Manifolds with singularities

Revision as of 14:35, 7 June 2010

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 $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; 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. </wikitex> == Definitions == ===Cone-like singularities=== <wikitex>; 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. Here is a precise definition. Let $P_1$ be a closed manifold. A manifold with a $P_1$-singularity (following {{cite|Baas1973}}) is a space of the form $$\overline{A} = A \cup_{A(1) \times P_1} A(1) \times C P(1)$$ $$\partial A = A(1) \times P_1 $$ Here, $A$ is a manifold with boundary $A(1)$. </wikitex> ===<wikitex>$\Sigma$-manifolds</wikitex>=== <wikitex> Following {{cite|Botvinnik2001}}, {{cite|Botvinnik1992}}, a definition can be given as follows. </wikitex> == Construction and examples == <wikitex>; ... </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}\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. Here is a precise definition. Let $P_1$ be a closed manifold. A manifold with a $P_1$-singularity (following [Baas1973]) is a space of the form

$$\displaystyle \overline{A} = A \cup_{A(1) \times P_1} A(1) \times C P(1)$$

$$\displaystyle \partial A = A(1) \times P_1$$

Here, $A$ is a manifold with boundary $A(1)$.

2.2 $\Sigma$-manifolds

Following [Botvinnik2001], [Botvinnik1992], a definition can be given as follows.

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
• [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