TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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$Deg : π_{1} (R / Z, [0]) \to (Z, +)$ is an isomorphism

Statement

$Deg : π_{1} (R / Z, [0]) \to (Z, +)$ is an isomorphism. Its inverse is

$n ⟼ [ω_{n}] .$

Facts & Assumptions

Given: The degree function on based loop classes of the quotient circle.

[L1]

$Deg : π_{1} (S^{1}, [0]) \to (Z, +)$ is a group homomorphism ( $Deg : π_{1} (S^{1}, [0]) \to (Z, +)$ is a group homomorphism).

[L2]

Two based circle loops are path-homotopic if and only if they have equal degree (Two based circle loops are path-homotopic if and only if they have equal degree).

[L3]

$\deg (ω_{n}) = n$ for every integer $n$ ( $\deg (ω_{n}) = n$ for every integer $n$ ).

[L4]

An isomorphism $f : G \to H$ is a bijective group homomorphism (Group isomorphisms, automorphisms and the set $Aut (G)$ ).

[L5]

The integers form a commutative ring, and hence $(Z, +)$ is a group (The integers form a commutative ring).

[L6]

$π_{1} (X, x_{0})$ consists of endpoint-fixed path-homotopy classes $[α]$ of based loops (Based loops and the fundamental group).

Proof

technique · direct

1.1

By [L1], $Deg$ is a group homomorphism into the additive group of integers supplied by [L5].

L1L5

1.2

If $Deg ([α]) = Deg ([β])$ , then $\deg (α) = \deg (β)$ , so [L2] makes $α$ and $β$ path-homotopic. Their classes are equal by [L6], and $Deg$ is injective.

L2L6

1.3

Let $n \in Z$ . By [L3], $Deg ([ω_{n}]) = \deg (ω_{n}) = n$ , so every integer is attained and $Deg$ is surjective. This includes $n = 0$ , $n = 1$ , and negative integers.

2.1

Steps 1.1, 1.2, and 1.3 show that $Deg$ is a bijective group homomorphism, hence an isomorphism by [L4]. Step 1.3 also shows that $n \mapsto [ω_{n}]$ is its inverse, since injectivity makes this preimage unique.

step 1.1step 1.2step 1.3L3L4∎

Depends on

Used by

The trigonometric loops give π₁({(x,y):x²+y²=1},(1,0))≅ℤ Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 64 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, Theorem 1.7 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Ch. 1, Section 5 (standard reference, not scraped)

Deg⁡:π1(R/Z,[0])→(Z,+)Deg:π1​(R/Z,[0])→(Z,+) is an isomorphism