Poincaré
Closed 3-manifold · homology sphereobject:M/poincare
- Area
- Area 3 · Closed 3-manifolds
- True vertices
- 5
- π₁
- —
- H₁
- 0
- Diagram
- Topology.jl · MST invariant examples, figure 2
Closed normal o-graph
gauss code · source presentation[[[4,-5,3,-4,2,-3,1,-2,5,-1]],[+,+,+,+,+]]- True vertices
- 5
π₁ presentation, unreduced
No presentation published for this object.
H₁ = 0
- Free rank
- 0
- Torsion invariant factors
- none
- Order
- 1
H₁ = 0 does not separate this example from S³. Closed 3-manifolds with trivial first homology are not unique.
Invariants
| π₁Unreduced presentation from the raw edge data | Uncomputed |
| H₁Abelianisation of the presentation | Computed |
| MST · F_p[x]/(xᵖ)MST invariant over the truncated polynomial algebra | Uncomputed |
| MST · U_q(sl₂)MST invariant over the quantum group | Uncomputed |
| KuperbergKuperberg invariant | Uncomputed |
- Diagram
- Topology.jl · MST invariant examples, figure 2
- Computation
- Recomputation, 4 September 2026 · Poincaré
Naming
Labels name the example. No independent identification of the object is claimed.
What would settle the identification
The Poincaré homology sphere has H₁ = 0 and fundamental group the binary icosahedral group of order 120, so the fundamental group distinguishes it from S³ where homology cannot.
The computation on this record stops at homology. An identification waits on π₁.