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.

Definition

Let ${\displaystyle \xi _{i}=(E_{i},M_{i},p_{i})}$ be a rank n vector bundle over an n-dimensional smooth manifold ${\displaystyle M_{i}}$ for i = 1,2. Denote by ${\displaystyle D(E_{i})}$ the total space of the associated (closed) disk bundle ${\displaystyle D(\xi _{i})}$and suppose that ${\displaystyle \xi _{i},M_{i}}$ and ${\displaystyle D(E_{i})}$are oriented in a compatible way. If we pick two points ${\displaystyle x_{i}\in M_{i}}$, i = 1,2, and consider a ball neighbourhood of ${\displaystyle x_{i}}$ in ${\displaystyle M_{i}}$, then we get neighbourhoods ${\displaystyle D_{i}^{n}\times D_{i}^{n}}$ of the fibre over ${\displaystyle x_{i}}$ in ${\displaystyle D(E_{i})}$. Let ${\displaystyle h:D_{1}^{n}\rightarrow D_{2}^{n}}$ and ${\displaystyle k:D_{1}^{n}\rightarrow D_{2}^{n}}$ be two diffeomorphisms (either both orientation preserving or reversing). The plumbing ^[2] of ${\displaystyle D(E_{1})}$ and ${\displaystyle D(E_{2})}$ at ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ is defined to be the quotient space ${\displaystyle P=D(E_{1})\cup _{f}D(E_{2})}$ where ${\displaystyle f:D_{1}^{n}\times D_{1}^{n}\rightarrow D_{2}^{n}\times D_{2}^{n}}$ is defined by ${\displaystyle f(x,y)=(k(y),h(x))}$. The smooth structure on the quotient is defined by "straightening the angles". ^[2]

Plumbing according to a tree

If the base manifold is an n-sphere ${\displaystyle S^{n}}$, then by iterating this procedure over several vector bundles over ${\displaystyle S^{n}}$ one can plumb them together according to a tree ^{[3] §8}. If ${\displaystyle T}$ is a tree, we assign to each vertex a vector bundle ${\displaystyle \xi }$ over ${\displaystyle S^{n}}$ 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 ${\displaystyle D(\tau _{S^{2k}})}$ denote the disk bundle associated to the tangent bundle of the 2k-sphere. If we plumb eight copies of ${\displaystyle D(\tau _{S^{2k}})}$ according to the diagram ${\displaystyle E_{8}}$, we obtain a 4k-dimensional manifold which certain authors ^{[4] [5]} call the Milnor manifold${\displaystyle M_{B}^{4k}}$ (see also E₈ manifold).

For ${\displaystyle k>1}$, the boundary ${\displaystyle \Sigma ^{4k-1}=\partial M_{B}^{4k}}$ is a homotopy sphere which generates ${\displaystyle \theta ^{4k-1}(\partial \pi )}$, the group of h-cobordism classes of homotopy spheres which bound π-manifolds (see also exotic spheres for more details). Its signature is ${\displaystyle sgn(M_{B}^{4k})=8}$ and there exists ^{[2] V.2.9} a normal map ${\displaystyle (f,b)}$ such that the surgery obstruction is ${\displaystyle \sigma (f,b)=1}$, where ${\displaystyle g:(M_{B}^{4k},\partial M_{B}^{4k})\rightarrow (D^{4k},S^{4k-1})}$ is a map of degree 1 and ${\displaystyle b:\nu _{M_{B}^{4k}}\rightarrow \xi }$ 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 ^{[2] II.1.3} (presented here in the simply connected case):

For all ${\displaystyle k>1,l\in \mathbb {Z} }$, there exists a 2k-dimensional manifold ${\displaystyle M}$ with boundary ${\displaystyle \partial M}$ and a normal map ${\displaystyle (g,c)}$ where ${\displaystyle g:(M,\partial M)\rightarrow (D^{2k},S^{2k-1})}$ is such that ${\displaystyle g|_{\partial M}}$ is a homotopy equivalence, ${\displaystyle c}$ is a bundle map into the trivial bundle and the surgery obstruction is ${\displaystyle \sigma (g,c)=l}$.

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

References

1. John Milnor, On simply connected 4-manifolds
2. William Browder, Surgery on simply-connected manifolds
3. Friedrich Hirzebruch, Thomas Berger, Rainer Jung, Manifolds and Modular Forms
4. Ib Madsen, R. James Milgram, The classifying spaces for surgery and cobordism of manifolds
5. 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