Topology Seminar: Rational Parametrised Stable Homotopy Theory

Rational homotopy theory is a simplification of homotopy theory in which all torsion phenomena are systematically ignored. Under some mild hypotheses, celebrated results of Quillen and Sullivan provide complete descriptions of the rational homotopy category in terms of algebraic data. Quillen's approach identifies the rational homotopy type of a 1-connected space with a dg coalgebra or, equivalently, with a dg Lie algebra, whereas Sullivan's approach identifies the rational homotopy theory of nilpotent spaces of finite type with finite type cochain algebras. Stably, the situation is much simpler: the stable rational homotopy category is identified with graded rational vector spaces.

In this talk, I present recent results on the rational homotopy theory of parametrised spectra which unify these established models for stable and unstable rational homotopy theory. A parametrised spectrum is a family of spectra parametrised by a fixed parameter space, representing a cohomology theory twisted by an unstable homotopy type. After discarding torsion, I demonstrate that both Quillen's and Sullivan's approaches to rational homotopy theory can be lifted to provide algebraic characterisations of the rational homotopy category of spectra parametrised by a 1-connected space. The underlying idea is that whereas a parametrised X-spectrum P is a family of spectra twisted by or acted upon by X, after disregarding torsion this becomes the information of a graded rational vector space acted upon by an algebraic avatar of X.

I conclude by discussing an application of these results to M-theory, where we obtain a rational lift to M-theory of the twisted K-theory classification of D-brane charges in 10-dimensional superstring theory.