Stratified manifolds with corners

TL;DR

This paper introduces categories of stratified manifolds with corners, called s-manifolds, which generalize smooth manifolds to include singularities, facilitating applications in symplectic geometry and related invariants.

Contribution

It defines the concept of s-manifolds and s-manifolds with corners, establishing their properties and potential for use in symplectic geometry and moduli space analysis.

Findings

s-manifolds can be very singular yet retain key manifold properties

transverse fiber products exist within s-manifolds category

oriented s-manifolds have fundamental classes in Steenrod homology

Abstract

We define categories of stratified manifolds (s-manifolds) and stratified manifolds with corners (s-manifolds with corners). An s-manifold $\bf X$ of dimension $n$ is a Hausdorff, locally compact topological space $X$ with a stratification $X=\coprod_{i\in I}X^i$ into locally closed subsets $X^i$ which are smooth manifolds of dimension $\le n$, satisfying some conditions. S-manifolds can be very singular, but still share many good properties with ordinary manifolds, e.g. an oriented s-manifold $\bf X$ has a fundamental class $[\bf X]_{\rm fund}$ in Steenrod homology $H_n^{St}(X,\mathbb Z)$, and transverse fibre products exist in the category of s-manifolds. S-manifolds are designed for applications in Symplectic Geometry. In future work we hope to show that after suitable perturbations, the moduli spaces $\mathcal M$ of $J$-holomorphic curves used to define Gromov-Witten invariants, Lagrangian Floer cohomology, Fukaya categories, and so on, can be made into s-manifolds or s-manifolds with corners, and their fundamental classes used to define Gromov-Witten invariants, Lagrangian Floer cohomology, ....