Handlebody decompositions of bordisms (Ex)

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

$$\displaystyle [\tilde{f}]=\pm\gamma\cdot[\phi^q_{i_0}] \in \pi_q(\tilde W). ??$$

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

$$\displaystyle [\tilde{f}]=\pm\gamma\cdot[\phi^q_{i_0}] \in \pi_q(\tilde W). ??$$