General Topology-I

Convex bodies: the brunn–minkowski theory

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Classical Descriptive Set Theory

Basic definitions and examples Algebraic theory Limit spaces Orbispaces Iterated monodromy groups Examples and applications Bibliography Index

Self-Similar Groups

This chapter and the next are adapted from a lecture published in the Proceedings of the Third Physics Workshop, held at the Institute of Theoretical and Experimental Physics, Moscow, in 1975. They focus on the uses of homotopy theory in physics, and especially on the calculation of homotopy groups. The next chapter summarizes briefly the ways in which homotopy theory can be applied to quantum field theory.

Elements of Homotopy Theory

We establish a link between rational homotopy theory and the problem which vector bundles admit a complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if C lies in the class and T is a torus of positive dimension, then "most" vector bundles over C x T admit no complete nonnegatively curved metrics.

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Obstructions to nonnegative curvature and rational homotopy theory

We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metrics.

Topological obstructions to nonnegative curvature

We establish a link between rational homotopy theory and the problem which vector bundles admit complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if C lies in the class and T is a torus of positive dimension, then "most" vector bundles over the product of C and T admit no complete nonnegatively curved metric.

We show that for any non--elementary hyperbolic group $H$ and any finitely presented group $Q$, there exists a short exact sequence $1\to N\to G\to Q\to 1$, where $G$ is a hyperbolic group and $N$ is a quotient group of $H$. As an application we construct a hyperbolic group that has the same $n$--dimensional complex representations as a given finitely generated group, show that adding relations of the form $x^n=1$ to a presentation of a hyperbolic group may drastically change the group even in case $n>> 1$, and prove that some properties (e.g. properties (T) and FA) are not recursively recognizable in the class of hyperbolic groups. A relatively hyperbolic version of this theorem is also used to generalize results of Ollivier--Wise on outer automorphism groups of Kazhdan groups.

Rips construction and Kazhdan property (T)

We apply various topological methods to distinguish connected components of moduli spaces of complete Riemannian metrics of nonnega- tive sectional curvature on open manifolds. The new geometric ingredient is that souls of nearby nonnegatively curved metrics are ambiently isotopic.

/pdf/moduli-spaces-of-nonnegative-sectional-curvature-and-non-14p7zt05jo.pdf

Igor Belegradek

Papers

Obstructions to nonnegative curvature and rational homotopy theory

Topological obstructions to nonnegative curvature

Obstructions to nonnegative curvature and rational homotopy theory

Rips construction and Kazhdan property (T)

Moduli spaces of nonnegative sectional curvature and non-unique souls