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}$$\newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \DeclareMathOrd\GL{GL} % general linear group \DeclareMathOrd\SL{SL} % special linear group \DeclareMathOrd\SO{SO} % special orthogonal group \DeclareMathOrd\SU{SU} % special unitary group \DeclareMathOrd\Spin{Spin} % Spin group \DeclareMathOrd\RP{\Rr\mathrm P} % real projective space \DeclareMathOrd\CP{\Cc\mathrm P} % complex projective space \DeclareMathOrd\HP{\Hh\mathrm P} % quaternionic projective space \DeclareMathOrd\Top{\mathrm{Top}} % topological category \DeclareMathOrd\PL{\mathrm{PL}} % piecewise linear category \DeclareMathOrd\Cat{\mathrm{Cat}} % any category \DeclareMathOrd\KS{KS} % Kirby-Siebenmann class \DeclareMathOrd\Hud{Hud} % Hudson torus\overline{A}$ that looks like a manifold outside of a compact 'singularity set', while the singularity set has a neighborhood that looks like the product of manifold and a cone. Here is a precise definition. Let $P_1$$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 $M$$M$ is a $\Sigma$$\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

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

.

3 Construction and examples

...

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.