---
date: 2022-02-23 18:45
modification date: Wednesday 23rd February 2022 18:45:08
title: "Poincare homology sphere"
aliases: [Poincare homology sphere]

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- Tags 
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- Refs:
	- #todo/add-references
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# Poincare homology sphere


- A [3-manifold](Unsorted/Three-manifolds%20MOC.md) and is the only $X\in \ZHS^n$ with finite $\pi_1$.

- Its [fundamental group](fundamental group) is order 120
- Proves that there exist $X\ni \ZHS^n$ where $X\not\cong_\Top S^n$.

- Constructions:
	- Glue faces of a dodecahedron
	- $\SO_3(\RR)/I$, for $I\cong A_5$ the symmetries of an isocashedron
	- $+1$ [trefoil](trefoil) 

![](attachments/Pasted%20image%2020220401212947.png)