This document discusses the generalization of Poincaré's conjecture to higher dimensions and its interpretation in terms of special relativity. It proposes that Poincaré's conjecture can be generalized to state that any 4-dimensional ball is topologically equivalent to 3D Euclidean space. This generalization has a physical interpretation in which our 3D space can be viewed as a "4-ball" closed in a fourth dimension. The document also outlines ideas for how one might prove this generalization by "unfolding" the problem into topological equivalences between Euclidean spaces.

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

2. Vasil Penchev
Bulgarian Academy of Sciences:
Institute for the Study of Societies and
Knowledge:
Dept. of Logical Systems and Models
vasildinev@gmail.com

3. “Space And Time:
An Interdisciplinary Approach”
3rd annual conference
September 26, 2019 - September 28, 2019
Institute of Philosophy, Vilnius University:
201, Universiteto 9/1, Vilnius, Lithuania

4. Project:
ДН 15/14 - 18.12.2017 to the "Scientific
Research" Fund, Bulgaria: "Non-Classical
Science and Non-Classical Logics.
Philosophical and Methodological
Analyses and Assessments"

5. Contains:
Background and prehistory
A generalization of Poincaré’s conjecture
Physical interpretation in terms of special relativity
An idea for proving the generalization

6. Background and prehistory

7. “Poincaré’s conjecture”
The French mathematician Henri Poincaré offered a
statement known as “Poincaré’s conjecture” without
a proof:
He wrote: “Mais cette question nous entraînerait
trop loin” - as the last sentence of his paper:
Poincaré, H. (1904) "Cinquième complément à
l'analysis situs" Rendiconti del Circolo Matematico
di Palermo 18, 45-110.

8. “Poincaré’s conjecture”
Poincaré formulated the problem in §6 (pp.
99-110) of his Fifth supplement to “Analysis
Citus”:
Meaning the context, he questioned: “Est-il
possible que le groupe fondamental de V se
réduise ál la substitution identique, et que
pourtant V ne soit pas simplement connexe?”

9. The main source [MS] further
Carlson, J. (ed.) 2014. The Poincaré conjecture:
Clay Mathematics Institute Research Conference,
resolution of the Poincaré conjecture, Institute Henri
Poincaré, Paris, France, June 8-9, 2010. Clay math.
proc.: vol. 19. Providence (RI): AMS for CMI
It is freely downloadable from:
http://www.claymath.org/library/proceedings/cmip19.pdf

11. A century of failures to be proved ...
Anyway, nobody managed neither to prove nor to
reject rigorously the conjecture about one century:
Morgan, J.W. 2014. 100 Years of Topology: Work
Stimulated by Poincaré’s Approach to Classifying
Manifolds. In MS, pp. 7-29 (esp. pp. 19-25)
“2.8. Why the Poincaré Conjecture has been so
tantalizing.”

12. One of the seven “Millennium Problems”
It was included even in the Millennium Problems
by the Clay Mathematics Institute and a prize of
$1,000,000 for its solution:
https://www.claymath.org/millennium-problems
“In 1904 the French mathematician Henri
Poincaré asked if the three dimensional sphere is
characterized as the unique simply connected
three manifold.”

13. CMI’s popular representation
“We say the surface of the apple is "simply
connected," but that the surface of the doughnut
is not. Poincaré, almost a hundred years ago,
knew that a two dimensional sphere is essentially
characterized by this property of simple
connectivity, and asked the corresponding
question for the three dimensional sphere.”

14. Grigoriy Perelman’s proof
It was proved by Grigoriy (Grigori, Grigory,
Grisha) Perelman in 2002-2003 published in
Arxiv (freely downloadable):
“The Clay Mathematics Institute hereby awards
the Millennium Prize for resolution of the
Poincaré conjecture to Grigoriy Perelman” (Press
Release of March 10, 2010. In MS, pp. vii-xv: vii)

15. Perelman’s “Нет!”
“He said nyet to $1 million. Grigory Perelman, a
reclusive Russian mathematics genius who made
headlines earlier this year for not immediately
embracing a lucrative math prize, has decided to
decline the cash” (Ritter, M. 1.7.10. "Russian
mathematician rejects $1 million prize"
https://phys.org/news/2010-07-russian-mathematici
an-million-prize.html )

