The shift map on Floer trajectory spaces

TL;DR

This paper provides a unified proof demonstrating that the shift map on Floer homology trajectory spaces is scale smooth across various types of Floer homologies, using Hilbert space valued Sobolev theory.

Contribution

It introduces a general proof technique for the scale smoothness of the shift map applicable to multiple Floer homology theories.

Findings

The shift map is scale smooth in Floer trajectory spaces.

The proof applies to periodic, Lagrangian, Hyperkähler, elliptic, and parabolic Floer homologies.

Uses Hilbert space valued Sobolev theory for the proof.

Abstract

In this article we give a uniform proof why the shift map on Floer homology trajectory spaces is scale smooth. This proof works for various Floer homologies, periodic, Lagrangian, Hyperk\"ahler, elliptic or parabolic, and uses Hilbert space valued Sobolev theory.