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
Simply-connected
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
,
.