Vanishing homology

TL;DR

This paper introduces a new homology theory for definable families in o-minimal structures, focusing on vanishing cycles and local invariants, with proven finite generation of homology groups.

Contribution

It develops a novel homology framework for o-minimal definable families, capturing vanishing cycles and deriving local metric invariants.

Findings

Homology groups are finitely generated.

The theory applies to semi-algebraic, subanalytic, and other o-minimal definable families.

Provides tools for studying local geometric invariants.

Abstract

In this paper we introduce a new homology theory devoted to the study of families such as semi-algebraic or subanalytic families and in general to any family definable in an o-minimal structure (such as Denjoy-Carleman definable or $ln-exp$ definable sets). The idea is to study the cycles which are vanishing when we approach a special fiber. This also enables us to derive local metric invariants for germs of definable sets. We prove that the homology groups are finitely generated.