How does Morse theory on non-compact manifolds differ from compact manifolds?

Question

What is the Morse homology of a non-compact manifold? When is it, as in the compact case, isomorphic to singular homology of the underlying manifold? What other constructions can be identified with the Morse homology of a non-compact manifold? Links to places where these questions are discussed would be appreciated.

Answer 1

The power of Morse homology on compact manifolds is that it doesn't depend on the Morse function $f$. This is not true for non-compact manifolds (see below). This "invariance" problem requires extra machinery to interpret the relationship between homology groups computed with different Morse functions. Ultimately, there is not usually a unique Morse homology assigned to a non-compact manifold, so we can't directly compare it to singular homology. This paper by Kang has a good discussion of the invariance problem. I'll include some other ideas below.

Locally we have all the necessary tools, e.g. Morse charts for nondegenerate critical points. As indicated above, the problems are global. Consider a non-compact manifold $M$ with a smooth Morse function $f:M \to \mathbb{R}$, i.e. one with no degenerate critical points. (Such a function is guaranteed to exist: Whitney's embedding theorem lets us embed $M$ into some $\mathbb{R}^N$, and the standard argument in, say, Milnor's Morse Theory lets us define Morse functions on submanifolds of $\mathbb{R}^N$.)

The first problem is that $M$ may be topologically interesting but contain no critical points. For example, this happens if $M$ is the open annulus encircling the $z$-axis in $\mathbb{R}^3$ and $f$ is the $z$-coordinate projection. If we instead embed $M$ as a tube encircling the parabola $\{(x,0,x^2)\in \mathbb{R}^3: x \in \mathbb{R}\}$, the natural way of defining Morse homology would give us $H^f_*(M)\cong H_*(S^1)$, as expected.

Suppose we do have critical points and can define nontrivial chain groups $C_k(f)$ generated by the critical points of index $k$. We want to define a differential that "counts" the number of gradient flow lines from $x \in C_k(f)$ to critical points in $C_{k-1}(f)$. There are two issues here:

1. Since $M$ is non-compact, the gradient flow might not exist for all time $t \in \mathbb{R}$. (That said, you might be able to approximate the vector field and get a complete flow that is still useful, so I won't harp on this point.)

2. Even if the gradient flow exists for all time $t \in \mathbb{R}$, gradient flow lines on a non-compact manifold don't necessarily limit to critical points, e.g. $M=\mathbb{R}$ and $f: M \to \mathbb{R}; x \mapsto -x^2$. So the differential may not be able to send a critical point to (a formal sum of) other critical points.

One way to get around these issues is to define Morse homology for functions whose gradient flow lines are compact. This is mentioned, for example, in the above paper of Kang's. The dependence on $f$ persists, of course.

Answer 2

Schwarz' book discusses Morse homology for coercive Morse functions on non-compact manifolds. I believe one recovers the singular homology.

I think compact manifolds with boundary are a good playground to get intuition for what is happening in the non-compact case. Suppose $(M,f,g)$ is a manifolds with boundary, Morse function, and metric such that the gradient flow of $f$ is always transverse to the boundary. Then the Morse homology is well defined and it computes the singular homology as follows $$ HM_*(M,f,g)\cong H_*(M,\partial M_-) $$ Here $\partial M_-$ is the part of the boundary where the gradient points outwards. If one can connect $(f_0,g_0)$ and $(f_1,g_1)$ using a homotopy where the gradient flow stays transverse to the boundary the Morse homology does not change.

I discuss this more in the introduction of my thesis (http://dare.ubvu.vu.nl/bitstream/handle/1871/52012/complete_dissertation.pdf?sequence=1). See the introduction and the section of local Morse homology.