Manifolds with singularities
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== Introduction == | == Introduction == | ||
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− | Manifolds with singularities are geometric objects | + | Manifolds with singularities are geometric objects in topology generalizing manifolds. They were introduced in {{cite|Sullivan1996}} and {{cite|Baas1973}}. Applications of the concept include representing cycles in homology theories with coefficients. |
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== Definitions == | == Definitions == | ||
+ | ===Cone-like singularities=== | ||
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A manifold with singularities of Baas-Sullivan type is a topological space $\overline{A}$ 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. | A manifold with singularities of Baas-Sullivan type is a topological space $\overline{A}$ 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. | ||
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== Construction and examples == | == Construction and examples == | ||
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Revision as of 14:21, 7 June 2010
Contents |
1 Introduction
Manifolds with singularities are geometric objects in topology generalizing manifolds. They were introduced in [Sullivan1996] 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
3 Construction and examples
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4 Invariants
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5 Classification/Characterization
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6 Further discussion
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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
- [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. |