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> </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 link <pre> [[Category:orientable]] </pre> at the end of the article. <!--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 Special:Categories. For example, an article describing an orientable manifold should contain the link

[[Category:orientable]]

at the end of the article.