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− | 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
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− | $$\phi^q \colon S^{q-1} \times D^{n-q} \to \partial_i W$$
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− | is an embedding then $W + (\phi^q)$ denotes the manifold of obtained from $W$ by attaching a $q$-handle along $\phi^q$:
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− | $$ W + (\phi^q) \cong W \cup_{\phi^q} (D^q \times D^{n-q}).$$
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− | {{beginthm|Exercise}}
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− | Let $(W; \partial_0 W, \partial_1 W)$ be an $n$-dimensional cobordism, and suppose that, relative to $\partial_0 W$, we have
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− | $$ W
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− | \cong \partial_0 W \times [0,1]
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− | + \sum_{i=1}^{p_0} (\phi^0_i)
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− | + \ldots
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− | + \sum_{i=1}^{p_n} (\phi^n_i).
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− | $$
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− | Show that there is another diffeomorphism, relative to $\partial_1W$, which is of the following form:
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− | $$
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− | W
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− | \cong \partial_1 W \times [0,1]
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− | + \sum_{i=1}^{p_n} (\psi^0_i)
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− | + \ldots
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− | + \sum_{i=1}^{p_0} (\psi^n_i).
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− | $$
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− | 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.
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− | {{endthm}}
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− | {{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.
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− | {{endrem}}
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| {{beginthm|Exercise}} | | {{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 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. |
The exercises and hints on this page were sent by Alex Koenen and Arkadi Schelling.