In this post, we will provide a description of the Poincare homology sphere as a plumbing of disk bundles.

### Plumbing Disk Bundles and Handle Decompositions

**Definition.** Let , be two disk bundles over surfaces and let , be disks in , respectively. A plumbing of two disk bundles over surfaces along , is the space where is given by

**Definition. **A space is a plumbing if it is obtained from finitely many such identification along disjoints disks that are in the bases.

**Definition. **A plumbing graph for a plumbing is a graph where vertices are labeled by the disk bundles and edges correspond to plumbing between corresponding disk bundles.

**Example. **Let . Following the convention that , we see that

.

We have two disk bundles over spheres and glued along . It is clear that the gluing is base-to-fiber. Therefore, is a plumbing with the plumbing graph consists of two vertices labeled and one edge connecting the two vertices. The space also has a handle diagram

By reversing the roles of the handles and considering as a 2-handle of , we can see that is obtained by attaching a 2-handle to another two handle . The attaching circle is isotopic to the meridian of the bundle surgery curve.

**Lemma. **The handle diagram for plumbing of disk bundles over the spheres with plumbing graph a tree is a link of unknots where each unknot corresponds to a vertex (and has framing number equals to the euler number of that vertex), and two unknots are linked when the corresponding vertices are joined by an edge.

*Proof. *We proceed by induction. When the plumbing graph has exactly one vertex, our plumbing is a disk bundle over one sphere. We have seen that the corresponding handle diagram for this plumbing consists of one circle labelled by the Euler number for the plumbing.

In general, suppose that we add a vertex and edge to our plumbing graph (in such a way that the graph remains a tree) where corresponds to a new disk bundle over a sphere with Euler number . In a similar fashion to the previous example, the plumbing can be achieved by attaching a 2-handle to the 2-handle for with framing . The attaching circle is again isotopic to the meridian. This completes the proof. ∎

We recall that a handle diagram for a 4-manifold is also a surgery diagram for its boundary. Therefore, the lemma leads us to a new description of the Poincare homology sphere.

### Poincare Homology Sphere as a Plumbing

Recall that the Poincare homology sphere can be described as the boundary of the 4-manifold with a handle diagram

It follows immediately from the previous lemma that the Poincare homology sphere is the boundary of the plumbing with the following plumbing graph.

Finally, we will give a description of the classification of disk bundles over surfaces.

### Classification of Oriented Disk Bundles over Surfaces

Throughout, will be a compact, closed, oriented surfaces. By restricting each fiber to the unit disk, we have a bijection between bundles over :

{Oriented -bundles} { -bundles}.

For each oriented -bundle , we can associated to it a -bundle in a canonical way. We take an oriented vector bundle atlas for and identify with via the formula

.

Therefore, we obtained a bijection between bundles over :

{Oriented -bundles} { -bundles}.

Therefore, we will give a classification of -bundles up to homotopy.

##### Projective Space and the Universal Line Bundle

The inclusion induces the inclusion . Taking the direct limit, we get

.

The canonical line bundle over is the bundle

**Definition.** The universal line bundle is the canonical line bundle over .

It is an important fact that every -bundle over a paracompact topological space is a pull-back of (unique up to homotopy). Furthermore,

**Theorem.** Two -bundles over a paracompact space are isomorphic if and only if the induced maps and are homotopic.

Consequently, the set of isomorphism classes of -bundle over is in bijection with the set of homotopy classes of maps . Since is a , the cohomology representation theorem says that homotopy classes of map is in bijection with . Therefore, -bundle over are classified by these cohomology classes.