Is there a long exact cofiber sequence for a homotopy pushout?
By David Jones •
Let $f:Z \to X$, $g: Z \to Y$ and let $M(f,g)$ be the corresponding mapping cylinder. Does the homotopy pushout diagram induce a long cofiber sequence? If so what does it look like?
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$\begingroup$What you can do is the following: if $W$ is an Eilenberg-MacLane space, or more generally an infinite loop space, then $[-, W]$ produces from this homotopy pushout diagram a homotopy pullback diagram of infinite loop spaces, and then this thing produces a long fiber sequence and hence a long exact sequence in cohomology. (Well, part of it: to get the rest you need to extend this long fiber sequence to a long cofiber sequence in the other direction, or equivalently to repeatedly deloop $W$.)
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