Related Concepts
The fundamental group measures the 1-dimensional hole structure of a space. For studying "higher-dimensional holes", the homotopy groups are used. The elements of the n-th homotopy group of X are homotopy classes of (basepoint-preserving) maps from Sn to X.
The set of loops at a particular base point can be studied without regarding homotopic loops as equivalent. This larger object is the loop space.
For topological groups, a different group multiplication may be assigned to the set of loops in the space, with pointwise multiplication rather than concatenation. The resulting group is the loop group.
Read more about this topic: Fundamental Group
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