[Binghamton Geometry/Topology Seminar] Geometry & Topology Seminar

[somnath at math.binghamton.edu]

BINGHAMTON GEOMETRY & TOPOLOGY SEMINAR

SPECIAL SUMMER EVENT

Pedro Ontaneda will give three lectures entitled

Riemannian Hyperbolization

Times:

Monday June 10: 3.30-4.30

Monday June 10:  4.45-5.45.

Tuesday June 11:  2-3

Abstract:  Negatively curved Riemannian manifolds are fundamental objects
in many areas of mathematics, but very few examples are known: besides the
hyperbolic ones, the other known examples are the Mostow-Siu examples
(1980, dimension 4), the Gromov-Thurston examples (1987), the exotic
Farrell-Jones examples (1989), and the three examples of Deraux (2005,
dimension 6). Hence, apart from dimensions 4 and 6, every known example of
a closed negatively curved Riemannian manifold is homeomorphic to either a
hyperbolic one or a branched cover of a hyperbolic one.
On the other hand, Charney-Davis strict hyperbolization (1995) produces a
rich and abundant class of (non-Riemannian) negatively curved spaces.
Charney-Davis hyperbolization builds on the hyperbolization process
introduced by Gromov in 1987 and later studied by Davis and Januszkiewicz
(1991). But the negatively curved manifolds constructed using the
Charney-Davis strict hyperbolization process are very far from being
Riemannian because the metrics have large and highly complicated sets of
singularities.
In three Lectures we will sketch how to remove all the singularities from
Charney-Davis hyperbolized manifolds, obtaining in this way a Riemannian
strict hyperbolization process. Hence, through this work, we now know that
the class of Riemannian negatively curved manifolds is also rich and
large. And we can say that, in some sense, Riemannian negative curvature
abounds in nature. Moreover we show we can do the Riemannian
hyperbolization in a pinched way, that is, with curvature as close to -1
as desired.
Here are two of the many direct consequences of Riemannian hyperbolization
that we will mention in the Lectures: (1) Every closed smooth manifold is
smoothly cobordant to a closed Riemannian manifold with curvatures
epsilon-close to -1, for every positive epsilon.. (2) Every closed almost
flat manifold is a cusp cross section of a finite volume pinched
negatively curved manifold.
In the first half of Lecture 1 we will state the main result and its
corollaries. In the second half of Lecture 1 and part of Lecture 2 we will
introduce three geometric processes: the two-variable warping trick (based
on the Farrell-Jones warping trick), warp forcing, and hyperbolic
extensions. Also in Lecture 2 we will discuss the construction of
extremely useful differentiable structures: normal differentiable
structures on cubical manifolds and on Charney-Davis hyperbolizations.
Finally in Lecture 3 we will sketch how to smooth metrics on hyperbolic
cones and sketch how to smooth Charney-Davis hyperbolized manifolds.