Poincaré–Hopf Theorem for singular analytic varieties (The Marie-Hélène Schwartz ideas in the Lipschitz framework)
Volume 128 / 2024
Abstract
The famous Poincarè–Hopf Theorem was proved by Poincarè (for surfaces in 1881) and Hopf (general case in 1926): Let $M$ be a compact differentiable manifold. Let $v$ be a (continuous) vector field tangent to $M$ with isolated zeroes $a_i$ of index $I(v,a_i)$. If $M$ has no boundary, the Euler–Poincar\xB2 characteristic of $M$ is equal to $\chi (M) = \sum_{a_i} I(v,a_i).$
It was unclear how to prove and even formulate a similar result for singular varieties. The first proof for singular varieties was given by Marie-H\xB2l\xB1ne Schwartz. She used the idea of radial extension of a vector field, in the context of Whitney stratifications, that needs complicated and delicate constructions.
In this paper, we use the Schwartz ideas in the framework of Lipschitz stratifications, which allows us to simplify the construction. We prove a Poincar\xB2–Hopf Theorem for Lipschitz radial vector fields in the context of Lipschitz stratifications.
