Geometry and Topology Seminar

Abstract: For any mathematical structure, one of the fundamental problems is to decompose more complicated examples into their simpler components. Given a topological space~$X$, consider the problem of deciding if $X$ is homotopy equivalent to some product $A\times B$ of non-contractible spaces. If such a decomposition exists for $X$ (or its loop space) it provides a lot of information about $X$, and certain types of problems, such as the classical ``Hopf Invariant One'' problem, can be reformulated in terms of whether or not some space admits such a decomposition. I will discuss product decompositions for spaces of the form~$X=\Omega\Sigma Y$. In this case the construction of such decompositions can be reduced to problems in the modular representation theory of the symmetric group, however explicit calculations, such as formulas for the homology of the factors in terms of~$H_*(Y)$, have not yet been found, since the corresponding problems in representation theory are major open questions. I will describe the reduction of the topological problem to the algebraic, and the possibilities for passing information (in one direction or the other) between the subjects.