I'm reading Audin and Damian's Morse Theory and Floer Homology; they say there is an analogous way to define nondegenerate critical points for 1-forms as well as pseudo-gradient fields but don't discuss how. The goal is to define such a vector field for a 1-form $\alpha$ and then lift it to a pseudo-gradient field for a function on a covering space.

Here's the context. Suppose I have a smooth closed 1-form $\alpha$ on a manifold $M$. If I consider a map $\phi:\pi_1(M) \to \mathbb{R}$ which is simply integrating $\alpha$ along a loop in $M$, then it is in fact a homomorphism. I can then consider $\ker \phi \subset \pi_1(M)$ and find a smooth covering space $p: \hat{M} \to M$ such that $p_*(\pi_1(\hat{M}))=\ker \phi$. This means that for all loops $\hat{\gamma} \in \pi_1(\hat{M})$, $$\int_{\hat{\gamma}}p^* \alpha = 0 $$

by construction. Thus, $p^* \alpha$ is exact ($= df$) for some function $f$. We also observe that $(df)_y = 0 \Leftrightarrow \alpha_{p(y)}=0$. Thus, they say that $f$ and $p^* \alpha$ share the same critical points which themselves share properties such as nondegeneracy and index. It seems a critical point for $\alpha$ is simply where it vanishes.

My questions: How are the notions of critical points, nondegeneracy, and pseudogradients defined for a 1-form? Can this be done for $k$-forms?

This paper by Latour is referenced but I can't read French: http://www.numdam.org/article/PMIHES_1994__80__135_0.pdf


1 Answer 1


I know this is an old question, hopefully you already figured it out. But if not:

The closed $1$-form $\alpha$ is, in any small neighbourhood, the differential of a function. The notions you ask about are local and does not depend on which function you take (since they only differ by a constant).

So a critical point of $\alpha$ is a point $x$ where it vanishes (since this is the definition when $\alpha = df$). This $x$ is nondegenerate if the corresponding intersection point $(x,0) \in T^*M$ of the zero section and the image of $\alpha$ is transverse, equivalently, if locally $\alpha = df$ then the Hessian of $f$ at $x$ is invertible (same definition as when $\alpha = df$).

Thus near a nondegenerate zero of $\alpha$ we have a local form given by a Morse chart of any representing function.

A vector field $X$ is a pseudo-gradient of a function $f$ if $df (X) <0$ except at critical points, and near any critical point there is a Morse chart such that $X$ is the gradient of $f$ for the Euclidean metric in the chart. Again these conditions do not depend on $f$ but only on $df$, and you may thus use them to define pseudo-gradients of $\alpha$.

I don't know how this could generalize to forms of higher degree.


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