Spaces of flattenings of spheres

TL;DR

This paper investigates the topological structure of flattening spaces of simplicial spheres, establishing their homotopy equivalence to the orthogonal group, which advances understanding of differentiable structures on spheres.

Contribution

The paper proves that certain flattening spaces of simplicial spheres have the homotopy type of the orthogonal group, linking combinatorial and geometric topology.

Findings

Flattening spaces of some simplicial spheres are homotopy equivalent to the orthogonal group.

Results contribute to the understanding of differentiable structures on spheres.

Establishes a new connection between combinatorial topology and classical Lie groups.

Abstract

The spaces of flattenings of a simplicial sphere played a key role in the study of existence and uniqueness of differentiable structures on a simplicial sphere. In this paper, we will establish that the spaces of flattenings of some simplicial spheres and show that they have the homotopy type of the orthogonal group.