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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 n-dimensional total space and produces another n-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 i \in  \{1,  \dots, k\}, let (p_i, q_i) be pairs of positive integers such that p_i + q_i = n and let \alpha_i be an oriented q_i-dimensional vector bundle over the p_i-dimensional oriented manifold M_i. We consider the corresponding disk bundles

\displaystyle  D^{q_i} \to D(\alpha_i) \to M_i.

Let G be a graph with vertices \{v_1, \dots, v_n\} such that the edge set between v_i and v_j, is non-empty only if p_i = q_j and i \neq j. We choose disjoint disks D_{ij} in M_i (one for each edge incident to v_i) and trivializations D(\alpha_i)|_{D_{ij}}\cong D^{p_i} \times D^{q_i} (preserving orientations). Finally we form the manifold W = W(G;\{\alpha_i\}) from the disjoint union of the D(\alpha_i) by identifying, for each edge of G, the corresponding D^{p_i} \times D^{q_i} \subseteq D(\alpha_i) and D^{p_j}  \times D^{q_j}\subseteq D(\alpha_j) with the standard diffeomorphism (x,y)\mapsto (y,x), which interchanges base and fiber of the two bundles.

The manifold W is the result of the plumbing, often one is mainly interested in its boundary \partial W.

[edit] 3 Invariants


[edit] 4 An important special case

If G is simply connected and all the base manifolds are spheres then

\displaystyle \Sigma(G, \{\alpha_i \}) : = \partial W

is often a homotopy sphere. We establish some notation for graphs, bundles and define

  • let T denote the graph with two vertices and one edge connecting them and define \Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\}),
  • let E_8 denote the E_8-graph,
  • let \tau_{n} \in \pi_{n-1}(SO(n)) denote the tangent bundle of the n-sphere,
  • let \gamma_{4s-1}^k \in \pi_{4s-1}(SO(k)) \cong \Zz, k > 4s, denote a generator,
  • let \gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz, denote a generator:
  • let S : \pi_k(SO(j)) \to \pi_k(SO(j+1)) be the suspension homomorphism,
    • S^2(\gamma'_{4k-1})  = \pm 2 \gamma_{4k-1}^{4k+1} for k = 1, 2 and S^2 (\gamma'_{4k-1}) =  \pm \gamma_{4k-1}^{4k+1} for k > 2,
  • let \eta_n : S^{n+1} \to S^n be essential.

The plumbing construction can be used to produce exotic spheres:

  • \Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\}) =: \Sigma_M, the Milnor sphere, generates bP_{4k}, k>1.
  • \Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K, the Kervaire sphere, generates bP_{4k+2}.
  • \Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}') is the inverse of the Milnor sphere for k = 1, 2.
    • For general k, \Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}') is exotic.
  • \Sigma^8(\gamma_3^5, \eta_3\tau_4), generates \Theta_8 = \Zz_2.
  • \Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8), generates \Theta_{16} = \Zz_2.

[edit] 5 References

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