Plumbing (mathematics)

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]

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.

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.

• 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