Point-deletion inclusion induces isomorphism on fundamental groups for manifold of dimension at least two

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Statement

Suppose M is a connected manifold of dimension at least two. Suppose p is a point in M. Note that the manifold M \setminus \{ p \} is still a connected manifold (because M has dimension at least two). Consider the inclusion map:

i: M \setminus \{ p \} \to M

This induces a homomorphism between the Fundamental group (?)s (note that it is not necessary to specify basepoints because both manifolds are path-connected):

\pi_1(i): \pi_1(M \setminus \{ p \}) \to \pi_1(M)

This induced map is an isomorphism. In particular, both the fundamental groups are isomorphic groups.

Facts used

  1. Seifert-van Kampen theorem