K-connectedness and the homotopy-lifting property

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Bacle
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Hi:
Is there a relation between k-connectedness ( meaning 1st, 2nd,..,k-th fundamental
groups are trivial.) and having the homotopy-lifting property.? . This may be
vaguely-related to being able to extend global sections from the j-th skeleton, to
the (j+1)-st skeleton (j<=k, obviously), but I am not sure.

How about k-connectedness for a pair (A,X) ( A a subspace of X).

Thanks.
 

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Could you be more specific? Are we fixing a triple, or are we asking about whether any surjective map is a fibration?