A manifold is simply connected if it has no ‘holes’: Image for post
(A) is a simply connected space (B) is not simply connected
An equivalent formulation of being simply-connected is that each loop can be continuously tightened to a point.
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(A) a loop in A can be tightened to a point (B) here a loop in B gets ‘caught’ on a hole and can’t be tightened to a point
Two manifolds are homeomorphic if you can deform one into the other and back again continously. Permissable deformations include stretching, squeezing and twisting, but not ripping, tearing and puncturing. This leads to the famous equivalence between a dougnut and a mug.
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How to deform a doughnut into a mug
In topology, we want to classify all manifolds into classes where all manifolds within a certain class are all homeomorphic to each other.
In two dimensions, it is (somewhat) easy to see that if a manifold is closed and without holes then it is equivalent to a sphere. As it turns out, this is sufficient to determine whether a 2-manifold is homeomorphic to the sphere (2-sphere).
Poincaré hypothesised at the start of the 20th century that this is true also in three dimensions, that is any closed, simply connected 3-manifold is homeomorphic to the 3-sphere.
In 2002, Grigori Perelman proved this to be true by using techniques such as Ricci flow and manifold surgery.