Plumbing (mathematics)

Way to create new manifolds out of disk bundles

Plumbing two 1-disk bundles to get a new 2-manifold.

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

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

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If the base manifold is an n-sphere  S n {\displaystyle S^{n}} , then by iterating this procedure over several vector bundles over  S n {\displaystyle S^{n}} one can plumb them together according to a tree[3]§8. If  T {\displaystyle T} is a tree, we assign to each vertex a vector bundle  ξ {\displaystyle \xi } over  S n {\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

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Let  D ( τ S 2 k ) {\displaystyle D(\tau _{S^{2k}})} denote the disk bundle associated to the tangent bundle of the 2k-sphere. If we plumb eight copies of  D ( τ S 2 k ) {\displaystyle D(\tau _{S^{2k}})} according to the diagram  E 8 {\displaystyle E_{8}} , we obtain a 4k-dimensional manifold which certain authors[4][5] call the Milnor manifold M B 4 k {\displaystyle M_{B}^{4k}} (see also E8 manifold).

For  k > 1 {\displaystyle k>1} , the boundary  Σ 4 k − 1 = ∂ M B 4 k {\displaystyle \Sigma ^{4k-1}=\partial M_{B}^{4k}} is a homotopy sphere which generates  θ 4 k − 1 ( ∂ π ) {\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  s g n ( M B 4 k ) = 8 {\displaystyle sgn(M_{B}^{4k})=8} and there exists[2] V.2.9 a normal map ( f , b ) {\displaystyle (f,b)} such that the surgery obstruction is  σ ( f , b ) = 1 {\displaystyle \sigma (f,b)=1} , where  g : ( M B 4 k , ∂ M B 4 k ) → ( D 4 k , S 4 k − 1 ) {\displaystyle g:(M_{B}^{4k},\partial M_{B}^{4k})\rightarrow (D^{4k},S^{4k-1})} is a map of degree 1 and  b : ν M B 4 k → ξ {\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

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

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

## References

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1. ^ John Milnor, On simply connected 4-manifolds
2. ^ a b c d 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