§ Simply connected spaces
- A space is simply connected iff fundamental group at all points is trivial.
- We usually don't want to talk about basepoint, so we assume that the space is path-connected. This means we can move the basepoint around, or not take about the basepoint.
- So, a path-connected space is simply connected iff the fundamental group is trivial.
§ Simply connected => all paths between two points are homotopic.
If are two points, then there is a single unique homotopy class of
points from to . Consider two paths from to called .
Since , we have that
. [ie, path is homotopic to trivial
path ]. compose by on the left: This becomes .
- This is pretty cool to be, because it shows that a simply connected space is forced to be path connected. Moreover, we can imagine a simply connected space as one we can "continuously crush into a single point".