Reducible Poincaré Complexes (Ex)
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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 | 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 | ||
$$ X \simeq X^\bullet \cup_\phi e^n$$ | $$ X \simeq X^\bullet \cup_\phi e^n$$ | ||
− | 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. | + | 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 $\nu_X$, the Spivak normal fibration of $X$, is trivial. |
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> |
Revision as of 23:34, 27 August 2013
Exercise 0.1. Let be a finite Poinaré complex of formal dimension with Spivak Normal Fibration . A Theorem of [Wall1967a, Theorem 2.4] states that may be written
where has dimension less than . Show that for some , the top cell of splits off, i.e. , if and only if , the Spivak normal fibration of , is trivial.