Following on from my previous post, I would like to add a little introduction to the Poincare homology sphere early on in Chapter 1. This gives an opportunity to introduce the \(E_8\) plumbing in a reasonably down-to-earth context, so it will not come as such a surprise when we bring it up with the exotic spheres.
So the historical introduction goes: classification of 2-manifolds; Poincaré and 3-manifolds, and the history of the Poincare conjecture; the Fifth Complement to Analysis Situs and the Poincaré homology sphere.
Poincaré’s discussion is phrased in terms of (what became known as) a Heegard splitting, but without too much difficulty this can be related to the calculations (using van Kampen’s theorem) to compute the fundamental group below…
We start with 2-disk bundles over the 2-sphere. These are classified by an Euler number. We take 8 copies of this bundle and plumb them together according to the \(E_8\) Dynkin diagram:
Each node of the Dynkin diagram represents a copy of our bundle, and two bundles are plumbed together if the two nodes are linked by an edge. The result (after rounding the corners) is a smooth 4-manifold with boundary; call the boundary \(M\), a closed 3-manifold.
The homology of \(M\) is computed from a complex whose only non-trivial differential is the \(E_8\) matrix. Because this matrix is unimodular, \(M\) is a homology sphere. This part of the computation is the same as in higher dimensions.
However, \(M\) is not simply connected. This can be seen as follows. Using van Kampen’s theorem, one computes that the fundamental group of \(M\) has one generator for each node of the Dynkin diagram, and also one relator for each node. Let \(x_k\) be the generator corresponding to node \(k\). The relator corresponding to node \(k\) is of the form
\[ x_k^2 x_{j_1}x_{j_2}\ldots = 1 , \]
where \(j_1,j_2,\ldots \) are the nodes adjacent to \(k\) in the Dynkin diagram. In the case of the diagram above this yields
\[ 1 = x_1^2x_2 = x_2^2x_1x_3 = x_3^2x_4x_5x_2 = x_4^2x_3 = x_5^2x_3x_6 = x_6^2x_5x_7=x_7^2x_8x_6 = x_8^2x_7. \]
Writing \(x=x_1\), \(y=x_8\) allows us to present the group \(\pi_1(M)\) as
\[ \langle x,y | x^3 = y^5 = (xy)^2 \rangle. \]
The permutations \( x = (531) \), \(y = (12345) \) generate the alternating group \(A_5\) and satisfy the above relations (with \(x^3 = y^5 = (xy)^2 = 1\).) Thus \(\pi_1(M) \) surjects onto \(A_5\) and in particular is not trivial. (In fact, it is known that the kernel is of order 2; \(\pi_1(M) \) is the binary icosahedral group of order 120.
Thus \(M\) is a homology sphere but not a homotopy sphere.
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