Dold manifold

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[edit] 1 Introduction

A Dold manifold is a manifold of the form

$\displaystyle P(m,n):= (S^m \times \mathbb {CP}^n)/\tau,$ $\displaystyle P(m,n):= (S^m \times \mathbb {CP}^n)/\tau,$

where $m>0$ $\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}m>0$ , and the involution $\tau$ $\tau$ sends $(x,[y])$ $(x,[y])$ to $(-x, [\bar y])$ $(-x, [\bar y])$ where $\bar y = (\bar y_0,...,\bar y_n)$ $\bar y = (\bar y_0,...,\bar y_n)$ for $y =(y_0,...y_n)$ $y =(y_0,...y_n)$ .

Dold used these manifolds in [Dold1956] as generators for the unoriented bordism ring.

[edit] 2 Construction and examples

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[edit] 3 Invariants

The fibre bundle $p:P(m,n) \to \mathbb {RP}^m$ has a section $s([x]) := [(x,[1,...,1])]$ and we consider the cohomology classes (always with $\mathbb Z/2$ -coefficients)

$\displaystyle c:= p^*(x) \in H^1(P(m,n)),$ $\displaystyle c:= p^*(x) \in H^1(P(m,n)),$

where $x$ is a generator of $H^1(\mathbb {RP}^m)$ , and

$\displaystyle d \in H^2(P(m,n)),$ $\displaystyle d \in H^2(P(m,n)),$

which is characterized by the property that the restriction to a fibre is non-trivial and $s^*(d)=0$ .

The classes $c \in H^1(P(m,n))$ and $d\in H^2(P(m,n))$ generate $H^*(P(m,n);\mathbb Z/2)$ with only the relations

$\displaystyle c^{m+1} =0$ $\displaystyle c^{m+1} =0$

and

$\displaystyle d^{n+1} =0.$ $\displaystyle d^{n+1} =0.$

The Steenrod squares act by

$\displaystyle Sq^0 =id, \,\, Sq^1(c) = c^2,\,\, Sq^1(d) = cd,\,\, Sq^2(d) =d^2,$ $\displaystyle Sq^0 =id, \,\, Sq^1(c) = c^2,\,\, Sq^1(d) = cd,\,\, Sq^2(d) =d^2,$

and all other Squares $Sq^i$ act trivially on $c$ and $d$ . On the decomposable classes the action is given by the Cartan formula.