Cell attachments for surgery (Ex)

Latest revision as of 09:14, 1 April 2012

The exercises on this page come directly from [Lück2001, pp. 70-71] and we use the notation found there.

Let $<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 and let $\omega \in \pi_{k+1}(f)$ be represented by the following commutative diagram $$\xymatrix{S^k \ar[r]^q \ar[d]_j & Y \ar[d]^f \ D^{k+1} \ar[r]^{Q} & X}.$$ Let $Y'$ be the push out of $q$: i.e. $Y'=Y \cup_f D^{l+1}$ is the space obtained by attaching an $(l+1)$-cell to $Y$ along $f$. There is an induced map $$ f' \colon Y' \to X.$$ {{beginthm|Exercise}} With notation above assume that $f$ is $k$-connected with $k \geq 3$. Prove that the kernel of the surjection $\pi_{k+1}(f) \to \pi_{k+1}(f')$ is the $\Zz \pi_1(Y)$-module generated by $\omega$. {{endthm}} {{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: # 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. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]]f \colon Y \to X$ be a map of CW-complexes and let $\omega \in \pi_{k+1}(f)$ be represented by the following commutative diagram

Let $Y'$ be the push out of $q$: i.e. $Y'=Y \cup_f D^{l+1}$ is the space obtained by attaching an $(l+1)$-cell to $Y$ along $f$. There is an induced map

$$\displaystyle f' \colon Y' \to X.$$

The exercises on this page were sent by Nicolas Ginoux and Carolina Neira-Jiménez.