The loop space is the space of based loops in . It is the homotopy fiber of the inclusion of the base point into .
The evaluation map from the based path space gives a fiber sequenceSince is contractible, it relates invariants of to those of its loop space.
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In mathematics, particularly in algebraic topology, the term "loop space" refers to a certain kind of space that captures the idea of loops in a given topological space. Specifically, the loop space of a pointed topological space \( (X, x_0) \) is the space of all loops based at the point \( x_0 \).