# Question about the Poincaré conjecture

• A
• donglepuss
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.

#### donglepuss

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?

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.

Office_Shredder
How can a methematical proof tell us anything about the physical universe?

Hornbein and dextercioby
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.

dextercioby and PeroK