Reducible Poincaré Complexes (Ex)
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− | Let $X$ be a finite Poinaré complex of formal dimension $n$ with Spivak Normal Fibration $\nu_X$. | + | Let $X$ be a finite Poinaré complex of formal dimension $n \geq 3$ with Spivak Normal Fibration $\nu_X$. A theorem of Wall, \cite{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 $\nu_X$, the Spivak normal fibration of $X$, 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. |
Latest revision as of 23:58, 27 August 2013
Exercise 0.1. Let be a finite Poinaré complex of formal dimension with Spivak Normal Fibration . A theorem of Wall, [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.
References
- [Wall1967a] C. T. C. Wall, Poincaré complexes. I, Ann. of Math. (2) 86 (1967), 213–245. MR0217791 (36 #880)