If we write a manifold or CW-complex $X$ as a subset of $\mathbb{R}^n$, in expression of coordinates, for example, \begin{multline} F(S^2,k+1)=\{(x_1,x_2,x_3,\cdots, x_{3k+1},x_{3k+2},x_{3k+3})\in\mathbb{R}^{3k+3}\\ \mid x_1^2+x_2^2+x_3^2=1,\cdots, x_{3k+1}^2+x_{3k+2}^2+x_{3k+3}^2=1,\\ \text{ for }i\neq j, x_{3i+1}\neq x_{3j+1} \text{ or } x_{3i+2}\neq x_{3j+2} \text{ or }x_{3i+3}\neq x_{3j+3} \}, \end{multline} is there any computer software or programming that can give the cohomology algebra automatically?

Can the computer give a very complicated simplicial complexes to approximate the manifold and compute the cohomology algebra?