[Paper Review] Collapsing topology of isolated singularities

Lev Birbrair, Alexandre Fernandes+1 more|arXiv (Cornell University)|Aug 21, 2012

Advanced Topology and Set Theory14 references3 citations

TL;DR

This paper establishes a topological thick and thin decomposition for closed definable isolated singularities in Euclidean space, proving that the thin zone captures homology collapsing faster than linearly. The key contribution is an equivalence between the existence of separating sets and non-injective homomorphisms in fast-contracting homology, linking metric non-conicality to topological obstructions in the thin zone.

ABSTRACT

We proof here the existence of a topological thick and thin decomposition of any closed definable thick isolated singularity germ in the spirit of the recently discovered metric thick and thin decomposition of complex normal surface singularities of [10]. Our thin zone catches exactly the homology of the family of the links collapsing faster than linearly. Simultaneously we introduce a class of rigid homeomorphisms more general than bi-Lipschitz ones, which map the topological thin zone onto the topological thin zone of its image. As a consequence of this point of view for the class of singularities we consider we exhibit an equivalent description of the notion of separating sets in terms of this fast contracting homology

Motivation & Objective

• To establish a topological thick and thin decomposition for closed definable isolated singularities, analogous to the metric decomposition in complex normal surface singularities.
• To characterize the thin zone as the locus where homology collapses faster than linearly with respect to radius.
• To generalize bi-Lipschitz geometry by introducing rigid homeomorphisms that preserve the topological thin zone structure.
• To provide a topological characterization of separating sets via fast-contracting homology in the thin zone.
• To show that the existence of separating sets is equivalent to non-injective homomorphisms in the (d−2)-th homology group of the thin zone

Proposed method

• Define the topological thin zone as the set of points where the tangent link lies in the locus of non-simple tangent directions.
• Use the notion of fast-contracting cycles (fc-cycles) to detect homology that collapses faster than linearly with shrinking radius.
• Introduce a new class of rigid homeomorphisms that preserve the topological thin zone, extending beyond bi-Lipschitz maps.
• Apply Birbrair-Brasselet metric homology and Valette vanishing homology to analyze the collapsing topology of links.
• Establish a homomorphism between the (d−2)-th homology of the thin zone and the ambient space, using inclusion-induced maps.
• Prove that non-injectivity of this homomorphism implies the existence of a separating set, via contradiction involving tangent link dimension and connected components

Experimental results

Research questions

• RQ1 Does every closed definable isolated singularity admit a topological thick and thin decomposition where the thin zone captures homology collapsing faster than linearly?
• RQ2 Can separating sets in isolated singularities be characterized via the non-injectivity of a homomorphism in fast-contracting homology?
• RQ3 How does the locus of non-simple tangent directions relate to the topological thin zone and separating sets?
• RQ4 What is the role of rigid homeomorphisms in preserving the structure of the thin zone under topological deformations?
• RQ5 Is the existence of a separating set equivalent to a non-trivial fast-contracting homology in the thin zone?

Key findings

• The topological thin zone is precisely the set of points whose tangent link lies in the locus of non-simple tangent directions.
• A separating set exists in a simply embedded model of a thick isolated singularity if and only if the locus of non-simple tangent directions satisfies condition (SC).
• The inclusion-induced homomorphism $ heta_*: ilde{ m H}_{d-2}({ m Thin}(X_1), extbf{0}) o ilde{ m H}_{d-2}(X_1, extbf{0})$ is not injective if and only if a separating set exists.
• The existence of a separating set implies that some $(d-2)$-cycle in the thin zone cannot be contracted in any small conical neighborhood of the non-simple tangent locus.
• The tangent link of the separating set has dimension at most $ ext{dim}(X_1) - 2$, contradicting purity of the ambient tangent link if the cycle were contractible in the thin zone.
• The thin zone captures all topology collapsing faster than linearly, and its structure is intrinsically linked to obstructions in homology that prevent local metric conicalness