On The Acyclic MacPhersonian

TL;DR

This paper presents a novel approach demonstrating that the acyclic MacPhersonian shares the same homotopy type as the affine Grassmannian, resolving a long-standing topological conjecture.

Contribution

It introduces a new method to prove the homotopy equivalence between the acyclic MacPhersonian and the affine Grassmannian, differing from previous approaches.

Findings

Proves the homotopy type of the acyclic MacPhersonian matches that of the affine Grassmannian.

Provides a different proof technique from earlier attempts.

Clarifies the topological structure of the MacPhersonian.

Abstract

In 2003 Daniel K. Biss published, in the Annals of Mathematics, what he thought was a solution of a long standing problem culminating a discovery by Gelfand and MacPherson. Six years later he was encouraged to publish an "erratum" of his prove, observed by Nikolai Mnev; up to now, the homotopy type of the MacPhersonian had remained a mistery... The aim of this lecture is to convince the attendee of the fact that, using a compleatly different aproach to those used before, we can prove that the acyclic MacPhersonian has the homotopy type of the affine Grassmannian.