Whitehead torsion V (Ex)
In the following we use the notation of [Lück2001, Section 1.1]. In particular, if is an -manifold with boundary component and
is an embedding then denotes the manifold of obtained from by attaching a -handle along :
Exercise 0.1. Let be an -dimensional cobordism, and suppose that, relative to , we have
Show that there is another diffeomorphism, relative to , which is of the following form:
The important part is that for each -handle in the first handlebody decomposition, we have an -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 is trivial.
The exercises and hints on this page were sent by Alex Koenen and Arkadi Schelling.