Manifolds with singularities
Contents |
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.
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 .
2.2 -manifolds
Following ([Botvinnik2001], [Botvinnik1992]), a more general 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 a partition , such that is a manifold for each , and such that
- for each there is 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 more general definition. In fact, they give the manifolds with a -singularity. For given a manifold , set . Then the manifold with boundary , which appears in the above definition, is a -manifold. The attachement of the cone-end now corresponds to the collapsing of the equivalence relation in .
3 Some examples
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 $
4 Invariants
...
5 Classification/Characterization
...
6 Further discussion
...
7 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
This page has not been refereed. The information given here might be incomplete or provisional. |
Here, is a manifold with boundary .
2.2 -manifolds
Following ([Botvinnik2001], [Botvinnik1992]), a more general 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 a partition , such that is a manifold for each , and such that
- for each there is 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 more general definition. In fact, they give the manifolds with a -singularity. For given a manifold , set . Then the manifold with boundary , which appears in the above definition, is a -manifold. The attachement of the cone-end now corresponds to the collapsing of the equivalence relation in .
3 Some examples
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 $
4 Invariants
...
5 Classification/Characterization
...
6 Further discussion
...
7 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
This page has not been refereed. The information given here might be incomplete or provisional. |
Here, is a manifold with boundary .
2.2 -manifolds
Following ([Botvinnik2001], [Botvinnik1992]), a more general 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 a partition , such that is a manifold for each , and such that
- for each there is 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 more general definition. In fact, they give the manifolds with a -singularity. For given a manifold , set . Then the manifold with boundary , which appears in the above definition, is a -manifold. The attachement of the cone-end now corresponds to the collapsing of the equivalence relation in .
3 Some examples
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 $
4 Invariants
...
5 Classification/Characterization
...
6 Further discussion
...
7 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
This page has not been refereed. The information given here might be incomplete or provisional. |
Here, is a manifold with boundary .
2.2 -manifolds
Following ([Botvinnik2001], [Botvinnik1992]), a more general 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 a partition , such that is a manifold for each , and such that
- for each there is 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 more general definition. In fact, they give the manifolds with a -singularity. For given a manifold , set . Then the manifold with boundary , which appears in the above definition, is a -manifold. The attachement of the cone-end now corresponds to the collapsing of the equivalence relation in .
3 Some examples
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 $
4 Invariants
...
5 Classification/Characterization
...
6 Further discussion
...
7 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
This page has not been refereed. The information given here might be incomplete or provisional. |