Mathematical Sciences: Topology of Arithmetic Groups

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9504264 Connolly Connolly will be studying obstructions to topological rigidity such as those provided by the group UNIL of S. Cappell. He will be investigating the problem of fitting a boundary onto a stratified space with a tame end. He will also be using these to attack the Quinn Rigidity Conjecture (ICM, 1986). He wishes to take the new "continuous control" K-theory and use it to create a Surgery theory which is adapted to these problems. He is also including 4-dimensional smooth manifolds in his project (Gauge Theory). In this project Connolly will be attempting to show that: Arithmetic Groups are Topologically Rigid. In less epigrammatic terms, this means that the extreme symmetry exhibited by some of the fundamental geometries of nature (these geometries are called Locally Symmetric Spaces by mathematicians and physicists), should eventually be shown to be based on purely topological properties of their fundamental groups. In highly oversimplified terms, it means that anything that is vaguely like a torus (or a Klein bottle) should be topologically equivalent to a torus (or a Klein bottle). An example of a topological equivalence is given by a circle and ellipse; they are not congruent figures in the sense of Euclidean geometry, but they are topologically equivalent, because a little stretching of the one produces the other. ***