Functor calculus and the discriminant method

TL;DR

This paper reformulates the discriminant method within functor calculus to better analyze the cohomology and homotopy types of spaces of smooth maps with singularities, especially under multilocal conditions.

Contribution

It introduces a functor calculus perspective to the discriminant method, enabling new applications in studying smooth maps with prescribed local behaviors.

Findings

Reformulation of the discriminant method using functor calculus

Enhanced ability to impose multilocal conditions on smooth maps

Potential for broader applications in topology of smooth maps

Abstract

The discriminant method is a tool for describing the cohomology, or the homotopy type, of certain spaces of smooth maps with uncomplicated singularities from a smooth compact manifold L to R^k. We recast some of it in the language of functor calculus. This reformulation allows us to use the discriminant method in a setting where we wish to impose conditions on the multilocal behavior of smooth maps f from L to R^k.