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.

## Contents

* 1 Definitions
* 2 Examples
* 3 Bordism theories
* 4 Connection to the theory of genera
* 5 Umkehr maps
* 6 References

## 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), where X is a topological space and C is a subalgebra of the continuous functions X->ℝ such that a function is in C if it is locally in C and g∘(f1,…,fn):X->ℝ is in C for g:ℝn->ℝ smooth and fi∈C. A simple example takes for X a smooth manifold and for C just the smooth functions.

For a general differential space (X,C) and a point x in X we can define as in the case of manifolds a tangent space TxX as the vector space of all derivations of function germs at x. Define strata Xi={x∈X:TxX has dimension i}. For an n-dimensional manifold M we have that Mn=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), where S is a locally compact Hausdorff space with countable base of topology. All skeleta should be closed. In addition we assume:

The suspension

1. The (Si,C|Si) are i-dimensional smooth manifolds.
2. For all x in S, restriction defines an isomorphism of stalks Cx->C∞(Si)x.
3. All tangent spaces have dimension ≤ k.
4. For each x in S and every neighbourhood U of x, there exists a function ρ:U->ℝ with ρ(x)≠0 and supp(ρ)⊂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) of topological spaces such that T−∂T is an n-dimensional stratifold and ∂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.

## Examples

There are plenty of examples of stratifolds. 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 M×(0,1) and it is locally constant around the cone point. The last condition is automatic by point 2 in the definition of a stratifold. We can substitute M by a stratifold S in this construction. The cone is oriented if and only if S is oriented and not zero-dimensional. 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.

## Bordism theories

An example of a bordism relation

In this section, we will assume all stratifolds to be regular. We call two maps S,S′->X 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 S->X is denoted by SHkX. 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. Clearly, SHk(point)=0 for k>0 since every oriented stratifold S is the boundary of its cone, which is oriented if dim⁡(S)>0. One can show that SH0(point)≅ℤ. Hence, by the Eilenberg-Steenrod uniqueness theorem, SHk(X)≅Hk(X) 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. Let G be a compact Lie group. 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. Denote by SHkG(X) the bordism classes. One can prove SHkG(X)≅Hk−dim⁡(G)G(X) for every X homotopy equivalent to a CW-complex.

## Connection to the theory of genera

A genus is a ring homomorphism from a bordism ring into another ring. For example, the Euler characteristic defines a ring homomorphism ΩO(point)->ℤ/2[t] from the unoriented bordism ring and the signature defines a ring homomorphism ΩSO(point)->ℤ[t] from the oriented bordism ring. Here t has in the first case degree 1 and in the second case degree 4, since only manifolds in dimensions divisible by 4 can have non-zero signature. The left hand sides of these homomorphisms are homology theories evaluated at a point. With the help of stratifolds it is possible to construct homology theories such that the right hand sides are these homology theories evaluated at a point, the Euler homology and the Hirzebruch homology respectively.

## Umkehr maps

Suppose, one has a closed embedding i:N↪M of manifolds with oriented normal bundle. Then one can define an umkehr map Hk(M)->Hk+dim⁡(N)−dim⁡(M)(N). One possibility is to use stratifolds: represent a class x∈Hk(M) by a stratifold f:S->M. 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 Hk+dim⁡(N)−dim⁡(M)(N). 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.

## References

* M. Kreck, Differential Algebraic Topology: From Stratifolds to Exotic Spheres, AMS (2010), ISBN 0-8218-4898-4
* The stratifold page
* Euler homology

* v
* t
* e

Manifolds (Glossary, List, Category)

Basic concepts|

* Topological manifold
* Atlas
* Differentiable/Smooth manifold
* Differential structure
* Smooth atlas
* Submanifold
* Riemannian manifold
* Smooth map
* Submersion
* Pushforward
* Tangent space
* Differential form
* Vector field

Main theorems (list)|

* Atiyah–Singer index
* Darboux's
* De Rham's
* Frobenius
* Generalized Stokes
* Hopf–Rinow
* Noether's
* Sard's
* Whitney embedding

Maps|

* Curve
* Diffeomorphism
* Local
* Geodesic
* Exponential map
* in Lie theory
* Foliation
* Immersion
* Integral curve
* Lie derivative
* Section
* Submersion

Types of
manifolds|

* Calabi–Yau
* Closed
* Collapsing
* Complete
* (Almost) Complex
* (Almost) Contact
* Einstein
* Fibered
* Finsler
* (Almost, Ricci-) Flat
* G-structure
* Hadamard
* Hermitian
* Hyperbolic
* (Hyper) Kähler
* Kenmotsu
* Lie group
* Lie algebra
* Manifold with boundary
* Nilmanifold
* Oriented
* Parallelizable
* Poisson
* Prime
* Quaternionic
* Hypercomplex
* (Pseudo-, Sub-) Riemannian
* Rizza
* Sasakian
* Stein
* (Almost) Symplectic
* Tame

Tensors|

| Vectors|

* Distribution
* Lie bracket
* Pushforward
* Tangent space
* bundle
* Torsion
* Vector field
* Vector flow

|

Covectors|

* Closed/Exact
* Covariant derivative
* Cotangent space
* bundle
* De Rham cohomology
* Differential form
* Complex
* Vector-valued
* One-form
* Exterior derivative
* Interior product
* Pullback
* Ricci curvature
* flow
* Riemann curvature tensor
* Tensor field
* density
* Volume form
* Wedge product

Bundles|

* Adjoint
* Affine
* Associated
* Cotangent
* Dual
* Fiber
* (Co-) Fibration
* Jet
* Lie algebra
* (Stable) Normal
* Principal
* Spinor
* Subbundle
* Tangent
* Tensor
* Vector

Connections|

* Affine
* Cartan
* Ehresmann
* Form
* Generalized
* Koszul
* Levi-Civita
* Principal
* Vector
* Parallel transport

Related|

* Classification of manifolds
* Gauge theory
* History
* Morse theory
* Moving frame
* Singularity theory

Generalizations|

* Banach
* Diffeology
* Diffiety
* Fréchet
* Hilbert
* K-theory
* Non-Hausdorff
* Orbifold
* Secondary calculus
* over commutative algebras
* Sheaf
* Stratifold
* Supermanifold
* Stratified space

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