Plumbing
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Contents |
[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
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[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.
- , 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