Homotopy Group π_k(X, x₀). Hopf Fibering. Serre Theorem on Exact Sequence of Homotopy Groups of a Fibering

Abstract

Surely, not in all Serre fiberings p: X → Y a connection exists, even though we have lifting of curves from the base Y to curves in the bundle space of given starting point x ₀: in order to do this, it suffices to take as the polynomial W the one point set {x ₀}. But let us stress the connection in the bundle p: X → Y is something more that the possibility of lifting (arbitrary) curves; the operation (of lifting) Γ must be continuous as well!