Loop (topology)

A loop in a topological space is a continuous map $\gamma :[0,1]\to X$ with $\gamma (0)=\gamma (1)$. Equivalently it is a map from the circle. A based loop also specifies the common endpoint as a chosen base point.

A based loop in $(X,x_0)$ is a path $\alpha :[0,1]\to X$ whose two endpoints are $x_0$.

A boundary loop travels around a boundary circle of a topological surface. To regard it as an element of the fundamental group at a base point away from the boundary, join that base point to the boundary by a path, go around the boundary, and follow the reverse path. Changing that joining path conjugates the element; reversing the boundary orientation inverts it.

A based homotopy between based loops keeps both endpoints fixed at the base point throughout the deformation.

The reverse of a path $\alpha$ is $\bar{\alpha }(t)=\alpha (1-t)$. Concatenating a path with its reverse is homotopic relative to endpoints to the constant path.