Let $M,N$ be closed manifolds. Given a differentiable map $f:M\rightarrow N$, I am interested in computing $f\_k:H\_k(M)\rightarrow H\_k(N)$, in Morse Homology. This problems seems difficult, and the only reference I have found is Schwarz' Morse Homology. His strategy is to factor $f$ as follows
$$
M\rightarrow M\times N\rightarrow \mathbb{R}^n\times N\rightarrow N.
$$ 
where the first map is the graph of $f$, and the second map is an embedding of $M$ into some large $\mathbb{R}^n$, and the third map is a projection to $N$. The first two maps are embeddings of submanifolds, and it is not hard to see what the induced maps must be. Something similar happens with the projection map. 


This seems difficult to compute, because we need to construct an embedding $M\rightarrow\mathbb{R}^n$. I believe that the second and third step can be a simplified a bit, by choosing a suitable function $a$ on $M$ (with one minimum) and a function $b$ on $N$, and constructing an explicit map $C^k(M\times N,a\oplus b)\rightarrow C^k(N,b)$, which descends to homology. 


Has this approach been studied somewhere? Is there any literature on these functioral properties that I missed? Is anything known (and written down) for manifolds with boundary?



Induced maps in Morse Homology
I want to mention an approach described in Kronheimer and Mrowka's book *Monopoles and Three-Manifolds*. In section 2, they give an outline of Morse theory, including Morse homology for manifolds with boundary and functoriality in Morse theory. The nice thing is we can recover the induced map $f\_\* : H\_\* (M) \rightarrow H\_\* (N)$ from a chain map between Morse complexes by counting something.


I'll try to give some idea. Suppose we have a smooth map $r : Z \rightarrow M\_1 \times M\_2$. The compositions $\pi\_i \circ r$ give an induced map $H\_\* (M\_1) \rightarrow H\_\* (M\_2)$, where $\pi\_i$ is the projection to $M\_i$ and we use Poincare duality to get a map from $H\_\* (M\_1)$ to $H\_\* (Z)$. In the special case when $Z$ is a graph of $f : M\_1 \rightarrow M\_2$, this gives back $f\_\*$. This construction is called "pull-up and push-down".


Now suppose that we have Morse complexes for $M\_1$ and $M\_2$. Let $a \in M\_1$ and $b \in M\_2$ be critical points and denote $U\_a, S\_b$ by unstable and stable manifolds. Consider a subspace
$$ Z(a,b) = \lbrace w \in Z \ | \ r(w) \in U\_a \times S\_b \rbrace .$$
Assuming transversality, this is a submanifold of $Z$ and we can count $|Z(a,b)|$ when its dimension is 0. Define a chain map by
$$ m(a) = \Sigma |Z(a,b)|b,$$
where $b$ ranges over all critical points of $M\_2$. It is claimed that this induces a map on homology described earlier. In their book, this idea is treated in the context of Floer homology.


I'm not sure if this construction has appeared elsewhere. I haven't tried doing an explicit calculation from this construction either, but I hope this will be helpful.


