Homotopy theory MOC

Covering

Let 𝑋 and ˜𝑋 be topological spaces. A covering¹ of 𝑋 with ˜𝑋 is a surjective map 𝑝 :˜𝑋 ↠𝑋 such that every 𝑥 ∈𝑋 has a a neighbourhood 𝑈 such that 𝑝−1𝑈 =∐𝑖∈𝐼𝑉𝑖 and 𝑝 ↾𝑉𝑖 :𝑉𝑖 →𝑈 is a homeomorphism for each 𝑖 ∈𝐼. topology A neighbourhood with these properties is called an evenly covered neighbourhood, each 𝑉𝑖 is called a sheet, and the discrete set 𝑝−1(𝑥) is called the fibre of 𝑥.²

Further terminology

• A covering (˜𝑋,𝑝) is called a (path-)connected covering iff ˜𝑋 (and therefore 𝑋) is (path-)connected.
• For a connected covering, the sheet number |𝐼| is the same everywhere. See Sheet number of a connected covering
• The coverings of a space 𝑋 form a Category of coverings 𝖢𝗈𝗏𝑋
• The Characteristic subgroup of a covering is its image on the fundamental group
• A Regular covering has a basepoint-invariant characteristic subgroup
• A Deck transformation 𝛾 is an automorphism of the covering space such that 𝑝𝛾 =𝑝.

Properties

• The identity map is a trivial covering
• A covering is a Local homeomorphism
• The Orbit space of a properly discontinuous group action comes with a covering
• Lift of a map to a covering space
• A covering is injective on the fundamental group
• Main theorem of coverings

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Footnotes

1. German Überlagerung ↩

2. Eine Umgebung mit diesen Eigenschaften heißt eine gleichmäßig überlagerte Umgebung, jedes 𝑉𝑖 heißt ein Blatt, und die diskrete Menge 𝜋−1(𝑥) heißt die Faser von 𝑥. ↩