# Generators of local homology groups of an isolated critical point

This is a basic Morse theory question:

Let $M$ be a smooth manifold, and $f:M\to\mathbb{R}$ a smooth function with an isolated critical point $x$. Set $c:=f(x)$. The local homology of $f$ is the relative homology group $C_{\star} (f,x):=H_{\star} ( \{f < c\}\cup\{x\},\{f < c\})$.

Assume that this group is non-trivial in some degree $d>0$. Let $\mathcal{G}$ be the family of (continuous) singular simplexes $\sigma:\Delta^d\to \{ f < c\}\cup\{x\}$ such that $f \circ \sigma (z) < c$ for all $z \in \partial \Delta^d$. Now, let $\mathcal{G}'\subset C_d(f,x)$ be the set of homology classes represented by the elements of $\mathcal{G}$.

Is $\mathcal{G}'$ a set of generators for $C_d(f,x)$?

The answer is clearly YES when $x$ is a non-degenerate critical point, or when it is a local maximum. With a tiny bit of help from Morse theory, I can also say that it is YES when $d$ is the sum of the Morse index and of the nullity of $x$. Finally, if $d$ is the minimal degree at which the local homology is non-zero, then the relative Hurewicz theorem implies that the local homology $C_{d} (f,x)$ is the same as the local homotopy $\pi_{d} ( \{f < c\}\cup\{x\},\{f < c\})$, and the answer is again YES.

How about the remaining cases? I was hoping that there would be a simple argument from algebraic topology that encompasses all the cases?

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