Induced maps in Morse Homology

Question

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?

Answer 1

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.

Answer 2

This is a hard problem closely related to the following. Suppose that $M, N$ are $CW$ complexes and $f: M\to N$ is a continuous map, not necessarily compatible with the cellular structures on $M$ and $N$. How do we compute the induced map $f_k: H_k(M)\to H_k(N)$? The difficulty lies in the fact that $f$ does not induce maps between the cellular cell complexes so you have to scramble.

There is a way, though quite impractical for Morse cohomology. Use the results of Blaine Lawson, Finite volume flows and Morse theory to exhibit a cochain homotopy equivalence between the Morse complex and the DeRham complex. Then if you are lucky, you can write down explicit closed forms that span the DeRham cohomologies of $M$ and $N$. Finally use Lawson's isomorphism to interpret this in the language of Morse cohomology. This is where it gets tricky.

Let me give you another simple example suggesting that you need to formulate the problem more carefully. Suppose that $M$ is a smooth manifold, and $F:M\to M$ is a diffeomorphism. If $f: M\to \mathbb{R}$ is a Morse function and $g$ is a metric on $M$ such that the gradient flow of $f$ is Morse-Smale, then $F^* f=f\circ F$ is another Morse function and the pullback $F^* g$ is a metric so that the gradient flow of $F^* f$ is Morse-Smale. We obtain two Morse complexes

$$ C_* (M, f, g),\;\;C_*(M,F^*f, F^*g). $$

These are equipped with natural bases, and with respect to these natural bases the induced map

$$ F_* :C_* (M,F^*f, F^* g) \to C_* (M, f, g) $$

is the identity map. However, $F$ may not induce the identity map in homology.

Answer 3

A direct description of functoriality in Morse homology is given by Abbondandolo and Schwarz in Appendix A.2 of "Floer homology of cotangent bundles and the loop product".

If $\varphi: M_1 \to M_2$ is a differentiable map between manifolds, $x$ is a critical point of the Morse function on $M_1$ and $y$ is a critical point of the Morse function on $M_2$, the chain map induced by $\varphi$ is then roughly given by making the intersections $$\varphi(W^u(x))\cap W^s(y)$$ transverse and counting elements of zero-dimensional intersections. Transversality can easily be achieved by perturbing the Riemannian metrics on $M_1$ and $M_2$.

Answer 4

In case anybody is interested we give more detailed proofs of the functoriality described in the other answers by direct analysis of appropriate moduli spaces here.