Stratifold

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.

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), where X is a topological space and C is a subalgebra of the continuous functions ${\displaystyle X\to \mathbb {R} }$ such that a function is in C if it is locally in C and ${\displaystyle g\circ (f_{1},\dots ,f_{n}):X\to \mathbb {R} }$ is in C for ${\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }$ smooth and ${\displaystyle f_{i}\in C}$. A simple example takes for X a smooth manifold and for C just the smooth functions.

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