Plumbing
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[edit] 1 Introduction
Plumbing is a gluing construction which takes as input some disk bundles over manifolds (frequently just spheres)
with -dimensional total space and produces another
-manifold with boundary.
It identifies fibers of one bundle with disks in the base manifold of the other bundle and vice versa.
[edit] 2 Construction
As special case of the following construction goes back at least to [Milnor1959].
Let , let
be pairs of positive integers such that
and let
be an oriented
-dimensional vector bundle over the
-dimensional oriented manifold
. We consider the corresponding disk bundles

Let be a graph with vertices
such that the edge set between
and
, is non-empty only if
and
. We choose disjoint disks
in
(one for each edge incident to
)
and trivializations
(preserving orientations).
Finally we form the manifold
from the disjoint union of the
by identifying, for each edge of
, the corresponding
and
with the standard diffeomorphism
, which interchanges
base and fiber of the two bundles.
The manifold is the result of the plumbing, often one is mainly interested in its boundary
.
[edit] 3 Invariants
...
[edit] 4 An important special case
If is simply connected and all the base manifolds are spheres then

is often a homotopy sphere. We establish some notation for graphs, bundles and define
- let
denote the graph with two vertices and one edge connecting them and define
,
- let
denote the
-graph,
- let
denote the tangent bundle of the
-sphere,
- let
,
, denote a generator,
- let
, denote a generator:
- let
be the suspension homomorphism,
for
and
for
,
- let
be essential.
The plumbing construction can be used to produce exotic spheres:
-
, the Milnor sphere, generates
,
.
-
, the Kervaire sphere, generates
.
-
is the inverse of the Milnor sphere for
.
- For general
,
is exotic.
- For general
-
, generates
.
-
, generates
.
[edit] 5 References
- [Milnor1959] J. Milnor, Differentiable structures on spheres, Amer. J. Math. 81 (1959), 962–972. MR0110107 (22 #990) Zbl 0111.35501