k-connectedness of simplicial complexes

Question

Let $\mathcal{C}$ be a finite simplicial complex. Is there an algorithm, that can tell wether $\mathcal{C}$ is simply connected? If not, are there restrictions to $\mathcal{C}$ under which such an algorithm exists?

Answer 1

No, there is no such algorithm. Here's why.

Given a finite simplicial complex $\mathcal{C}$ and a choice of vertex $v$, there is an algorithm to write down a finite presentation of $\pi_1(\mathcal{C},v)$. Conversely, given a finite group presentation $\langle g_i \,|\, r_j\rangle$ there is an algorithm to construct a finite simplicial complex $\mathcal{C}$ and vertex $v$ such that $\pi_1(\mathcal{C},v)$ has the given presentation.

Thus, an algorithm as you have asked for exists if and only if an algorithm exists to decide whether, given a group presentation, the group is trivial.

But no such algorithm exists: https://berstein2015.wordpress.com/2015/02/17/just-about-any-property-of-finitely-presented-groups-is-undecidable/

Roughly speaking, the reason no such algorithm exists is that you can algorithmically encode the halting problem for Turing machines into the problem for deciding whether a group presentation presents the trivial group. Hence, if you could decide the latter problem, then you could decide the former, which cannot be done.