Let $X$ be a reasonable topological space (say one that has the homotopy type of a finite CW complex) and consider a subset $\Gamma$ of $X \times X$ so that the projection $p:\Gamma \to X$ onto the first factor is surjective (making no assumptions on the other projection $q$):


$$ X \stackrel{p}{\gets} \Gamma \stackrel{q}{\to} X$$


Let's call a function $f:X \to X$ *supported* by $\Gamma$ if $(x,f(x)) \in \Gamma$ for each $x$ in $X$. In other words, the graph of $f(x)$ must be a subset of $\Gamma$. Here's my question:



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> Is there a "classification" of all possible endomorphisms $H\_\bullet f: H\_\bullet X \to H\_\bullet X$ on (integral or field) homology induced by continuous $f$'s which are supported by (a fixed) $\Gamma$?
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As an initial example, note that if the first projection $p$ has contractible point-inverses, then there is a unique endomorphism supported by $\Gamma$. I'd also be happy with answers to the "opposite" question



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> Is there a (homological?) obstruction to a given endomorphism $\phi: HX \to HX$ being induced by a $\Gamma$-supported $f$?
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Thanks!



Classifying space for homology endomorphisms supported on a graph?
One obvious condition is that $H\_\*f$ must be a composition of a right inverse of $H\_\*p$ with $H\_\*q$, since $f$ will be the composition of a section of $p$ with $q$.


