Handlebody decompositions of bordisms (Ex)

In the following we use the notation of [Lück2001, Section 1.1]. In particular, if $W$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}W$ is an $n$$n$-manifold with boundary component $\partial_i W$$\partial_i W$ and

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

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

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

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

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

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

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

The important part is that for each $q$$q$-handle in the first handlebody decomposition, we have an $(n-q)$$(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$$W$ be an $n$$n$-dimensional manifold whose boundary $\partial W$$\partial W$ is the disjoint sum $\partial_0 W\sqcup\partial_1 W$$\partial_0 W\sqcup\partial_1 W$ and let $\phi^q:S^{q-1} \times D^{n-q}\to \partial_1 W$$\phi^q:S^{q-1} \times D^{n-q}\to \partial_1 W$ be a trivial embedding i.e. an embedding which is given by the restriction of an embedding of the disk $D^{n-1}\to \partial_1 W$$D^{n-1}\to \partial_1 W$ via a fixed standard embedding $S^{q-1} \times D^{n-q} \to D^{n-1}$$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))$$\phi^{q+1}:S^q\times D^{n-1-q}\to \partial_1(W+(\phi^q))$, such that $\phi^{q+1}(S^q\times\{0\})$$\phi^{q+1}(S^q\times\{0\})$ meets the transverse sphere of the handle $(\phi^q)$$(\phi^q)$ transversally in exactly one point. Conclude by the Cancellation Lemma [Lück2001, Lemma 1.12] that $W$$W$ and $W+(\phi^q)+(\phi^{q+1})$$W+(\phi^q)+(\phi^{q+1})$ are diffeomorphic relative to $\partial_0 W$$\partial_0 W$.

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

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

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

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

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