Homotopy of paths
A homotopy of paths in a space is a family , , such that
(i) The endpoints and are independent of .
(ii) The associated map defined by is continuous.
When two paths and are connected in this way by a homotopy , they are said to be homotopic. The notation for this is .
Example: Linear Homotopies
Any two paths and in having the same endpoints and are homotopic via the homotopy
A space is called simply-connected if it is path-connected and has trivial fundamental group.
A space is simply-connected iff there is a unique homotopy class of paths connecting any two parts in .
Path-connectedness is the existence of paths connecting every pair of points, so we need to be concerned only with the uniqueness of connecting paths.
() Suppose . If and are two paths from to , then since the loops and are each homotopic to constant loops, due to .
() Conversely, if there is only one homotopy class of paths connecting a basepoint to itself, then all loops at are homotopic to the constant loop and .
is isomorphic to if and are path-connected.
A basic property of the product topology is that a map is continuous iff the maps and defined by are both continuous.
Hence a loop in based at is equivalent to a pair of loops in and in based at and respectively.
Similarly, a homotopy of a loop in is equivalent to a pair of homotopies and of the corresponding loops in and .
Thus we obtain a bijection , . This is clearly a group homomorphism, and hence an isomorphism.
Note: The condition that and are path-connected implies that , .