Stratified manifolds with corners

2025/07/29 by Joyce, Dominic
#Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2507.21775

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∈ I}Xⁱ into locally closed subsets Xⁱ which are smooth manifolds of dimension ≤ 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, ....

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