Cell attachments for surgery (Ex)
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(Created page with "<wikitex>; The exercises on this page come directly from {{citeD|Lück2001|pp. 70-17}} and we use the notation found there. Let $f \colon Y \to X$ be a map of CW-complexes an...") |
Revision as of 14:31, 24 March 2012
The exercises on this page come directly from [Lück2001, pp. 70-17] and we use the notation found there.
Let be a map of CW-complexes and let be represented by the following commutative diagram
Let be the push out of : i.e. is the space obtained by attaching an -cell to along . There is an induced map
Exercise 0.1. With notation above assume that is -connected with . Prove that the kernel of the surjection is the -module generated by .
Exercise 0.2 [Lück2001, Lemma 3.55]. Let be a -connected map of CW-complexes for some where is connected. Consider a lift on the universal coverings. Show the following:
- The homology groups of are finitely generated modules.
- One can make -connected by attaching finitely many cells.