On configuration spaces of stable maps

TL;DR

This paper explores the topology of the space of smooth, stable genus 0 curves in a Riemannian manifold using polyfold structures, revealing homology injection results and introducing the concept of q-complete symplectic manifolds.

Contribution

It introduces a polyfold framework for analyzing stable maps and defines q-complete symplectic manifolds, connecting Gromov-Witten theory with homology of stable map spaces.

Findings

Rational homology of spherical mapping space injects into stable curves space for X=BU

Polyfold structures facilitate topological analysis of stable maps

Definition of q-complete symplectic manifolds linking Gromov-Witten theory to homology

Abstract

We study here some aspects of the topology of the space of smooth, stable, genus 0 curves in a Riemannian manifold $X$, i.e. the Kontsevich stable curves, which are not necessarily holomorphic. We use the Hofer-Wysocki-Zehnder polyfold structure on this space and some natural characteristic classes, to show that for $X=BU,$ the rational homology of the spherical mapping space injects into the rational homology of the space of stable curves. We also give here a definition of what we call $q$-complete symplectic manifolds, which roughly speaking means Gromov-Witten theory captures all information about homology of the space of smooth stable maps.