The 30th Annual
Workshop in Geometric Topology
Calvin College, Grand Rapids, Michigan
The featured speaker is Pedro Ontaneda of Binghamton University. He will give a series of three one-hour lectures on Riemannian Hyperbolization.
Participants are invited to contribute talks; time will be allotted each day for 20-minute talks by participants. Abstracts may be entered on the registration form or sent directly to G. Venema (send email).
Financial support, provided by the National Science Foundation, will be available to help defer the travel and living expenses of participants who do not have other funding for their research. Such support can be requested on the registration form. Requests for support must be received by May 3, 2013. Graduate students and recent PhDs in geometric topology are especially encouraged to apply.
The workshop is supported by a grant from the National Science Foundation and by the Calvin College Department of Mathematics and Statistics.
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 ε-close to -1, for every ε > 0. (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.
Fredric Ancel, University of Wisconsin-Milwaukee
Dennis Garity, Oregon State University
Craig Guilbault, University of Wisconsin-Milwaukee
Eric Swenson, Brigham Young University
Frederick Tinsley, Colorado College
Gerard Venema, Calvin College
David Wright, Brigham Young University
Contact Gerard Venema (send email) if you have questions about the workshop or comments on this web site.