A stochastic approach to the Euler-Poincaré number of the loop space of a developable orbifold
This dense and difficult paper deals with the set \(H_ g\) of paths \(\gamma : [0,1] \to M\) (a compact manifold) such that \(\gamma(1) = g \cdot \gamma(0)\) (\(g\) being one isometry of \(M\)), endowed with its Wiener-like measure \(Y_ g\) (for which integration by parts is established); a diffusion process \(w_ t\) is defined on \((H_ g, Y_ g)\) via a Dirichlet form, and is shown to verify an exponential localisation property for \(t \downarrow 0\). Then a differential calculus for \(n\)-forms and a regularised version of the de Rham-Hodge-Kodaira operator on \(H_ g\) are defined, and a semiclassical approximation of this operator is performed, such that the index of the limit operator equals the ``orbit Euler characteristic'' of physicists.
scientific article; zbMATH DE number 763310
scientific article