If f:E->B is a fiber bundle with B a paracompact topological space, then f satisfies the homotopy lifting property with respect to all topological spaces. In other words, if g:[0,1]×X->B is a homotopy from g_0 to g_1, and if g_0^' is a lift of the map g_0 with respect to f, then g has a lift to a map g^' with respect to f. Therefore, if you have a homotopy of a map into B, and if the beginning of it has a lift, then that lift can be extended to a lift of the homotopy itself.

A fibration is a map between topological spaces f:E->B such that it satisfies the homotopy lifting property.

See also

Fiber Bundle, Fiber Space

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Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 323, 1976.

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Cite this as:

Weisstein, Eric W. "Fibration." From MathWorld--A Wolfram Web Resource.

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