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 .
- 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.