Manifold Atlas:Structure of a Manifolds page

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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.
  • 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.

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