The Poincaré homology sphere

We've seen the Poincaré homology sphere bounds a smooth 4-manifold M.

To get M, we can also start with a 4-ball B⁴ and attach 8 copies of D² × D² to its boundary ∂B⁴ = S³. We attach these along solid tori as shown here:

E₈ again!

This is no coincidence: it's just a consequence of things we've already seen.

For a proof of the claim here see Chapter IV here:

Klaus Lamotke, Regular Solids and Isolated Singularities, Vieweg & Sohn, Braunschweig, 1986.

Robion Kirby and Martin Scharlemann, Eight faces of the Poincaré homology 3-sphere, Uspekhi Matematicheskikh Nauk 37 (1982), 139–159.

The picture of linked circles is from Section 8.3 here:

Alexandru Scorpan, The Wild World of 4-Manifolds, American Mathematical Society, Providence, RI, 2005.

This is a good place to learn more about the Poincaré homology sphere and also the technique of building manifolds via 'surgery on links'. For example, the number 2 labelling each circle in the link above means that we glue each D² × D² onto the 4-ball with 2 twists in the solid torus D² × S¹ ⊂ D² × D².

Another interesting thing you can learn about in Scorpan's book: the Poincaré homology sphere also bounds a topological (not smooth) 4-manifold with boundary that is contractible. Gluing this to M, we get a topological 4-manifold without any boundary. This is called the E₈ manifold since the intersection pairing on its 2nd homology is the Cartan matrix of E₈. This topological manifold cannot be smoothed!