| Topology Atlas Document # zaaa-39 | Topology Atlas Invited Contributions vol. 1 issue 1 (1996) p. 5 |
University of Catania, Catania, ItalyMy invited contribution to the Topology Atlas, made in 1995, was the following conjecture ([1],[2]), where <.,.> is the scalar product in Rn:
Conjecture.
Let X be the closed unit ball of Rn (n > 1),
let f: X --> X be a continuous function,
with f(x) not equal to x for some x in X,
and let e
be a positive number small enough.
Denote by Le
the set of all continuous functions
g: X × Rn --> R such that, for each x in X,
g(x, . ) is Lipschitzian in Rn, with Lipschitz constant
less than or equal to e.
Consider
Le
equipped
with the relativization of the strongest vector topology on the space
R(X × R^n).
Then, the set
{ (g,x,y) in Le × X × Rn : < f(x)-x, y> = g(x,y) }is disconnected.
The reason for this conjecture was the perspective of finding a drastically novel proof of the Brouwer fixed point theorem, in view of the following result:
Theorem ([1], Theorem 22).
Let X be a connected and locally
connected topological space,
E a real Banach space,
F: X --> E* a
(strongly)
continuous operator, with closed range. For each
e,
denote by
Le
the set of all continuous functions
g: X × E --> R
such that, for each x in X, g(x, . ) is
Lipschitzian in E, with Lipschitz constant less than or equal to e.
Consider Le
equipped with the relativization
of the strongest vector topology on the space
RX × E, and assume that the set
{ (g,x,y) in Le × X × E : F(x)(y) = g(x,y) }is disconnected. Then, F vanishes at some point of X.
Thus, in case the conjecture was true, only a proof independent of Brouwer's theorem would be really interesting.
From personal contacts, I know of at least two extremely talented topologists who tried to attack the conjecture, but without success.
Date published: December 7, 1995. Revised September 29, 2000.