# Computing homology of subvarieties of Euclidean spaces by persistent homology

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?