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:

==<sectioncount/><sectioncount/> Introduction ==

==<sectioncount/><sectioncount/> Construction and examples ==

==<sectioncount/><sectioncount/> Invariants ==

==<sectioncount/><sectioncount/> Classification/Characterization (if available) ==

==<sectioncount/><sectioncount/> Further discussion ==

==<sectioncount/><sectioncount/> 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$$<wikitex>; 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: <!--<pre> == Introduction == == Construction and examples == == Invariants == == Classification/Characterization (if available) == == Further discussion == == References == </pre> --> <pre> Introduction Construction and examples Invariants Classification/Characterization (if available) Further discussion References </pre> </wikitex> == Introduction == <wikitex>; Briefly orient the reader to manifolds to be discussed: ideally specify the default category being used. This section may be omitted. </wikitex> == Construction and examples == <wikitex>; Present the construction(s) of the manifolds of intereset. Following that give examples of manifolds falling under the construction(s). </wikitex> == Invariants == <wikitex>; 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. <!-- ** Stiefel-Whitney classes. ** Pontrjagin classes. ** Chern classes.--> * Orientability. </wikitex> == Classification and/or characterisation == <wikitex>; * 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. </wikitex> == Further topics == <wikitex>; These could include * Open problems related to this collection of manifolds: ideally write a [[:Category: Problems|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 $\Rr^n$. * Mapping class groups. * [[Chirality]]. * Group actions or other interesting self-maps. </wikitex> == Categories == <wikitex>; To enable searching for manifolds with given properties, manifolds pages will contain links to [[Special:Categories|categories]]. For example, an article describing an orientable manifold should contain the following link at the end of the article. <pre> [[Category:orientable]] </pre> <!--This is of course only a suggestion, and different families of manifolds may require different structures of their articles. --> </wikitex>\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]]