Cell attachments for surgery (Ex)
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{{beginthm|Exercise|{{citeD|Lück2001|Lemma 3.55}}}} | {{beginthm|Exercise|{{citeD|Lück2001|Lemma 3.55}}}} | ||
Let $f \colon Y \to X$ be a $(k-1)$-connected map of CW-complexes for some $k \geq 2$ where $X$ is connected. Consider a lift $\tilde f \colon \tilde Y \to \tilde X$ on the universal coverings. Show the following: | Let $f \colon Y \to X$ be a $(k-1)$-connected map of CW-complexes for some $k \geq 2$ where $X$ is connected. Consider a lift $\tilde f \colon \tilde Y \to \tilde X$ on the universal coverings. Show the following: | ||
− | # The natural map $\pi_k(\tilde f) \ | + | # The natural map $\pi_k(\tilde f) \to \pi_k(f)$ is $\pi_1$-equivariant and bijective. |
# The homology groups of $H_k(\tilde f)$ are finitely generated $\Zz\pi_1(Y)$ modules. | # The homology groups of $H_k(\tilde f)$ are finitely generated $\Zz\pi_1(Y)$ modules. | ||
# One can make $f$ $k$-connected by attaching finitely many cells. | # One can make $f$ $k$-connected by attaching finitely many cells. |
Revision as of 14:50, 24 March 2012
The exercises on this page come directly from [Lück2001, pp. 70-71] 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 natural map is -equivariant and bijective.
- The homology groups of are finitely generated modules.
- One can make -connected by attaching finitely many cells.
The exercises on this page were sent by Nicolas Ginoux and Carolina Neira Jim\'enez.