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.
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
where X is a topological space and C is a subalgebra of the continuous functions
and a point x in X we can define as in the case of manifolds a tangent space
as the vector space of all derivations of function germs at x.
where S is a locally compact Hausdorff space with countable base of topology.
In addition we assume: 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.
is an (n − 1)-dimensional stratifold, together with an equivalence class of collars.
This is a condition which is fulfilled in most stratifold one usually encounters.
The first example to consider is the open cone over a manifold M. We define a continuous function from S to the reals to be in C if and only if it is smooth on
The last condition is automatic by point 2 in the definition of a stratifold.
If we consider the (closed) cone with bottom, we get a stratifold with boundary S. Other examples for stratifolds are one-point compactifications and suspensions of manifolds, (real) algebraic varieties with only isolated singularities and (finite) simplicial complexes.
In this section, we will assume all stratifolds to be regular.
from two oriented compact k-dimensional stratifolds into a space X bordant if there exists an oriented (k + 1)-dimensional compact stratifold T with boundary S + (−S') such that the map to X extends to T. The set of equivalence classes of such maps
The sets have actually the structure of abelian groups with disjoint union as addition.
One can develop enough differential topology of stratifolds to show that these define a homology theory.
for every space X homotopy-equivalent to a CW-complex, where H denotes singular homology.
For other spaces these two homology theories need not be isomorphic (an example is the one-point compactification of the surface of infinite genus).
There is also a simple way to define equivariant homology with the help of stratifolds.
We can then define a bordism theory of stratifolds mapping into a space X with a G-action just as above, only that we require all stratifolds to be equipped with an orientation-preserving free G-action and all maps to be G-equivariant.
For example, the Euler characteristic defines a ring homomorphism
The left hand sides of these homomorphisms are homology theories evaluated at a point.
One possibility is to use stratifolds: represent a class
Then make ƒ transversal to N. The intersection of S and N defines a new stratifold S' with a map to N, which represents a class in
It is possible to repeat this construction in the context of an embedding of Hilbert manifolds of finite codimension, which can be used in string topology.
