Conner, Greg; A history of some unexpected examples in low dimensional topology

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A history of some unexpected examples in low dimensional topology

Greg Conner

Brigham Young University, Utah, ZDA

21. februar 2008

In mathematics we often try to understand a given field by understanding a number of generic examples. Sometimes, though, it's a lot more fun and profitable to study the wacky and strange examples that appear on the edge of what current technology can handle. At first these example appear as anomalies, but over time they become themselves standard examples of certain types of behaviors. In this talk I will present a sequence of beautiful examples of strange behavior in low dimensional spaces and how studying these examples allowed researchers to make progress in that field.

We will start with Poincare's discovery of an example of a 3-manifold in 1904 that had the homology of a 3-sphere but was not the 3-sphere (which disproved one of his own published results), and his subsequent construction of the notion of the fundamental group to distinguish between his example and the 3-sphere. We will go on to study examples discovered by Hopf (1930's), Higman (1940's), Griffiths (1950's), Barratt & Milnor(1960's), Cannon (1970's), Zastrow (1990's), Cannon & C. (2000's) and Repovs & Eda (2000's) and how these have led to a better understanding of low dimensional topology.

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