Path space

The space of the fibre space , which is called the path fibre space. Here is a path-connected space with a distinguished point , is the set of paths (cf. Path) in starting from and is the mapping associating to each path its end-point. Moreover, is considered to have the compact-open topology. The fibre of this fibre space (which is a Serre fibration) is the loop space — the set of all loops (cf. Loop) in at . A path space can be contracted within itself to a point, so the homotopy groups , and the homotopy sequence of the path fibre space degenerates into the so-called Hurewicz isomorphisms:

References

[a1]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 75ff, 99ff