Plumbing (mathematics)

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 call the Milnor manifold ${\displaystyle M_B^{4k}}$ (see also E₈ manifold).

For ${\displaystyle k>1}$, the boundary ${\displaystyle {\mathrm{\Sigma }}^{4k-1}=\mathrm{\partial }{M}_B^{4k}}$ is a homotopy sphere which generates ${\displaystyle {\theta }^{4k-1}(\mathrm{\partial }\pi )}$, 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 ^V.2.9 a normal map ${\displaystyle (f,b)}$ such that the surgery obstruction is ${\displaystyle \sigma (f,b)=1}$, where ${\displaystyle g:(M_B^{4k},\mathrm{\partial }{M}_B^{4k})\to (D^{4k},S^{4k-1})}$ is a map of degree 1 and ${\displaystyle b:{\nu }_{M_B^{4k}}\to \xi }$ 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 ^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 ${\displaystyle \mathrm{\partial }M}$ and a normal map ${\displaystyle (g,c)}$ where ${\displaystyle g:(M,\mathrm{\partial }M)\to (D^{2k},S^{2k-1})}$ is such that ${\displaystyle g{|}_{\mathrm{\partial }M}}$ 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.