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 and subsequently used extensively in surgery theory to produce manifolds and normal maps with given surgery obstructions. 
Definition.
Let formula_1 be a rank "n" vector bundle over an "n"-dimensional smooth manifold formula_2 for "i" = 1,2. Denote by formula_3 the total space of the associated (closed) disk bundle formula_4and suppose that formula_5 and formula_3are oriented in a compatible way. If we pick two points formula_7, "i" = 1,2, and consider a ball neighbourhood of formula_8 in formula_2, then we get neighbourhoods formula_10 of the fibre over formula_8 in formula_3. Let formula_13 and formula_14 be two diffeomorphisms (either both orientation preserving or reversing). The plumbing of formula_15 and formula_16 at formula_17 and formula_18 is defined to be the quotient space formula_19 where formula_20 is defined by formula_21.
Plumbing according to a tree.
If the base manifold is an "n"-sphere formula_22, then by iterating this procedure over several vector bundles over formula_22 one can plumb them together according to a tree§8. If formula_24 is a tree, we assign to each vertex a vector bundle "formula_25" over formula_22 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.
Milnor manifolds.
Let formula_27 denote the disk bundle associated to the tangent bundle of the "2k"-sphere. If we plumb eight copies of formula_27 according to the diagram formula_29, we obtain a "4k"-dimensional manifold which certain authors call the Milnor manifold formula_30 (see also E8 manifold). 
For formula_31, the boundary formula_32 is a homotopy sphere which generates formula_33, the group of "h"-cobordism classes of homotopy spheres which bound π-manifolds (see also exotic spheres for more details). Its signature is formula_34 and there exists V.2.9 a normal map formula_35 such that the surgery obstruction is formula_36, where formula_37 is a map of degree 1 and formula_38 is a bundle map from the stable normal bundle of the Milnor manifold to a certain stable vector bundle.
The plumbing theorem.
A crucial theorem for the development of surgery theory is the so-called "Plumbing Theorem" II.1.3 (presented here in the simply connected case):
For all formula_39, there exists a "2k"-dimensional manifold formula_40 with boundary formula_41 and a normal map formula_42 where formula_43 is such that formula_44 is a homotopy equivalence, formula_45 is a bundle map into the trivial bundle and the surgery obstruction is formula_46.
The proof of this theorem makes use of the Milnor manifolds defined above.