Suppose we have a compact $m$-dimensional submanifold $M$ of $\mathbb{R}^N$ and we want to know the cohomology ring $H^*(M;\mathbb{Z})$.

Let $\epsilon>0$ and a $m$-dimensional finite simplicial complex $K$ be a trangulation of $M$ such that all the simplices of $K$ have diameters less than $\epsilon$. Then since $M$ is compact, there exists $\epsilon_0>0$ such that for any $\epsilon<\epsilon_0$, $$ H^*(K;\mathbb{Z})=H^*(M;\mathbb{Z}). $$ There is a standard algorithm to compute $H^*(K;\mathbb{Z})$. Hence we have an algorithm to compute $H^*(M;\mathbb{Z})$.

**Question:** are there any software / programming that can give the cohomology ring $H^*(M)$ automatically for any compact submanifold of Euclidean spaces, given by finitely many equations of coordinates?

linear equationsthe answer would be yes. There are affirmative answers for reasonable algebraic equations as well, although the extent to which the algorithms have been implemented in software depends on the exact nature of your question, which you have left a little vague. $\endgroup$