Characteristic subgroup of a covering

Let $𝑝 : (˜ 𝑋, ˜ 𝑥 0) \to (𝑋, 𝑥 0)$ be a covering. The characteristic subgroup¹ is a subgroup of the fundamental group of $(𝑋, 𝑥 0)$ given by #m/def/homotopy

i m 𝜋 1 𝑝 = 𝜋 1 𝑝 (𝜋 1 (˜ 𝑋, ˜ 𝑥 0))

Properties

A covering is injective on the fundamental group
The Sheet number of a connected covering is the index of the characteristic subgroup within the entire fundamental group
Two coverings are equivalent iff their characteristic subgroups match
Basepoint independent: Characteristic conjugacy class of a path-connected covering

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

This term does not appear to be remotely established in the literature. My guess is it is a coinage by Frank Loose. ↩

Characteristic subgroup of a covering

Properties

Footnotes