Question about the Poincaré conjecture

In summary, although the Poincaré conjecture deals with closed simply connected 3-manifolds, it cannot be used to determine if the universe is the surface of a 3-sphere. This is because the conjecture does not take into account non-compact manifolds or manifolds with non-empty boundaries, and our observations of the universe being locally simply connected do not necessarily mean it is simply connected as a whole.
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TL;DR Summary
Does Perelman’s proof of the Poincaré conjecture imply that the universe is the surface of a 3 sphere?
Does Perelman’s proof of the Poincaré conjecture imply that the universe is the surface of a 3 sphere?
 
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donglepuss said:
TL;DR Summary: Does Perelman’s proof of the Poincaré conjecture imply that the universe is the surface of a 3 sphere?

Does Perelman’s proof of the Poincaré conjecture imply that the universe is the surface of a 3 sphere?
No.
 
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How can a methematical proof tell us anything about the physical universe?
 
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You haven't provided your reasoning for why you would be curious about this, so I'm left to assume that it's because from our local observations the universe appears to be a simply connected 3-manifold. There are two main reasons this doesn't imply the universe is the 3-sphere:

1) The Poincare conjecture takes as its premise closed simply connected 3-manifolds. These are compact manifolds without boundary. There are an abundance of simply connected 3-manifolds that aren't homeomorphic to the 3-sphere, but they are also non-compact (3-dimensional Euclidean space) or have non-empty boundary (the 3-ball). It is possible that the universe is not a closed manifold.

2) Our observations imply the universe is locally simply connected (i.e. simply connected within some neighborhood of a point). Every manifold is locally simply connected because every manifold is locally Euclidean. However, not every manifold is simply connected.

Hope this helped.
 
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