Title: Stratified manifolds with corners Show Chinese title

Title: 带角的分层流形

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, .... Show Chinese abstract

Abstract: 我们定义分层流形（s-流形）和带角的分层流形（带角的s-流形）。 一个维度为$n$的 s-流形$\bf X$是一个豪斯多夫、局部紧致的拓扑空间$X$，它通过分层$X=\coprod_{i\in I}X^i$被划分为局部闭子集$X^i$，这些子集是维度为$\le n$的光滑流形，满足一些条件。 s-流形可能非常奇异，但仍与普通流形有许多良好的性质，例如 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, ....