Plumbing (mathematics)

In the mathematical field of geometric topology, among the techniques known as surgery theory, the process of plumbing is a way to create new manifolds out of disk bundles.

It was first described by John Milnor[1] and subsequently used extensively in surgery theory to produce manifolds and normal maps with given surgery obstructions.

be a rank n vector bundle over an n-dimensional smooth manifold

the total space of the associated (closed) disk bundle

are oriented in a compatible way.

If we pick two points

, i = 1,2, and consider a ball neighbourhood of

be two diffeomorphisms (either both orientation preserving or reversing).

is defined to be the quotient space

The smooth structure on the quotient is defined by "straightening the angles".

[2] If the base manifold is an n-sphere

, then by iterating this procedure over several vector bundles over

one can plumb them together according to a tree[3]§8.

is a tree, we assign to each vertex a vector bundle

and we plumb the corresponding disk bundles together if two vertices are connected by an edge.

One has to be careful that neighbourhoods in the total spaces do not overlap.

denote the disk bundle associated to the tangent bundle of the 2k-sphere.

If we plumb eight copies of

, we obtain a 4k-dimensional manifold which certain authors[4][5] call the Milnor manifold

is a homotopy sphere which generates

, the group of h-cobordism classes of homotopy spheres which bound π-manifolds (see also exotic spheres for more details).

and there exists[2] V.2.9 a normal map

is a map of degree 1 and

is a bundle map from the stable normal bundle of the Milnor manifold to a certain stable vector bundle.

A crucial theorem for the development of surgery theory is the so-called Plumbing Theorem[2] II.1.3 (presented here in the simply connected case): For all

, there exists a 2k-dimensional manifold

and a normal map

is a homotopy equivalence,

is a bundle map into the trivial bundle and the surgery obstruction is

The proof of this theorem makes use of the Milnor manifolds defined above.