Manifolds with singularities

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 $\overline{A}$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; Manifolds with singularities are geometric objects in topology generalizing manifolds. They were introduced in ({{cite|Sullivan1996}},{{cite|Sullivan1967}}) and {{cite|Baas1973}}. Applications of the concept include representing cycles in homology theories with coefficients. </wikitex> == Definitions == ===Cone-like singularities=== <wikitex>; A manifold with singularities of Baas-Sullivan type is a topological space $\overline{A}$ that looks like a manifold outside of a compact 'singularity set', while the singularity set has a neighborhood that looks like the product of manifold and a cone. Here is a precise definition. Let $P_1$ be a closed manifold. A manifold with a $P_1$-singularity (following {{cite|Baas1973}}) is a space of the form $$\overline{A} = A \cup_{A(1) \times P_1} A(1) \times C P(1)$$ $$\partial A = A(1) \times P_1 $$ Here, $A$ is a manifold with boundary $A(1)$. </wikitex> ===<wikitex>$\Sigma$-manifolds</wikitex>=== <wikitex> Following ({{cite|Botvinnik2001}}, {{cite|Botvinnik1992}}), a definition can be given as follows. For closed manifolds $P_1 , ..., P_k$ set $\Sigma := (P_1, ... , P_k)$. $\Sigma$ may be empty. For a subset $I = \{i_1,..., i_q\} \subset \{1,...,k\}$ define $P^I := P_{i_1} \times ...\times P_{i_q}$. {{beginthm|Def}}A manifold $M$ is a $\Sigma$-Manifold if # there is a partition $\partial M = \partial_0 M \cup ... \cup \partial_k M$, such that $\partial_I M := \partial_{i_1} \cap ... \cap \partial_{i_q} M$ is a manifold for each $I = \{i_1,...,i_q\} \subset \{0,...,k\}$, and such that $$ \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M $$. # </wikitex> == Construction and examples == <wikitex>; ... </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}\overline{A}$ that looks like a manifold outside of a compact 'singularity set', while the singularity set has a neighborhood that looks like the product of manifold and a cone. Here is a precise definition. Let $P_1$$P_1$ be a closed manifold. A manifold with a $P_1$$P_1$-singularity (following [Baas1973]) is a space of the form

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

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

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

2.2 $\Sigma$$\Sigma$-manifolds

Following ([Botvinnik2001], [Botvinnik1992]), a definition can be given as follows. For closed manifolds $P_1 , ..., P_k$$P_1 , ..., P_k$ set $\Sigma := (P_1, ... , P_k)$$\Sigma := (P_1, ... , P_k)$. $\Sigma$$\Sigma$ may be empty. For a subset $I = \{i_1,..., i_q\} \subset \{1,...,k\}$$I = \{i_1,..., i_q\} \subset \{1,...,k\}$ define $P^I := P_{i_1} \times ...\times P_{i_q}$$P^I := P_{i_1} \times ...\times P_{i_q}$.

Def 2.1.A manifold

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$M$ is a $\Sigma$$\Sigma$-Manifold if

1. there is a partition $\partial M = \partial_0 M \cup ... \cup \partial_k M$$\partial M = \partial_0 M \cup ... \cup \partial_k M$, such that $\partial_I M := \partial_{i_1} \cap ... \cap \partial_{i_q} M$$\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\}$$I = \{i_1,...,i_q\} \subset \{0,...,k\}$, and such that

$\displaystyle \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M$ $$\displaystyle \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M$$

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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

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