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 a 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 $g|_U = f|_U$,

2. let $f_1,\ldots,f_n$ be elements of $C$ and $\phi : R^n\to R$ a smooth function. Then $\phi \circ (f_1,\ldots,f_n) : S\to R$ 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 $C= C^\infty(S)$. For a differentiable space $S = (S,C)$ a tangent space $T_x S$ can be defined at each $x\in S$. Define $S^i = \{x\in S | \mathrm{dim} T_x S = i\}$. By construction, $S$ decomposes into a disjoint union $S = \biguplus_{i=0}^\infty S^i$. 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,

• $T_x S \leq k$ 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 $C(S)_x\to C(S_i)_x = C^\infty(S^i)_x$ 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

• The stratifold page

• Matthias Kreck, Differential Algebraic Topology (pdf)