nLab
stratifold

Idea

Stratifolds are a generalization of smooth manifolds – a notion of generalized smooth space – which were introduced by Kreck; see his lecture notes.

Stratifolds comprise one of the many variants of the concept of a stratified space, and they may include some types of singularities.

Definition

Stratifolds form a class of differential modules, which are pairs $(S,C)$ of a topological space $S$ together with an subalgebra $C$ of the algebra of continuous real-valued functions $S\to R$ such that

1. $C$ is locally detectable, i.e. for all continuous functions $f:S\to R$, $f$ is in $C$ iff for every $x\in S$ there exist an open neighborhood $U\ni x$ and $g\in C$ such that $g{\mid }_U=f{\mid }_U$,

2. let $f_1,\dots ,f_n$ be elements of $C$ and $\varphi :R^n\to R$ a smooth function. Then $\varphi \circ (f_1,\dots ,f_n):S\to R$ is in $C$.

Local detectability is equivalent to requiring that $C$ is an algebra of global sections of a given subsheaf of the sheaf of all continuous functions on $S$; in particular germs at every point can be defined.

For a manifold $C=C^{\mathrm{\infty }}(S)$. For a differentiable space $S=(S,C)$ a tangent space $T_{x}S$ can be defined at each $x\in S$. Define $S^i=\{x\in S\mid \dim T_{x}S=i\}$. By construction, $S$ decomposes into a disjoint union $S={⨄}_{i=0}^{\mathrm{\infty }}{S}^i$. There is an induced stratifold structure on the topological subspace $S^i\subset S$, which we denote by $(S^i,C(S^i))$.

Definition

A $k$-dimensional stratifold $(S,C)$ is a differential space such that

• $S$ is locally compact Hausdorff space with countable basis,

• $T_{x}S\le k$ for all $x\in S$ (i.e. $S={\bigcup }_{i=0}^k{S}^i$),

• $(S^i,C(S^i))$ is isomorphic to a smooth manifold,

• the restriction map $C(S)\to C(S^i)$ induces an isomorphism of stalks of germs $C(S{)}_x\to C(S_i{)}_x=C^{\mathrm{\infty }}(S^i{)}_x$ in all points $x\in S^i$,

• for all $y\in S$, and all $U\ni y$ open, there is a “bump function” $\rho \in C$ nonvanishing at $y$, but whose support is contained in $U$.

References

• webpage on stratifolds

• Matthias Kreck, Differential Algebraic Topology (pdf)