MIT Topology Seminar
Monday, May 8, 2006
Room 1-134, 3:15pm
Vigleik Angeltveit will speak on:
Noncommutative Ring Spectra
Abstract: Let A be an A \infty ring spectrum. We give an explicit construction of topological Hochschild homology and cohomology of A using the Stasheff associahedra and another family of polyhedra called cyclohedra. Using this construction we can then study how THH(A) varies over the moduli space of A \infty structures on A, a problem which seems largely intractible using strictly associative replacements of A. We study how topological Hochschild cohomology of any 2-periodic Morava K-theory varies over the moduli space of A\infty structures and show that in the generic case, when a certain matrix describing the multiplication is invertible, the result is the corresponding E-theory. If this matrix is not invertible, the result is some finite extension of Morava E-theory, and exactly which extension we get depends on the A \infty structure.
To make sense of our constructions, we first set up a general framework for enriching a subcategory of the category of noncommutative sets over a category C using products of the objects of a non-\Sigma operad P in C. By viewing the simplicial category as a subcategory of the category of noncommutative sets in two different ways, we obtain two generalizations of simplicial objects. For the operad given by the Stasheff associahedra we obtain a model for the 2-sided bar construction in the first case and the cyclic bar and cobar construction in the second case. Using either the associahedra or the cyclohedra in place of the geometric simplices we can define the geometric realization of these objects.