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Let $Y$ be a CW-complex and $X$ its universal cover. Could you give me a proof (or a referece) for the following fact:

$X$ is contractible $\Leftrightarrow$ $H_i(X)=0$ $\forall i\geq2$ $\Leftrightarrow$ $\pi_iY=0$ $\forall i\geq2$.

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    I can't understand why $\pi_iY=0$ $\forall i\geq2$ implies$X$contractible. From Whitehead it's clear if I have $\forall i\geq 1$. Other thing that I don't understand is why $H_i(X)=0$ $\forall i\geq2$ implies X contractible (or why it implies the condition on the homotopy groups).2011-09-07

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