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Poincare homology sphere

• A 3-manifold and is the only \(X\in {\mathbf{Z}}\operatorname{HS}^n\) with finite \(\pi_1\).

• Its fundamental group is order 120

• Proves that there exist \(X\ni {\mathbf{Z}}\operatorname{HS}^n\) where \(X\not\cong_{\mathsf{Top}}S^n\).

• Constructions:

  • Glue faces of a dodecahedron
  • \({\operatorname{SO}}_3({\mathbf{R}})/I\), for \(I\cong A_5\) the symmetries of an isocashedron
  • \(+1\) trefoil