Manifolds with singularities
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== Introduction == | == Introduction == | ||
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− | 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. | + | Manifolds with singularities are geometric objects in topology generalizing manifolds. They were introduced in ({{cite|Sullivan1996}},{{cite|Sullivan1967}}) and {{cite|Baas1973}}. Applications of the concept include representing cycles in homology theories with coefficients. |
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===<wikitex>$\Sigma$-manifolds</wikitex>=== | ===<wikitex>$\Sigma$-manifolds</wikitex>=== | ||
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− | Following {{cite|Botvinnik2001}}, {{cite|Botvinnik1992}}, a definition can be given as follows. | + | Following ({{cite|Botvinnik2001}}, {{cite|Botvinnik1992}}), a definition can be given as follows. For closed manifolds $P_1 , ..., P_k$ set $\Sigma := (P_1, ... , P_k)$. $\Sigma$ may be empty. For a subset $I = \{i_1,..., i_q\} \subset \{1,...,k\}$ define $P^I := P_{i_1} \times ...\times P_{i_q}$. |
+ | {{beginthm|Def}}A manifold $M$ is a $\Sigma$-Manifold if | ||
+ | # there is a partition $\partial M = \partial_0 M \cup ... \cup \partial_k M$, such that $\partial_I M := \partial_{i_1} \cap ... \cap \partial_{i_q} M$ is a manifold for each $I = \{i_1,...,i_q\} \subset \{0,...,k\}$, and such that | ||
+ | $$ \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M $$. | ||
+ | # | ||
+ | |||
+ | |||
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== Construction and examples == | == Construction and examples == |
Revision as of 15:07, 7 June 2010
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 definition can be given as follows. For closed manifolds set . may be empty. For a subset define .
Def 2.1.A manifold is a -Manifold if
- there is a partition , such that is a manifold for each , and such that
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
- [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. |