Suppose that we have an oriented $n$-manifold M and an oriented $i$-manifold $X.$ Consider the mapping space (or function space) $F(X,M)$ equipped with the compact open topology. Now, let $f:M\to M$ be a function of degree $n.$ The $\mathbb{Z}$-homology of mapping spaces seems to be hard. But I was wondering if we could say anything about the action of $F(X, f)$ on the homology groups.
Is $F(X, f)_\ast: H_{n-i}(F(X,M)) \to H_{n-i}(F(X,M))$ just multiplication by $n$? If not, is there some other homological identity it must satisfy? Is there also some non-trivial behavior in the higher homotopy groups? I started the computation directly using chains but got nowhere.
There's an easy case: when $X$ is also a $n$-manifold, $F(X, f)_\ast: H_0(F(X,M)) \to H_0(F(X,M))$ is just sends the connected component corresponding to elements of degree $a$ to $na$.