# Manifold Atlas:Structure of a Manifolds page

The Manifold Atlas emphasises constructions and examples of manifolds. We suggest, but do not insist, that an article in the Manifolds chapter has the following structure:

Introduction

Construction and examples

Invariants

Classification/Characterization (if available)

Further discussion

References


## 1 Introduction

Briefly orient the reader to manifolds to be discussed: ideally specify the default category being used. This section may be omitted.

## 2 Construction and examples

Present the construction(s) of the manifolds of intereset. Following that give examples of manifolds falling under the construction(s).

## 3 Invariants

List and where necessary describe invariants of intereset. Where possible, record the known values of these invariants. Here are some simple suggestions for invariants.

• (Co)Homology groups and rings.
• Homotopy groups.
• Euler characteristic.
• Characteristic classes.
• Orientability.

## 4 Classification and/or characterisation

• Report what is known concerning completeness of the invariants listed for this class of manifolds or manifold.
• Statements here will typically depend upon the category chosen: the more classification results over more diverse categories, the better.

## 5 Further topics

These could include

• Open problems related to this collection of manifolds: ideally write a Problems page and link to it.
• Occurences of the manifolds in or other branches of mathematics.
• Existence and properties of further structures.
• Curvature and metric properties.
• Embedding/immersions into $\Rr^n$$\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}{^}\Rr^n$.
• Mapping class groups.
• Chirality.
• Group actions or other interesting self-maps.

## 6 Categories

To enable searching for manifolds with given properties, manifolds pages will contain links to categories. For example, an article describing an orientable manifold should contain the following link at the end of the article.

[[Category:orientable]]