Characteristic subgroup of a covering

Characteristic conjugacy class of a path-connected covering

Let be a path-connected covering with characteristic subgroup . Then a subgroup is conjugate to iff it is the characteristic subgroup of with respect to a different basepoint . homotopy

Proof

For the reverse direction, let and be the same path-connected covering considered with different basepoints, with characteristic subgroups and respectively Let be a path from to . We can define an isomorphism

Then

hence the characteristic groups are conjugate.

For the forward direction, let and for some closed loop . Then has a unique lift with . Then is the characteristic group of with basepoint .

Hence a covering without choice of basepoint corresponds to a conjugation class of subgroups of .

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