Mathematical Sciences: Function Spaces, Homotopy Theory, and Group Cohomology

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There are two faculty investigators in the projects described below, F. R. Cohen and S. Gitler, as well as two of their graduate students involved in thesis research. All of the research concerns algebraic techniques in topology. F. R. Cohen intends to consider problems in classical homotopy theory and some offshoots to the cohomology of certain discrete groups. Three of the main projects involve (1) improving his bounds on the exponent of a mod-2-to-the-r Moore space, r at least 2, and trying to extend some results to the case when r is 1, (2) a study of the Whitehead product involving spaces of rational functions together with their interplay with Cayley-Dickson algebras, and (3) the cohomology and stable structure for the classifying space of certain discrete groups called mapping class groups. Some of these problems involve configuration spaces and are part of joint work with S. Gitler and L. Taylor (University of Notre Dame). S. Gitler intends to consider several problems centered on mapping spaces, their cohomology, and their relationship to homotopy theory. For example, he intends to study maps (X,Y) and their cohomology for "good" spaces X and Y. Setting Y equal to a space in the omega-spectrum for Morava K-theory, K(n), one gets useful examples of spaces with periodic homotopy groups. Gitler intends to study applications of these spaces.