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It seems pretty clear to me that if you fix $n$ and look at finite simply connected n-dimensional simplicial complexes then the (rational) homotopy equivalence problem is decidable. It's pretty clear that construction of (rational) Postnikov towers is algorithmic. Comparing two Postnikov towers is a sequence of obstruction problems, each decidable. And you don't need to compare full Postnikov towers, it's enough to compare up to height $n$ (the rest are determined automatically). I have a suspicion that the homeomorphism problem is also decidable.
It seems pretty clear to me that if you fix $n$ and look at finite simply connected n-dimensional simplicial complexes then the (rational) homotopy equivalence problem is decidable. It's pretty clear that construction of (rational) Postnikov towers is algorithmic. Comparing two Postnikov towers is a sequence of obstruction problems, each decidable. And you don't need to compare full Postnikov towers, it's enough to compare up to height $n$ (the rest are determined automatically). I have a suspicion that the homeomorphism problem is also decidable.