Covering

Characteristic subgroup of a covering

Let 𝑝 :(˜𝑋,˜𝑥0) →(𝑋,𝑥0) be a covering. The characteristic subgroup¹ is a subgroup of the fundamental group of (𝑋,𝑥0) given by homotopy

im⁡𝜋1𝑝=𝜋1𝑝(𝜋1(˜𝑋,˜𝑥0))

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

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tidy | en | SemBr

Footnotes

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