# Difference between revisions of "Dodecahedral space"

From Encyclopedia of Mathematics

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− | The first example of a [[Poincaré space|Poincaré space]]. Constructed by H. Poincaré in 1904. It is obtained by identifying the opposite faces of a dodecahedron after they have been rotated by an angle | + | The first example of a [[Poincaré space|Poincaré space]]. Constructed by H. Poincaré in 1904. It is obtained by identifying the opposite faces of a dodecahedron after they have been rotated by an angle $\frac \pi 5$ relative to each other. The dodecahedral space is a manifold of genus 2 with a [[Seifert fibration|Seifert fibration]] and is the only known Poincaré space with finite fundamental group. A dodecahedral space is the orbit space of the free action of a binary icosahedron group on the three-dimensional sphere. |

====References==== | ====References==== | ||

<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Seifert, W. Threlfall, "Lehrbuch der Topologie" , Chelsea, reprint (1980)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Seifert, W. Threlfall, "Lehrbuch der Topologie" , Chelsea, reprint (1980)</TD></TR></table> |

## Revision as of 04:31, 16 March 2012

The first example of a Poincaré space. Constructed by H. Poincaré in 1904. It is obtained by identifying the opposite faces of a dodecahedron after they have been rotated by an angle $\frac \pi 5$ relative to each other. The dodecahedral space is a manifold of genus 2 with a Seifert fibration and is the only known Poincaré space with finite fundamental group. A dodecahedral space is the orbit space of the free action of a binary icosahedron group on the three-dimensional sphere.

#### References

[1] | H. Seifert, W. Threlfall, "Lehrbuch der Topologie" , Chelsea, reprint (1980) |

**How to Cite This Entry:**

Dodecahedral space.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Dodecahedral_space&oldid=21689

This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article