Indifferential topology, a branch ofmathematics, astratifold is a generalization of adifferentiable manifold where certain kinds ofsingularities are allowed. More specifically a stratifold is stratified into differentiable manifolds of (possibly) different dimensions. Stratifolds can be used to construct newhomology theories. For example, they provide a new geometric model for ordinary homology. The concept of stratifolds was invented byMatthias Kreck. The basic idea is similar to that of atopologically 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: Adifferential space (in the sense of Sikorski) is a pair${\displaystyle (X,C),}$ whereX is a topological space andC is a subalgebra of the continuous functions${\displaystyle X\to \mathbb{R}}$ such that a function is inC if it is locally inC and${\displaystyle g\circ \left(f_1,\dots ,f_n\right):X\to \mathbb{R}}$ is in C for${\displaystyle g:{\mathbb{R}}^n\to \mathbb{R}}$ smooth and${\displaystyle f_i\in C.}$ A simple example takes forX a smooth manifold and forC just the smooth functions.

For a general differential space${\displaystyle (X,C)}$ and a pointx inX we can define as in the case of manifolds atangent space${\displaystyle T_{x}X}$ as thevector space of allderivations of functiongerms at x. Define strata${\displaystyle X_i=\{x\in X:T_{x}X}$ has dimension i${\displaystyle \}.}$ For ann-dimensional manifoldM we have that${\displaystyle M_n=M}$ 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:

Ak-dimensionalstratifold is a differential space${\displaystyle (S,C),}$ whereS is alocally compactHausdorff space withcountable base of topology. All skeleta should be closed. In addition we assume:

1. The${\displaystyle \left(S_i,C{|}_{S_i}\right)}$ arei-dimensional smooth manifolds.
2. For allx inS, restriction defines anisomorphism ofstalks${\displaystyle C_x\to C^{\mathrm{\infty }}(S_i{)}_x.}$
3. All tangent spaces have dimension ≤ k.
4. For eachx inS and every neighbourhoodU ofx, there exists a function${\displaystyle \rho :U\to \mathbb{R}}$ with${\displaystyle \rho (x)\ne 0}$ and${\displaystyle \text{supp}(\rho )\subset U}$ (a bump function).

An-dimensional stratifold is calledoriented if its (n − 1)-stratum is empty and its top stratum is oriented. One can also define stratifolds with boundary, the so-calledc-stratifolds. One defines them as a pair${\displaystyle (T,\mathrm{\partial }T)}$ of topological spaces such that${\displaystyle T-\mathrm{\partial }T}$ is ann-dimensional stratifold and${\displaystyle \mathrm{\partial }T}$ is an (n − 1)-dimensional stratifold, together with an equivalence class ofcollars.

An important subclass of stratifolds are theregular stratifolds, which can be roughly characterized as looking locally around a point in thei-stratum like thei-stratum times a (n − i)-dimensional stratifold. This is a condition which is fulfilled in most stratifold one usually encounters.

There are plenty of examples of stratifolds. The first example to consider is the opencone over a manifoldM. We define a continuous function fromS to the reals to be inCif and only if it is smooth on${\displaystyle M\times (0,1)}$ 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 substituteM by a stratifoldS in this construction. The cone is oriented if and only ifS 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 areone-point compactifications andsuspensions of manifolds, (real) algebraic varieties with only isolated singularities and (finite) simplicial complexes.

In this section, we will assume all stratifolds to be regular. We call two maps${\displaystyle S,S^{\prime }\to X}$ from two oriented compactk-dimensional stratifolds into a spaceXbordant if there exists an oriented (k + 1)-dimensional compact stratifoldT with boundaryS + (−S') such that the map toX extends to T. The set of equivalence classes of such maps${\displaystyle S\to X}$ is denoted by${\displaystyle S{H}_{k}X.}$ 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 ahomology theory. Clearly,${\displaystyle S{H}_k(\text{point})=0}$ for${\displaystyle k>0}$ since every oriented stratifoldS is the boundary of its cone, which is oriented if${\displaystyle \dim (S)>0.}$ One can show that${\displaystyle S{H}_0(\text{point})\cong \mathbb{Z}.}$ Hence, by theEilenberg–Steenrod uniqueness theorem,${\displaystyle S{H}_k(X)\cong H_k(X)}$ for every spaceX homotopy-equivalent to aCW-complex, whereH denotessingular 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 defineequivariant homology with the help of stratifolds. LetG be a compactLie group. We can then define a bordism theory of stratifolds mapping into a spaceX with aG-action just as above, only that we require all stratifolds to be equipped with an orientation-preserving freeG-action and all maps to be G-equivariant. Denote by${\displaystyle S{H}_k^G(X)}$ the bordism classes. One can prove${\displaystyle S{H}_k^G(X)\cong H_{k-\dim (G)}^G(X)}$ for every X homotopy equivalent to a CW-complex.

Agenus is a ring homomorphism from a bordism ring into another ring. For example, theEuler characteristic defines a ring homomorphism${\displaystyle {\mathrm{\Omega }}^O(\text{point})\to \mathbb{Z}/2[t]}$ from theunoriented bordism ring and thesignature defines a ring homomorphism${\displaystyle {\mathrm{\Omega }}^{SO}(\text{point})\to \mathbb{Z}[t]}$ from theoriented bordism ring. Heret has in the first case degree1 and in the second case degree4, since only manifolds in dimensions divisible by4 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.

Suppose, one has a closed embedding${\displaystyle i:N\hookrightarrow M}$ of manifolds with oriented normal bundle. Then one can define anumkehr map${\displaystyle H_k(M)\to H_{k+\dim (N)-\dim (M)}(N).}$ One possibility is to use stratifolds: represent a class${\displaystyle x\in H_k(M)}$ by a stratifold${\displaystyle f:S\to M.}$ Then makeƒ transversal to N. The intersection ofS andN defines a new stratifoldS' with a map toN, which represents a class in${\displaystyle H_{k+\dim (N)-\dim (M)}(N).}$ It is possible to repeat this construction in the context of an embedding ofHilbert manifolds of finite codimension, which can be used instring topology.

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

• Generalized manifolds
• Homology theory
• Stratifications

• Articles with short description
• Short description matches Wikidata