Whitehead torsion V (Ex)

In the following we use the notation of [Lück2001, Section 1.1]. In particular, if $W$$<wikitex>; In the following we use the notation of {{citeD|Lück2001|Section 1.1}}. In particular, if $W$ is an $n$-manifold with boundary component $\partial_i W$ and $$\phi^q \colon S^{q-1} \times D^{n-q} \to \partial_i W$$ is an embedding then $W + (\phi^q)$ denotes the manifold of obtained from $W$ by attaching a $q$-handle along $\phi^q$: $$ W + (\phi^q) \cong W \cup_{\phi^q} (D^q \times D^{n-q}).$$ {{beginthm|Exercise}} Let $(W; \partial_0 W, \partial_1 W)$ be an $n$-dimensional cobordism, and suppose that, relative to $\partial_0 W$, we have $$ W \cong \partial_0 W \times [0,1] + \sum_{i=1}^{p_0} (\phi^0_i) + \ldots + \sum_{i=1}^{p_n} (\phi^n_i). $$ Show that there is another diffeomorphism, relative to $\partial_1W$, which is of the following form: $$ W \cong \partial_1 W \times [0,1] + \sum_{i=1}^{p_n} (\psi^0_i) + \ldots + \sum_{i=1}^{p_0} (\psi^n_i). $$ The important part is that for each $q$-handle in the first handlebody decomposition, we have an $(n-q)$-handle in the second, dual handlebody decomposition. {{endthm}} {{beginrem|Comment}} If one approaches this exercise using Morse functions (and their relation to handlebody decompositions), the above is almost trivial (Question: Why?). The actual intention of this exercise is to go through the details of the rather direct approach outlined in {{citeD|Lück2001|pp.17-18}}. While this is a bit tedious, it provides a good opportunity to get more familiar with handlebody attachments and the like. {{endrem}} {{beginthm|Exercise}} * Show that the operations (1)-(5) from {{citeD|Lück2001|Section 1.4}} used in the definition of the Whitehead group in fact yield equivalence classes of invertible matrices of arbitrary size. * Show that $Wh(e)$ is trivial. {{endthm}} </wikitex> [[Category:Exercises]]W$ is an $n$$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

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

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

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

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.

• Show that the operations (1)-(5) from [Lück2001, Section 1.4] used in the definition of the Whitehead group in fact yield equivalence classes of invertible matrices of arbitrary size.
• Show that $Wh(e)$$Wh(e)$ is trivial.