16. Perelman’s explanation
“But the Interfax news agency quoted Perelman
as saying he believed the prize was unfair.
Perelman told Interfax he considered his
contribution to solving the Poincare conjecture no
greater than that of Columbia University
mathematician Richard Hamilton” (Ritter, M. July
1, 2010, see the previous slide)

17. Perelman’s explanation in original
"Если говорить совсем коротко, то главная
причина - это несогласие с организованным
математическим сообществом. Мне не нравятся
их решения, я считаю их несправедливыми. Я
считаю, что вклад в решение этой задачи
американского математика Гамильтона ничуть
не меньше, чем мой"
(https://www.interfax.ru/russia/143603 )

18. A generalization of
Poincaré’s conjecture

19. From a “3-sphere” to a “4-ball”
One can mean not only a “3-sphere” in
4-Euclidean space, but the “internality”
furthermore: a “4-ball” bounded by the 3-sphere
The 4-ball is more relevant for the intended
physical interpretation by Einstein’s special
relativity and Minkowski space as its
mathematical model

20. More about the generalization
If one “unfolds” a 4-ball in 3 dimensions,
the 3-sphere would be “unfolded” in 2 dimensions
The 2-dimensional boundary of the 3-dimensional
unfolding of the 4-ball would be the 3-sphere
unfolded
Further, we mean the generalization as to the
space-time physical interpretation

21. Our space as a “4-ball”, topologically
The generalization would state that any
4-dimensional ball (or “3-sphere”) is equivalent to
3-dimensional Euclidean space topologically:
Though Euclidean space is open “by itself”, as if
it can be “closed” in an additional, fourth
dimension at least in a topological sense is what
the generalized conjecture supposes

22. What the generalization means
A continuous mapping exists so that it maps the
former ball into the latter space one-to-one
A visualization: one might deform gradually (i.e.
mathematically continuously, or even smoothly)
the closed ball transforming it into the open space

23. Seemingly paradoxical
At first glance, that seems to be too paradoxical for
a few mismatches
The suggested visualization might illustrate mis-
matches forcing the conjecture to seem paradoxical:
One might cancel the fourth dimension of the ball
gradually transforming it into the openness of the
space

24. Both mismatches
The former is 4-dimensional and as if “closed”
unlike the latter, 3-dimensional and as if “open”
according to common sense
The conjecture “equates” the two misfits in a way
to compensate each other absolutely at least
topologically
Openness is equivalent to a new dimension closed

25. Discreteness and continuity
So, any mapping seemed to be necessarily
discrete to be able to overcome those
mismatches ...
… and being discrete, this implies for
the conjecture to be false
As if: continuity in an additional dimension and
discreteness without it might be the same ...

26. Physical interpretation
in terms of special relativity

27. A unit 4-ball unfolded in 3 dimensions
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

28. Unfolding a 4-ball
One would obtain a 3-dimensional unfolding of
a 4-ball as follows:
1. Cutting the ball by a 3-dimensional “knife” in
an arbitrary 3-dimensional (usual) ball
2. A (well-)ordered set of 3-balls parameterized
from “-r” to “+r” (where “r” is the radius of both
4-ball and 3-ball is the unfolding at issue)

29. The case of infinite radius ...
Then, the unfolding would be an ordered set of
3-balls
Further, each ball is topologically equivalent to the
internality (i.e. without the surface) of a “finite” ball
with its radius equal to its parameter
The unfolding of an “infinite” 4-ball is topologically
equivalent to any domain of Minkowski space
without the light cone

30. What remains
1. The “knife” as to the topological difference of
an “infinite” 4-ball and its 3-unfolding
2. The “light cone” as to the “half” of Minkowski
space
Might they be the same?
Topologically, obviously yes!

