High-Dimensional Topology vs. Low-Dimensional Topology: What are the hard questions in the former?

Question

This is a somewhat vague/non-technical question.

I've heard a lot about how the topology of manifolds (smooth or otherwise) is simpler in dimension at least 5, due to the applicability of surgery theory and things like the h-cobordism theorem. In particular, questions such as classification of simply-connected manifolds are much more easily solved in dimension at least 5. Is this a general phenomenon in (geometric) topology? Or are there some natural questions about manifolds which have easier answers in dimension 3 and 4 than in dimension greater than 5? If so, what is an example?

(I've heard that (Riemannian) geometry can be harder in high dimension, but I'm curious if there are more purely topological questions)

Thanks!

Answer 1

1. Algorithmic PL homeomorphism problem for compact manifolds is solvable in dimensions $<4$, is unsolvable in higher dimensions.

2. Borel conjecture holds in dimensions $<4$, is wide open in higher dimensions.

3. Smith conjecture is proven in dimension 3, fails in higher dimensions.

4. A good classification of compact manifolds is known in dimensions $<4$, is hopeless in higher dimensions.

Answer 2

I'm far from an expert on this, but apparently the question "Does there exist a compact $n$-manifold not homeomorphic to a simplicial complex?" fits your criteria.

In dimension 3 or less, the answer is "no", and this is well known.

In dimension 4, the answer is "yes", for example the $E_8$ manifold, and this is a difficult result (due to Freedman) but it has been known since the '80s.

In dimension 5 or greater it was not known until last year, when Manolescu answered in the negative.

For a bit of history on this problem, wikipedia is a good place to start: http://en.wikipedia.org/wiki/Hauptvermutung