Handlebody decompositions of bordisms (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.
Let be an -dimensional manifold whose boundary is the disjoint sum and let be a trivial embedding i.e. an embedding which is given by the restriction of an embedding of the disk via a fixed standard embedding .
Show that there exists an embedding , such that meets the transverse sphere of the handle transversally in exactly one point. Conclude by the Cancellation Lemma [Lück2001, Lemma 1.12] that and are diffeomorphic relative to .
Exercise 0.4. Let be an h-cobordism with , which is written as follows:
For let be the result of attaching all the -handles of to for . Fix and assume that is an embedding which meets the transverse sphere of transversally in exactly one point and is disjoint from the transverse spheres of the handles for . Show that there is with
The exercises and comments on this page were sent by Alex Koenen, Farid Madani, Mihaela Pilca and Arkadi Schelling.