Manifolds with singularities
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 1 Introduction
Manifolds with singularities are geometric objects in topology generalizing manifolds. They were introduced in ([Sullivan1996],[Sullivan1967]) and [Baas1973]. Applications of the concept include representing cycles in homology theories with coefficients. All manifolds in this article are understood to be smooth.
 2 Definitions
 2.1 Cone-like singularities
A manifold with singularities of Baas-Sullivan type is a topological space that looks like a manifold outside of a compact 'singularity set', while the singularity set has a neighborhood that looks like the product of manifold and a cone. Here is a precise definition. Let be a closed manifold. A manifold with a -singularity (following [Baas1973]) is a space of the form
Here, is a manifold with boundary .
More complex singularities occur if, instead of taking a cone over only one manifold , we allow a collection of several closed manifolds. In this case, we define a a manifold with a -singularity to be a (second-countable and Hausdorff) topological space locally homeomorphic to one of the spaces .
An alternative approach to manifolds with singularities would be to remove the singular set and to define an equivalence relation on the remaining manifold that 'remembers' the singularities. This view is taken in ([Botvinnik1992],[Botvinnik2001]). We describe it in the next section.
 2.2 -manifolds
Following ([Botvinnik2001], [Botvinnik1992]), an alternative definition can be given. Let be a (possibly empty) collection of closed manifolds and denote by the set containing only one point. Then define . For a subset define .
Definition 2.1.A manifold is a -Manifold if there is given
- a partition , such that is a manifold for each , and such that
- for each a manifold and a diffeomorphism
such that if and is the inclusion, then the composition
restricts to the identity on the factor in . The diffeomorphisms are called product structures.
On a -manifold , there is a canonical equivalence relation : two points are defined to be equivalent if there is an such that and , where is the projection. Now we can give a general definition: a manifold with a -singularity is a topological space of the form
for a -manifold .
The spaces defined above as manifolds with a -singularity are contained in this new definition. Given manifolds , set . Removing a neighborhood of the cone-tips in a manifold with -singularity gives a -manifold . Now the collapsing of the equivalence relation in corresponds to the re-attachement of the cone-ends.
When dealing with manifolds with singularities it is convenient to work with the underlying -manifold and make sure that all operations one performs on them are compatible with the equivalence relation.
 3 Some examples and constructions
 3.1 Intersecting spheres
A basic example is presented by two spheres intersecting each other in a sphere of lower dimension. Choose an embedding . Then define . Outside of the intersecting sphere this is an -dimensional manifold. The intersecting sphere itself has a neigborhood homeomorphic to the product of and a cone over . We can write .
 3.2 Inverse images of critical points
Let be a Morse-function with as a single critical point. We can suppose that near . Setting , we see that the cone provides a neighborhood of in . It follows that is of the form .
 3.3 Structures on manifolds with singularities
Geometric and topological structures that exist for ordinary manifolds can also be defined for manifolds with singularities. This is done in the following way. A manifold with -singularity with underlying -manifold carries the structure in question if all manifolds involved in carry this structure and the product diffeomorphisms preserves it.
For example, is orientable if as well as all manifolds in and the manifolds are orientable and the product diffeomorphisms are orientation-preserving. As another example, becomes a Riemannian manifold with singularities if we put a Riemannian metric on as well as on the manifolds in and on the manifolds in such a way that the product diffeomorphism are isometries.
 3.4 Bundles on manifolds with singularities
As usual, we define a bundle on a manifold with singularities as a bundle on the underlying manifold subject to the following additional condition: there are bundles over the manifolds and bundle equivalences covering the product diffeomorphisms.
 4 Bordism theory for manifolds with singularities
A -manifold induces the structure of a -manifold on . We call the boundary of . Given a manifold with a -singularity , we define to be its boundary. A theory of bordism with -singularities can now be developed just as for ordinary manifolds.
For illustration we pick up the case of a -singularity considered above. is bordant to zero if there exists , such that
 5 Invariants
 6 Classification/Characterization
 7 Further discussion
 8 References
- [Baas1973] N. A. Baas, On bordism theory of manifolds with singularities, Math. Scand. 33 (1973), 279–302 (1974). MR0346824 (49 #11547b) Zbl 0281.57027
- [Botvinnik1992] B. I. Botvinnik, Manifolds with singularities and the Adams-Novikov spectral sequence, Cambridge University Press, Cambridge, 1992. MR1192127 (93h:55002) Zbl 0764.55001
- [Botvinnik2001] B. Botvinnik, Manifolds with singularities accepting a metric of positive scalar curvature, Geom. Topol. 5 (2001), 683–718 (electronic). MR1857524 (2002j:57045) Zbl 1002.57055
- [Sullivan1967] D. Sullivan, On the Hauptvermutung for manifolds, Bull. Amer. Math. Soc. 73 (1967), 598–600. MR0212811 (35 #3676) Zbl 0153.54002
- [Sullivan1996] D. P. Sullivan, Triangulating and smoothing homotopy equivalences and homeomorphisms. Geometric Topology Seminar Notes, 1 (1996), 69–103. MR1434103 (98c:57027) Zbl 0871.57021