Local invariant for scale structures on mapping spaces

TL;DR

This paper shows that the local invariant for scale structures on mapping spaces depends solely on the domain's dimension, providing a complete classification using spectral analysis techniques.

Contribution

It establishes that scale structures on mapping spaces are fully determined by the domain dimension and describes the local invariant accordingly.

Findings

Scale structures depend only on domain dimension

Complete description of Frauenfelder's local invariant

Analysis of product and relative mapping spaces

Abstract

Scale structures were introduced by H. Hofer, K. Wysocki, and E. Zehnder as a new concept of a smooth structure in infinite dimensions. We prove that scale structures on mapping spaces are completely determined by the dimension of domain manifolds. As a consequence, we give a complete description of the local invariant introduced by U. Frauenfelder for mapping spaces. Product mapping spaces and relative mapping spaces are also studied. Our approach is based on the spectral resolution of Laplace type operators together with the eigenvalue growth estimate.