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It is known in elementary algebraic topology that a covering map induces an isomorphism of higher homotopy groups.

Is there any relation of the higher homotopy groups of the total space of a ramified covering in the sense of topology (a map which is a covering map outside of a nowhwere dense set) and its base space? I am interested in base spaces which are manifolds with mild singularities.

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    $\begingroup$ Probably there is nothing at all one can say. Think of the torus, a ramified cover of the 2-sphere. $\endgroup$
    – Dan Petersen
    Commented Jan 10, 2018 at 16:46

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