ ?\to W\wedge C_f$. It's also in the middle of a cofiber sequence $(Z\wedge C_f)\vee (C_g\wedge X)\to \ ?\to C_g\wedge C_f$. We could probably come up with more. But no formula in the sense that I think you mean. It may be instructive to think of the special case where $X$ and $W$ are both contractible.
• @TomGoodwillie: Consider the localizations maps $X \to P^n X$, $Y \to P^n Y$ and $X \wedge Y \to P^n(X \wedge Y )$. Then can we have a map from $P^n(X) \wedge P^n(Y) \to P^n(X \wedge Y)?$ Here $P^n$ is the Postnikov section. Jun 1, 2020 at 8:23