Mathematical Sciences: Symmetries of Low-Dimensional Manifolds

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The existence and classification of finite groups of symmetries of manifolds of dimensions three and four will be studied. Recently there has been a tremendous growth in our understanding of low-dimensional manifolds, based on techniques pioneered by W. Thurston, M. Freedman, and S. Donaldson. These new techniques and results have provided the tools to investigate much more systematically the symmetries of these manifolds and to sharpen significantly the questions that one can hope to address. Existence of symmetries of 4-manifolds: Freedman showed how to classify all simply connected, closed topological 4-manifolds in terms of their intersection forms and stable triangulation obstructions. It is then appropriate to ask what symmetries these manifolds admit. Results of the Principal Investigator show that all these 4-manifolds admit locally linear Zn actions for all odd n and that all except possibly the fake complex projective plane admit Z actions that are not necessarily locally linear. All the actions above were constructed so as to be homologically trivial, inducing the identity on homology. One is led to ask: What automorphisms of the intersection form of a closed simply connected topological 4-manifold can be realized by a periodic homeomorphism of the manifold?