Euler characteristics of collapsing Alexandrov spaces

TL;DR

This paper proves a formula relating the Euler characteristic of collapsing Alexandrov spaces to the Euler characteristics of their strata and fibers, confirming a conjecture by Semyon Alesker.

Contribution

It establishes a new formula for the Euler characteristic in collapsing Alexandrov spaces, linking topology of the space, its strata, and fibers, thus advancing understanding of their geometric structure.

Findings

Euler characteristic equals sum over strata of product of Euler characteristics of strata and fibers.

Confirms Semyon Alesker's conjecture on Euler characteristics in collapsing Alexandrov spaces.

Provides a topological invariant relation in the context of collapsing geometric spaces.

Abstract

We prove that the Euler characteristic of a collapsing Alexandrov space (in particular, a Riemannian manifold) is equal to the sum of the products of the Euler characteristics with compact support of the strata of the limit space and the Euler characteristics of the fibers over the strata. This was conjectured by Semyon Alesker.