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The answer to your question is yes, of course. The theory has been around at least since the late 60s! See Wasserman's paper A Wasserman. Equivariant differential topology, Topology 1969; 8(2):12 …

Your question is precisely the subject of Justin Curry's recent preprint. Bottom line: if you agree to identify functions $f,g:[0,1] \to \mathbb{R}$ whenever they have the same merge-tree, then ther …

Let $f:X \to \mathbb{R}$ be a Morse function on some compact submanifold $X \subset \mathbb{R}^n$, and assume that $p \in X$ is not a critical point of $f$. For some $\epsilon > 0$ let $D_\epsilon(p)$ …

Let $\mu:M \to \mathbb{R}$ be a fixed surjective smooth function on a smooth manifold $M$. Let $N$ be a smooth compact manifold that embeds smoothly into $M$ via $\iota:N \to M$. What conditions o …

Consider a compact sub-manifold $X \subset \mathbb{R}^n$ of Euclidean space and let $f:X \to \mathbb{R}$ be any smooth function. Recall that $x \in X$ is a critical point of $f$ if the gradient $\nabl …

This is a rapidly developing area, and there are many short-cuts if all you want to do is compute the homology of sub-level sets of $f$. To answer your main question, as Liviu has already mentioned: t …

This might be ridiculously obvious, but... For each $n \in \mathbb{N}$, let $M_n$ denote the manifold of $n \times n$ matrices with real entries. It is well known that the $n$-dimensional determinant …

The answer to your question as stated is no. What discrete Morse theory gives you, starting from a finite regular CW complex $X$ and a discrete Morse function $f:X \to \mathbb{R}$ (with discrete vec …

I realize that I am several months late to the Cerf theory party, but this paper of Chari and Joswig might be of interest to the original poster and certainly deserves a mention in the context of this …