Stratified manifolds with corners

Abstract

We define categories of stratified manifolds (s-manifolds) and stratified manifolds with corners (s-manifolds with corners). An s-manifold $\mathbf{X}$ 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 $\mathbf{X}$ has a fundamental class $[\mathbf{X}{]}_{\mathrm{fund}}$ 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, ....

74 pages. Final version, to appear in the Quarterly Journal of Mathematics