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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?

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  • $\begingroup$ This is not explicit in the question, perhaps on purpose, but the very specific space $F(S^2,k+1)$ is also known as the configuration space of $k+1$ points on $S^2$. Googling around shows that there supposedly already is a way (Totaro 1996) to compute the rational cohomology algebra of this space using spectral sequence methods. I guess the thrust of the question is whether an automated approach to this computation (if one is possible) would be advantageous to this alternative. $\endgroup$
    – Igor Khavkine
    Mar 11, 2015 at 13:08

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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.

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