31. To whom it was not obvious
The light cone is an ordered set of spherical
surfaces parametrized by its radius from “minus
infinity” to “plus infinity”
So, the light cone in turn is a 2-unfolding of the
3-”knife” assigned to a corresponding ball by the
same parameter (= their radius) after unfolding

32. An idea for proving Poincare’s conjecture
Indeed, here is a series of topological equivalences:
1. The “knife” space “is” Euclidean space
2. The light cone “is” the “knife” space
3. The 3-sphere “is” the light cone
4. Consequently, the 3-sphere “is” Euclidean space
“Is” means ‘is topologically equivalent to’

33. The idea seen by elastic deformations
1. One “pulls” Euclidean space in the fourth
dimension transforming it into the “light cone” of
the future: the point of pulling is the present
2. One “deforms” that light cone to
an infinite 3-sphere (or 3-hemisphere):
the deformation is continuous
3. One “shrinks” the infinite 3-sphere to a unit one

34. The physical meaning of the generalization
Then, the generalization 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
Anyway, causality is irreversible, and continuity is not

35. The irreversibility of causality
Time’s arrow is what implies the irreversibility of
cause and effect
However, Minkowski space as the model of
special relativity means time to be space-like and
thus reversible just as continuity is
Time’s arrow is a consistent complement to
Minkowski space within special relativity

36. “Time’s arrow”
Time’s arrow is absent in both Minkowski space
and generalized conjecture
It is represent in both by continuity
It can be added consistently to both as a
complemental and restricting condition if one
considers each of them as a mathematical
construction relevant to special relativity

37. Causality as topological equivalence
Causality means continuity in a physical sense, and
the topological equivalence conserves it
Indeed, а continuous series of shrinking
neighborhoods links the cause to the effect
That is the set-theory and topological interpretation
of a continuous series of logical implications between
them

38. In other words ...
So, Grisha Perelman proving Poincare’s conjecture
has proved furthermore the adequacy of Minkowski
space as a model of the physical 3-dimensional
space rigorously
A model containing any topological mismatch
would not conserve causality: that causal violation
would reject special relativity

39. A mathematical proof of causality
Of course, all experiments confirm the same
empirically, but not mathematically as Perelman
did
Perelman’s proof excludes any experimental
refusal in future and in principle as to causality
Furthermore and rather shocking: the other “half”
of Minkowski space is not less relevant causally

40. An idea for proving
the generalization

41. “Unfolding” the problem
Topologically seen, the problem turns out to be
reformulated so:
One needs a proof of the topological equivalence
of the “infinite” 4-ball and its unfolding by 3-balls +
the “knife”:
That is what the “half” of Minkowski space is,
topologically

42. “Unfolding” the problem (2)
Poincare’s conjecture means a finite or “unit”
3-sphere rather than an “infinite” one
Its generalization means an infinite 4-ball in
general and implicitly, an infinite 3-sphere
The finite and infinite ones of the same kind
are equivalent topologically to each other
Thus, the mismatch is not essential

43. Indeed ...
The meant “half” of Minkowski space is equivalent
to a continuous interval of Euclidean spaces
The number of its elements is: “infinity (for the
“unfolding”) plus one (for the “knife”)”
A continuous “interval” of Euclidean spaces is
equivalent topologically to a single one as both are
continuous and their set theory power is the same

44. Indeed (2) ...
One can call “not-knife” that single Euclidean
space (topologically equivalent to the “unfolding”)
What remains to be proved is the topological
equivalence of both “not-knife” and “knife”
Euclidean spaces (discrete to each other) to an
Euclidean space

45. Indeed (3) ...
One can divide Euclidean space into two disjunctive
subspaces:
For the example: by the parameter of any dimension:
the one, “less and equal than any constant of the
parameter”; the other, “greater than it”
Homeomorphism refers only to open subsets and
can ignore all closed sets containing the border

46. Indeed (4) ...
Then, the union of both disjunctinctive
subspaces is the Euclidean space itself
Each of the subspaces is topologically
equivalent to one of the “knife” and “not-knife”
Euclidean spaces correspondingly

47. Conclusion
Consequently, the “knife” and “not-knife” Euclidean
spaces are equivalent topologically to one single
Euclidean space
An idea about proving the generalization of
Poincaré’s conjecture is sketched
An idea for the Poincaré conjecture itself was
sketched a few slides ago in the same framework

48. Thank you for your kind attention!
Any questions or comments
are welcome!