The French mathematician Henri Poincaré offered a statement known as “Poincaré’s conjecture” without a proof. It states that any 4-dimensional ball is equivalent to 3-dimensional Euclidean space topologically: a continuous mapping exists so that it maps the former ball into the latter space one-to-one.

At first glance, it seems to be too paradoxical for the following mismatches: the former is 4-dimen-sional and as if “closed” unlike the latter, 3-dimensional and as if “open” according to common sense. So, any mapping seemed to be necessarily discrete to be able to overcome those mismatches, and being discrete impies for the conjecture to be false.

Anyway, nobody managed neither to prove nor to reject rigorously the conjecture about one century. It was included even in the Millennium Prize Problems by the Clay Mathematics Institute .

It was proved by Grigory Perelman in 2003 using the concept of information.

One may notice that the 4-ball is almost equivalent topologically to the “imaginary domain” of Minkowski space in the following sense of “almost”: that “half” of Minkowski space is equivalent topologically to the unfolding of a 4-ball. Then, the conjecture means the topological equivalence of the physical 3-space and its model in special relativity. In turn, that topological equivalence means their equivalence as to causality physically. So, Perelman has proved the adequacy of Minkowski space as a model of the physical 3-dimensional space rigorously. Of course, all experiments confirm the same empirically, but not mathematically as he did.