In differential topology, a branch of mathematics, a stratifold is a generalization of a differentiable manifold where certain kinds of singularities are allowed. More specifically a stratifold is stratified into differentiable manifolds of (possibly) different dimensions. Stratifolds can be used to construct new homology theories. For example, they provide a new geometric model for ordinary homology. The concept of stratifolds was invented by Matthias Kreck. The basic idea is similar to that of a topologically stratified space, but adapted to differential topology.
Definitions Before we come to stratifolds, we define a preliminary notion, which captures the minimal notion for a smooth structure on a space: A differential space (in the sense of Sikorski) is a pair ( X , C ) , {\displaystyle (X,C),} where X is a topological space and C is a subalgebra of the continuous functions X → R {\displaystyle X\to \mathbb {R} } such that a function is in C if it is locally in C and g ∘ ( f 1 , … , f n ) : X → R {\displaystyle g\circ \left(f_{1},\ldots ,f_{n}\right):X\to \mathbb {R} } is in C for g : R n → R {\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} } smooth and f i ∈ C . {\displaystyle f_{i}\in C.} A simple example takes for X a smooth manifold and for C just the smooth functions. For a general differential space ( X , C ) {\displaystyle (X,C)} and a point x in X we can define as in the case of manifolds a tangent space T x X {\displaystyle T_{x}X} as the vector space of all derivations of function germs at x. Define strata X i = { x ∈ X : T x X {\displaystyle X_{i}=\{x\in X:T_{x}X} has dimension i } . {\displaystyle \}.} For an n-dimensional manifold M we have that M n = M {\displaystyle M_{n}=M} and all other strata are empty. We are now ready for the definition of a stratifold, where more than one stratum may be non-empty: A k-dimensional stratifold is a differential space ( S , C ) , {\displaystyle (S,C),} where S is a locally compact Hausdorff space with countable base of topology. All skeleta should be closed. In addition we assume:
The ( S i , C | S i ) {\displaystyle \left(S_{i},C|_{S_{i}}\right)} are i-dimensional smooth manifolds. For all x in S, restriction defines an isomorphism of stalks C x → C ∞ ( S i ) x . {\displaystyle C_{x}\to C^{\infty }(S_{i})_{x}.}
All tangent spaces have dimension ≤ k. For each x in S and every neighbourhood U of x, there exists a function ρ : U → R {\displaystyle \rho :U\to \mathbb {R} } with ρ ( x ) ≠ 0 {\displaystyle \rho (x)\neq 0} and supp ( ρ ) ⊂ U {\displaystyle {\text{supp}}(\rho )\subset U} (a bump function). A n-dimensional stratifold is called oriented if its (n − 1)-stratum is empty and its top stratum is oriented. One can also define stratifolds with boundary, the so-called c-stratifolds. One defines them as a pair ( T , ∂ T ) {\displaystyle (T,\partial T)} of topological spaces such that T − ∂ T {\displaystyle T-\partial T} is an n-dimensional stratifold and ∂ T {\displaystyle \partial T} is an (n − 1)-dimensional stratifold, together with an equivalence class of collars. An important subclass of stratifolds are the regular stratifolds, which can be roughly characterized as looking locally around a point in the i-stratum like the i-stratum times a (n − i)-dimensional stratifold. This is a condition which is fulfilled in most stratifold one usually encounters.
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