DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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The degree of a based circle loop

Definition

Let $α : I \to R / Z$ be a based loop at $[0]$ . Since $p : R \to R / Z$ is a covering map ( $p : R \to R / Z$ is a covering map with translated interval sheets), path lifting (Existence and uniqueness of path lifts through a covering map) gives a unique lift $\tilde{α} : I \to R$ with $\tilde{α} (0) = 0$ and $p \circ \tilde{α} = α$ ; this is a lift in the sense of Lifts of maps, paths, and homotopies through a covering map.

Because $α (1) = [0]$ , its terminal value satisfies $\tilde{α} (1) \in p^{- 1} ([0]) = Z$ by The circle as $S^{1} = R / Z$ with basepoint $[0]$ . Define $\deg (α) = \tilde{α} (1)$ .

This defines degree on based loops. Its independence from a representative of a path-homotopy class is proved separately before degree is used on $π_{1} (S^{1}, [0])$ .

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 71 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Allen Hatcher, Algebraic Topology, Ch. 1, Section 1.1 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Ch. 1, Section 5 (standard reference, not scraped)