Daniel A. Ramras  

Drafts and Work in Progress
  • Stable representation theory of crystallographic groups. In progress, 2009.

    We study deformation K-theory of crystallographic groups using results of Lawson, which relate deformation K-theory spectra to spaces of irreducible representations. This provides new examples relating deformation K-theory to topological K-theory.

  • (With Tom Baird) Gauge theory, maps between classifying spaces, and Quillen-Lichtenbaum failure in deformation K-theory. In progress, 2008.

    Using Chern-Weil theory, we explain how the rational cohomology of a manifold M can be used to produce non-trivial homotopy classes in the space of flat connections on principal U(n)-bundles over M. As an application, we show that when M is aspherical, the natural map Hom(pi_1 M, U(n)) -> Map_* (Bpi_1 M, BU(n)) fails to be an equivalence below the rational cohomological dimension of M minus 1, and in fact deformation K-theory of pi_1 M and topological K-theory of M fail to agree below this dimension. This failure reflects the substantial geometric content of deformation K-theory, and is precisely analagous to the low-dimensional failure of the Quillen-Lichtenbaum conjecture in algebraic K-theory (relating the algebraic and etale K-theories of a scheme).

  • Deformation K-theory of surface bundles over surfaces. In progress, 2008.

    We study the unitary representation spaces associated to fundamental groups of "virtually trivial" surface bundles over surfaces (and somewhat more general groups). We show that deformation K-theory is rationally periodic above the rational cohomological dimension of the group. This follows, via work of T. Lawson, from a finiteness result for the stable moduli space of representations. This finiteness result is a consequence of the author's work on surface groups, and involves the study of transfer maps.

  • Stability for representation spaces of surface groups. In preparation.

    Using gauge theory, we explicitly determine the homotopy type of the representation spaces $\Hom(\pi_1 \Sigma, U(n))$ (where $\Sigma$ is any surface), after stabilizing with respect to the rank $n$. Additionally, determine precise formulas for the connectivity of the inclusions $\Hom(\pi_1 \Sigma, U(n)) \injects \Hom(\pi_1 \Sigma, U(n+1))$.

  • Homotopy invariance of deformation K-theory. Posted 10/19/2006. Abstract