Weakly Smooth Structures in Gromov-Witten Theory

TL;DR

This paper develops a framework for analyzing stratified topological Banach manifolds in Gromov-Witten theory, focusing on the existence of smooth functions and sections despite only having continuous transition maps.

Contribution

It introduces methods to establish smooth structures and analyze weakly smooth sections on stratified Banach manifolds in Gromov-Witten and Floer theories.

Findings

Existence of smooth functions and sections in admissible charts

Analysis of weakly smooth sections

Framework for Fredholm theory on stratified manifolds

Abstract

In order to establish Fredholm theory on stratified topological Banach manifolds in Gromov-Witten theory, we have introduced flat structures on such manifolds in [L4]. Such a structure is obtained from local flat coordinate charts. The transformations between these charts are only continuous in general. The purpose of this paper is two-fold : firstly to show that on the topological Banach manifolds and bundles appeared in Gromov-Witten and Floer type theories, there are enough smooth functions and sections viewed in any admissible charts and trivializations; secondly to demonstrate some finer aspects about the weakly smooth sections.