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Let $M$ be a submanifold of the Euclidean space $\mathbb{R}^n$. Let $G$ be a finite group acting on $M$ freely. I want to compute the homology (or even the cohomology ring) of $M/G$.

Suppose the coordinates of all points in $M$ are given by some polynomial equations (or inequalities) of the $n$-variables of coordinates of $\mathbb{R}^n$. And the free $G$-action on $M$ is also given by polynomial equations of the $n$-variables of coordinates of $\mathbb{R}^n$. Could we use persistent homology and computer programming codes to compute the homology of $M/G$?

How to do it? Are there any programmes or codes that can be applied?

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This is probably feasible but not easily. The paper

Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time (with P. Bürgisser and P. Lairez). J. of the ACM, 66, pp. 5:1-5:30, 2019.

which as far as I understand solves the question when there's no group action, might indicate the level of difficulty of the problem.

If on top of that you have a group action, you might try to build a G-invariant sample of the manifold and do persistent homology on the simplicial complex built from that sample, after identifying congruent simplices.

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