Mathematical Sciences: Geometric and Algebraic Topology

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9626111 Hatcher The two investigators will continue their research into homological properties of automorphism groups of free groups, in particular, to determine whether there is nontrivial rational homology in the stable dimension range and whether these automorphism groups are virtual duality groups. The homotopy type of related complexes on which the automorphism groups act, for example the complex of free factors of a free group, will also be investigated. Another problem which will be studied is to determine the homotopy type of the space of all embeddings of a link in 3-space, extending known results for knots. In particular, it should be possible to answer Kontsevich's question whether this space has finite homotopy type. In more general terms, this involves determining all the essentially different ways of removing unnecessary tangles from a knotted piece of string or from several pieces which are knotted and linked together. With regard to the other part of the project, automorphisms of free groups play a fundamental role in group theory (the general study of symmetry) somewhat analogous to matrices in linear algebra, and the goal is to understand just how close this parallel is: When one takes a geometric view of both sorts of algebraic objects, is the picture qualitatively the same, or qualitatively different? ***