The space-time interpretation of Poincare’s conjecture proved by G. Perelman

Background and prehistory :

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.
Physical interpretation in terms of special relativity:

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.

An idea of another proof of the conjecture based on that physical interpretation :

Topologically seen, the problem turns out to be reformulated so: one needs a proof of the topological equivalence of a 4-ball and its unfolding by 3-balls (what the “half” of Minkowski space is, topologically).

If one adds a complementary, second unfolding to link both ends of the first unfolding, the problem would be resolved: 4-ball would be equivalent to two 3-spaces topologically. Two 3-spaces are equivalent to a single one as follows: one divides a 3-space into two parts by a certain plane (that plane does not belong to any of them). Any part is equivalent topologically to a 3-space for any open neighborhood is transformed into an open one by the mapping of each part (excluding the boundary of the plane) into the complete 3-space.

That idea is linked to the original proof of Perelman by the concept of information. It means that any bit of information interpreted physically conserves causality. In other words, the choice of any of both states of a bit (e.g. designated as “0” and “1” recorded in a cell) does not violate causality (the cell, either “0” or “1”, or both “0” and “1” are equivalent to each other topologically and to a 3-space).

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