Point-deletion inclusion induces isomorphism on fundamental groups for manifold of dimension at least two
Suppose is a connected manifold of dimension at least two. Suppose is a point in . Note that the manifold is still a connected manifold (because has dimension at least two). Consider the inclusion map:
This induces a homomorphism between the Fundamental group (?)s (note that it is not necessary to specify basepoints because both manifolds are path-connected):
This induced map is an isomorphism. In particular, both the fundamental groups are isomorphic groups.