Plumbing (Ex)

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The exercises below are about plumbing manifolds. For the details of the construction, see the page Plumbing.

In this page we use slightly different notation. For i = 1, \dots, n let

\displaystyle  \zeta_i \colon E_i \to S^q_i

be an oriented D^{q}-bundle over S^q. Let G be a graph with verticies \{v_1, \dots, v_k \}.

Starting from the dijoint union of the total spaces E_i we form the plumbing manifold X(G; \{\zeta_i\} as follows: given an edge in G connecting v_i and v_j, for k = i and j, let x_k \in E_k and let D^{q} \times D^{q} \subseteq E_k be a neighbourhood of x_k, such that D^{q} \times \{0\}\subseteq S^q_k and such that y \times D^{q} if the fiber of E_k \to S^q_k. Let h_{\pm}: D^{q} \rightarrow D^{q} and k_{\pm}: D^{q} \rightarrow D^{q} be orientation preserving (resp. reversing) diffeomorphisms. We define the plumbing E_i \diamond E_j of E_i and E_j at x_i and x_j by taking E_i \sqcup E_j and identifying D^q \times D^q \subset E_i and D^q \times D^q \subset E_j via

\displaystyle I_\pm(x,y)=(k_{\pm}(y), h_{\pm}(x)).

Proceeding in this way for each edge of G we obtain the plumbing manifold

\displaystyle  X := X(G; \{\zeta_i\}).

Exercise 0.1. Let X = X(G; \{\zeta_i \}) be connected. Show the following:

  1. \pi_1(\partial X)\cong\pi_1(X) is free.
  2. H_i(\partial X)=H_i(X)=0 for 1<i<q-1

Hint 0.2. The statement is trivial for X, since X is homotopy equivalent to a wedge of 1-spheres and q-spheres. Now use van Kampen's theorem for \pi_1(\partial X) and for H_i(\partial X) use the Mayer-Vietories Sequence with \partial E_i\backslash (D_i^q\times S^{q-1}) and show that all components involved are (q-2) connected.

Exercise 0.3. Choose S^1\subseteq \partial X representing a generator of \pi_1(\partial X) and let V be the trace of a surgery on this S^1. Define X':=X\cup_{\partial X}V. Show the following:

  1. H_i(X')\cong H_i(X) for i\neq 1.
  2. H_i(\partial X')\cong H_i(\partial X) for 1<i<2q-2.

Hint 0.4. For (1) use the long exact sequence of the pair (X',X) and X'\simeq X\cup D^2. For (2) use the long exact sequence of the pair (V,\partial X') as well as Poincaré Duality and the Universal Coefficient Theorem.

Exercise 0.5. Assume now that each bundle \zeta_i is some multiple of \tau_{S^q}, the unit disc bundle of the tangent bundle of the q-sphere. Show that there is a degree 1 normal map (f,b),f:(X',\partial X')\rightarrow (D^{2q},S^{2q-1}).

Hint 0.6. Use that X is homotopy equivalent to a wedge of 1-spheres and q-spheres and that the tangent bundle of X is stabily trivial (since it is so on every component of the wedge), so that the normal bundle of (X,\partial X)\subseteq (D^{2q+k},S^{2q+k-1}) is trivial for k large. Then show that the map can be extend over V.

The exercises and hints on this page were sent by Fabian Hebestreit, Daniel Kasprowski and Christoph Winges.
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