The topology of Birkhoff varieties

TL;DR

This paper proves that Birkhoff varieties are homotopy equivalent to their inclusion in the affine Grassmannian and constructs tubular neighborhoods, providing new insights into their topological structure and equivariant cohomology.

Contribution

It establishes the homotopy equivalence of Birkhoff varieties with their ambient affine Grassmannian and introduces tubular neighborhood constructions for these varieties.

Findings

Birkhoff varieties are homotopy equivalent to their inclusion in the affine Grassmannian

Constructed tubular neighborhoods for Birkhoff and Schubert varieties

Provided observations on torus-equivariant cohomology

Abstract

Our main theorem is that the inclusion of a Birkhoff variety in the affine Grassmannian is a homotopy equivalence. We also construct analogues of tubular neighborhoods for Birkhoff and Schubert varieties. We include some observations on torus-equivariant cohomology.