How does the germ of a singular space look like?

Manifolds are locally rectifiable (at each point) to R^n or C^n. The local Geometry, Topology, Algebra of singular spaces is much richer. Such a germ X is homeomorphic to the cone over Link[X]. In ‘most cases’ this homeomorphism cannot be chosen differentiable. This brings various pathologies.

The Lipschitz equivalence of space-germs has been under investigation in the last 30 years. It excludes various pathologies of homeomorphisms, but is ‘rough enough’ to prevent moduli.

The first natural question is whether/when the homeomorphism X ~ Link[X] can be chosen bi-Lipschitz. The first obstructions to this are fast vanishing cycles on Link[X]. We detect lots of fast cycles. This gives countable (multi-index) series of `exotic Lipschitz structures’ on the germ (R^n,o), all realizable as complex-analytic hypersurface germs.