Rational Orthogonal Calculus

TL;DR

This paper develops a model category framework for rational orthogonal calculus, enabling the construction of functor towers based on rational homology, with layers classified by torsion modules over certain cohomology rings.

Contribution

It introduces a model category approach to rational orthogonal calculus, linking functor approximations to rational homology and classifying layers via torsion modules.

Findings

Constructed a tower of functor approximations depending on rational homology.

Classified layers using torsion modules over cohomology rings.

Connected orthogonal calculus with rational homotopy theory.

Abstract

We show that one can use model categories to construct rational orthogonal calculus. That is, given a continuous functor from vector spaces to based spaces one can construct a tower of approximations to this functor depending only on the rational homology type of the input functor, whose layers are given by rational spectra with an action of $O(n)$. By work of Greenlees and Shipley, we see that these layers are classified by torsion $H^*(B SO(n))[O(n)/SO(n)]$-modules.