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A covering map $p:X\to Y$ between topological spaces can be viewed as a fiber bundle $\Sigma\to X\to Y$ with a discrete group $\Sigma=Gal(X/Y)$ as fiber. Such a fiber bundle leads to a long exact sequence of homotopy groups. In this case, if $Y$ is contractible then, of course, so is $X$.

I'm wondering what happens if the covering map $p$ is ramified. Is there any relation between the homotopy groups of $\pi_n(X),\pi_n(Y)$ and $\Sigma$? I'm guessing that perhaps the fixed set $X^\Sigma$ might be involved.

I'm particularly interested in two cases:

1) When $\Sigma=\Sigma_2$, the two-element group. This occurs often in toric topology.

2) What conditions can force $Y$ to be contractible (or just weakly null-homotopic).

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Is there a long exact sequence associated to a ramified covering?

A covering map $p:X\to Y$ between topological spaces can be viewed as a fiber bundle $\Sigma\to X\to Y$ with a discrete group $\Sigma=Gal(X/Y)$ as fiber. Such a fiber bundle leads to a long exact sequence of homotopy groups. In this case, if $Y$ is contractible then, of course, so is $X$.

I'm wondering what happens if the covering map $p$ is ramified. Is there any relation between the homotopy groups of $\pi_n(X),\pi_n(Y)$ and $\Sigma$? I'm guessing that perhaps the fixed set $X^\Sigma$ might be involved.

I'm particularly interested in two cases:

1) When $\Sigma=\Sigma_2$, the two-element group. This occurs often in toric topology.

2) What conditions can force $Y$ to be contractible.
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