Handlebody decompositions of bordisms (Ex)

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In the following we use the notation of [Lück2001, Section 1.1]. In particular, if W is an n-manifold with boundary component \partial_i W and

\displaystyle \phi^q \colon S^{q-1} \times D^{n-q} \to \partial_i W

is an embedding then W + (\phi^q) denotes the manifold of obtained from W by attaching a q-handle along \phi^q:

\displaystyle  W + (\phi^q) \cong W \cup_{\phi^q} (D^q \times D^{n-q}).

Exercise 0.1. Let (W; \partial_0 W, \partial_1 W) be an n-dimensional cobordism, and suppose that, relative to \partial_0 W, we have

\displaystyle  W      \cong \bigl(\partial_0 W \times [0,1] \bigr)     + \sum_{i=1}^{p_0} (\phi^0_i)     + \ldots     + \sum_{i=1}^{p_n} (\phi^n_i).

Show that there is another diffeomorphism, relative to \partial_1W, which is of the following form:

\displaystyle      W     \cong \bigl( \partial_1 W \times [0,1] \bigr)     + \sum_{i=1}^{p_n} (\psi^0_i)     + \ldots     + \sum_{i=1}^{p_0} (\psi^n_i).

The important part is that for each q-handle in the first handlebody decomposition, we have an (n-q)-handle in the second, dual handlebody decomposition.

Comment 0.2. If one approaches this exercise using Morse functions (and their relation to handlebody decompositions), the above is almost trivial (Question: Why?). The actual intention of this exercise is to go through the details of the rather direct approach outlined in [Lück2001, pp.17-18]. While this is a bit tedious, it provides a good opportunity to get more familiar with handlebody attachments and the like.

Exercise 0.3.

Let W be an n-dimensional manifold whose boundary \partial W is the disjoint sum \partial_0 W\sqcup\partial_1 W and let \phi^q:S^{q-1} \times D^{n-q}\to \partial_1 W be a trivial embedding i.e. an embedding which is given by the restriction of an embedding of the disk D^{n-1}\to \partial_1 W via a fixed standard embedding S^{q-1} \times D^{n-q} \to D^{n-1}.

Show that there exists an embedding \phi^{q+1}:S^q\times D^{n-1-q}\to \partial_1(W+(\phi^q)), such that \phi^{q+1}(S^q\times\{0\}) meets the transverse sphere of the handle (\phi^q) transversally in exactly one point. Conclude by the Cancellation Lemma [Lück2001, Lemma 1.12] that W and W+(\phi^q)+(\phi^{q+1}) are diffeomorphic relative to \partial_0 W.

Exercise 0.4. Let (W,\partial_0 W,\partial_1 W) be an h-cobordism with n\geq 6, which is written as follows:

\displaystyle W\cong \bigl(\partial_0W\times[0,1] \bigr) + \underset{i=1}{\overset{p_2}{\sum}}(\phi_i^2)+\cdots +\underset{i=1}{\overset{p_n}{\sum}}(\phi_i^n).

For 0 \leq q \leq n let W_q \subset W be the result of attaching all the j-handles of W to \partial_0W \times [0, 1] for j \leq q. Fix 2\leq q\leq n-3 and assume that f:S^q\to \partial_1 W_q is an embedding which meets the transverse sphere of (\phi^q_{i_0}) transversally in exactly one point and is disjoint from the transverse spheres of the handles (\phi^q_i) for i\neq i_0. Show that there is \gamma \in \pi_1(W) with

\displaystyle [\tilde{f}]=\pm\gamma\cdot[\phi^q_{i_0}]\in H_q(\widetilde{W}_q, \widetilde{W}_{q-1})\cong \pi_q(\widetilde{W}_q, \widetilde{W}_{q-1})

The exercises and comments on this page were sent by Alex Koenen, Farid Madani, Mihaela Pilca and Arkadi Schelling.

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