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As well known, we can prove Poincare duality in terms of morse theory. (By comparing two chain complexes obtained from two morse functions, $f\colon M\to \mathbb{R}$,$-f\colon M\to \mathbb{R}$ for smooth closed manifold $M$)

Of course, I can prove Alexander duality theorem by using usual Lefschetz-Poincare duality theorem and some excision arguments.

But, I want to prove Alexander duality theorem by using a morse function defined on $S^{n}-X$, for a submanifold $X$ of $S^n$.

Are there any references or proofs about this?

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1 Answer 1

There are various versions of stratified Morse theory out there, the granddaddy being the Goresky-Macpherson work. But for these purposes you're fine just using Morse theory on manifolds with boundary, which is a straightforward generalization of plain old Morse theory. In this setting you get Alexander duality "seen" at the chain-complex level in some sense, kind of analogous to Poincare's proof using dual cell decompositions to a triangulation, of the Morse theory/handle proof you refer to.

It's a little different though. The idea is to take the standard height function on $S^n$, and to perturb it so that it restricts to a Morse function on $M \subset S^n$, and on the boundary of a smooth tubular neighbourhood to $M$. Critical points to the height function, restricted to $M$ split into two kinds -- the kinds that contribute cells to the decomposition of the tubular neighbourhood, and the kinds that contribute to the decomposition of the complement. So a $k$-cell for $M$ corresponds to a $(n-k-1)$-cell for the complement. And so on.

I can say more later.

edit: The key idea is that if a function $f : S^n \to \mathbb R$ is morse on a submanifold $M \subset S^n$, then on the boundary of a small tubular neighbourhood of $M$, it's also Morse. Moreover, critical points of $f$ on $M$ correspond to opposite pairs of critical points on the boundary of the tubular neighbourhood (one over the other, in pairs). Think of Milnor's Morse Theory example of a torus in $\mathbb R^3$, but now think of the torus as the boundary of a tubular neighbourhood of a circle. In stratified Morse theory, the critical points of $f$ on the boundary of the tubular neighbourhood break up into two types of cell attachments: (1) cells for the tubular neighbourood, and (2) cells for the complement. The way you can tell (1) and (2) apart, is that the "down" direction pointing outwards for which ever object the cell is contributing to. This is because in stratified Morse theory you modify the flow so that it stays in your manifold with boundary. If the vector field is orthogonal to the boundary and pointing outwards, there's no natural way to modify the flow, so you get a cell. On the other hand if it points inwards, you certainly can.

So if $M \subset S^n$ is a closed submanifold, let $\nu M$ be a compact tubular neighbourhood of $M$. Let $p \in M$ be a critical point of $f$ that contributes a cell of dimension $i$ to $M$ using the downwards flowline. There are two critical points to $f$ above in $\pi^{-1}(p)$ where $\pi : \partial \nu M \to M$ is bundle projection. The "top" critical point contributes an $(n-m-1+i)$-cell to $\partial \nu M$, also to $\overline{S^n \setminus \nu M}$. The "bottom" point contributes an $i$-cell to $\partial \nu M$, also to $\nu M$.

So if for the purpose of generating $M$'s CW-decomposition we use the upward flowlines, then at a critical point corresponding to an $m-i$ cell on $M$, we have an $n-1-(m-i)$ cell attachment for $\overline{S^n \setminus \nu M}$ using the downward flowlines for $f$. These cells are not only complementary dimensional in the Alexander duality sense, but you can write out very explicitly the bounding cells that intersect in a single point.

I hope that helps. It's best to draw a picture and keep track of the dimensions in a few examples before trying the general picture.

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@Ryan: Could you say a little more? And perhaps about how the flowlines are behaving. Thanks! –  Chris Gerig Jul 2 '12 at 16:51

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