CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-17

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Degree defines a function $Deg : π_{1} (S^{1}, [0]) \to Z$

Statement

Degree defines a function $Deg : π_{1} (S^{1}, [0]) \to Z$ by

$Deg ([α]) = \deg (α) .$

Facts & Assumptions

Given: The based circle loops and path-homotopy classes defining $π_{1} (S^{1}, [0])$ .

[L1]

Path-homotopic based circle loops have the same degree (Path-homotopic based circle loops have the same degree).

[L2]

The fundamental group set of $X$ at $x_{0}$ is the set of endpoint-fixed path-homotopy classes of based loops at $x_{0}$ (Based loops and the fundamental group).

Proof

technique · direct

1.1

If $[α] = [β]$ in the set of [L2], then $α$ and $β$ are path-homotopic relative to their endpoints, so [L1] gives $\deg (α) = \deg (β)$ .

L1L2

2.1

Therefore the displayed rule is independent of the representative and defines one function on $π_{1} (S^{1}, [0])$ . No representative-selection function is used.

step 1.1L2∎

Depends on

Used by

Deg:π₁(S¹,[0])→(ℤ,+) is a group homomorphism Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 27 results over 11 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

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

Degree defines a function Deg⁡:π1(S1,[0])→ZDeg:π1​(S1,[0])→Z