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.
Let be a rank n vector bundle over an n-dimensional smooth manifold for i = 1,2. Denote by the total space of the associated (closed) disk bundle and suppose that and are oriented in a compatible way. If we pick two points , i = 1,2, and consider a ball neighbourhood of in , then we get neighbourhoods of the fibre over in . Let and be two diffeomorphisms (either both orientation preserving or reversing). The plumbing of and at and is defined to be the quotient space where is defined by .
Plumbing according to a tree
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§8. If is a tree, we assign to each vertex a vector bundle over 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.
Let denote the disk bundle associated to the tangent bundle of the 2k-sphere. If we plumb eight copies of according to the diagram , we obtain a 4k-dimensional manifold which certain authors call the Milnor manifold (see also E8 manifold).
For , the boundary 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). Its signature is and there exists V.2.9 a normal map such that the surgery obstruction is , where 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.
The plumbing theorem
For all , there exists a 2k-dimensional manifold with boundary and a normal map where is such that 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.
- John Milnor, On simply connected 4-manifolds
- William Browder, Surgery on simply-connected manifolds
- Friedrich Hirzebruch, Thomas Berger, Rainer Jung, Manifolds and Modular Forms
- Ib Madsen, R. James Milgram, The classifying spaces for surgery and cobordism of manifolds
- Santiago López de Medrano, Involutions on Manifolds
- Browder, William (1972), Surgery on simply-connected manifolds, Springer-Verlag, ISBN 978-3-642-50022-0
- Milnor, John (1956), On simply connected 4-manifolds, Symposium Internal de Topología Algebráica, México
- Hirzebruch, Friedrich; Berger, Thomes; Jung, Rainer (1994), Manifolds and Modular Forms, Springer-Verlag, ISBN 978-3-528-16414-0
- Madsen, Ib; Milgram, R. James (1979), The classifying spaces for surgery and cobordism of manifolds, Princeton University Press, ISBN 978-1-4008-8147-5
- López de Medrano, Santiago (1971), Involutions on Manifolds, Springer-Verlag, ISBN 978-3-642-65014-7