I think that often the topology of things in low dimensions ends up interestingly different to in high dimensions—roughly when your dimensionality gets big enough (often 3, 4, or 5 is “big enough”) there’s enough space to do the things you want without things getting in the way.
One of the proofs I know takes advantage of the fact that f D3×S1 (which is not simply connected) has boundary S2×S1 , which is also the boundary of D2×S2 (which is simply connected); there isn’t room for the analogous trick a dimension down.
I think that often the topology of things in low dimensions ends up interestingly different to in high dimensions—roughly when your dimensionality gets big enough (often 3, 4, or 5 is “big enough”) there’s enough space to do the things you want without things getting in the way.
One of the proofs I know takes advantage of the fact that f D3×S1 (which is not simply connected) has boundary S2×S1 , which is also the boundary of D2×S2 (which is simply connected); there isn’t room for the analogous trick a dimension down.