# Defining a pseudo-gradient field for a $1$-form

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

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 $$\alpha$$.