On Isolated Real Singularities I

TL;DR

This paper investigates the topology of real isolated hypersurface singularities, showing how handle attachments relate to critical points and Euler characteristic, and providing conditions for homology group isomorphisms with bouquets of spheres.

Contribution

It introduces a handle attachment framework for real Milnor fibres, linking critical points to topology, and establishes conditions for homology groups to match those of bouquets of spheres.

Findings

Milnor fibres become contractible after handle attachments.

Handles correspond to critical points of morsifications.

Homology groups can be isomorphic to those of bouquets of spheres under certain conditions.

Abstract

This article and its successor concern the topology of real isolated hypersurface singularities. We prove that after attaching a certain number of handles the real Milnor fibres become contractible, with each handle corresponding to a critical point of a morsification; in particular one recovers the classical formula of Khimshiashvili for the Euler characteristic of the Milnor fibres. We then give sufficient conditions for having that the integer homology groups of the real Milnor fibres are isomorphic to the homology groups of a bouquet of spheres. This is followed by a discussion on real vanishing cycles which we define and then demonstrate that, under the validity of our assumptions, they uniquely determine the homology groups of the real Milnor fibres.