Euler-Poincar\'e obstruction for pretzels with long tentacles \`a la Cantor-Nyikos

TL;DR

This paper introduces a new Euler obstruction concept for foliated structures on certain non-metric surfaces, extending classical obstruction theory beyond metric spaces, with historical insights into Poincaré's 1885 argument.

Contribution

It develops an Euler-Poincaré obstruction framework for non-metric surfaces, broadening the scope of classical geometric obstruction theories.

Findings

Obstruction propagates beyond metric spaces

Extension of classical obstruction theory to non-metric surfaces

Historical connection to Poincaré's 1885 argument

Abstract

We present an avatar of the Euler obstruction to foliated structures on certain non-metric surfaces. This adumbrates (at least for the simplest 2D-configurations) that the standard mechanism---to the effect that the devil of algebra sometimes barricades the existence of angelic geometric structures (obstruction theory more-or-less)---propagates slightly beyond the usual metrical proviso. Alas, the game is much more conservative than revolutionary: in particular we enjoyed retrospecting at Poincar\'e's argument of 1885 (announced in 1881).