Reducible Poincaré Complexes (Ex)

Exercise 0.1. Let $X$$<wikitex>; {{beginthm|Exercise}} Let $X$ be a finite Poinaré complex of formal dimension $n$ with Spivak Normal Fibration $\nu_X$. A Theorem of {{citeD|Wall1967a|Theorem 2.4}} states that $X$ may be written $$ X \simeq X^\bullet \cup_\phi e^n$$ where $X^\bullet$ has dimension less than $n$. Show that for some $k$, the top cell of $X$ splits off, i.e. $\Sigma^k X \simeq S^{n+k} \vee \Sigma^k X^\bullet$, if and only if $\nu_X$ is trivial. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]X$ be a finite Poinaré complex of formal dimension $n$$n$ with Spivak Normal Fibration $\nu_X$$\nu_X$. A Theorem of [Wall1967a, Theorem 2.4] states that $X$$X$ may be written

$\displaystyle X \simeq X^\bullet \cup_\phi e^n$ $$\displaystyle X \simeq X^\bullet \cup_\phi e^n$$

where $X^\bullet$$X^\bullet$ has dimension less than $n$$n$. Show that for some $k$$k$, the top cell of $X$$X$ splits off, i.e. $\Sigma^k X \simeq S^{n+k} \vee \Sigma^k X^\bullet$$\Sigma^k X \simeq S^{n+k} \vee \Sigma^k X^\bullet$, if and only if $\nu_X$$\nu_X$ is trivial.