A covering is injective on the fundamental group

Let be a covering. Then the induced homomorphism is a group monomorphism. #m/thm/homotopy

Proof

Let . Then . But and are the lifts of and respectively, so by Third lemma Lifts of homotopies of paths , Therefore and thus is a Group monomorphism.

Therefore the characteristic subgroup of the covering is isomorphic to .

#state/tidy | #lang/en | #SemBr