This question is inspired by a paper by B. Poonen that appeared on the arxiv some time ago: http://arxiv.org/abs/1204.0299. The paper gives a sample of algorithmically unsolvable problems from various areas of mathematics.

The topology part however contains only two such problems: the homeomorphism problem for 4-manifolds, which was shown to be undecidalbe by Markov in 1958, and the problem of recognizing $S^n,n\geq 5$ up to homeomorphism. The indecidability in both cases basically boils down to the undecidability of the group isomorphism problem.

Note that both the above problems become decidable if one restricts one's attention to simply-connected PL-manifolds. This follows in the first case from the fact that simply-connected PL 4-manifolds are determined up to homeomorphism by the integral cohomology and in the second case from the generalized Poincare conjecture.

This makes one wonder what happens if one imposes some natural topological restrictions like simple connectedness. So I would like to ask if the following problems are decidable for simply-connected finite simplicial complexes, maybe under some further restrictions (e.g. for those those simplicial complexes that are homeomorphic to smooth or PL-manifolds):

the homeomorphism problem

the homotopy equivalence problem

the rational (or mod a prime $p$) homotopy equivalence problem

Personally, I do not hold out much hope that any of these turns out to be algorithmically decidable. For instance, the rational homotopy type of a space $X$ can be seen as an infinite collection of maps $H^{\otimes n}\to H$ of degrees $2-n$ (where $H=H^*(X,\mathbb{Q})$) subject to some condition, up to an equivalence relation, and it looks plausible that all the components in this collection matter. However, it is not completely clear to me how to prove this.