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?



cohomology algebra of submanifold in euclidean space
It surely depends on what expressions you want to allow, 
but what comes to my mind is that even the much more basic question whether 
a given set is empty or not


* is NP-complete, so don't expect an effective algorithm,
* is computable by Tarski's theorem.


Maybe a logician could say more.


