TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-17

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

Statement

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

Facts & Assumptions

Given: Loop classes $[α], [β] \in π_{1} (S^{1}, [0])$ .

[L1]

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

[L2]

Degree sends concatenation to addition, reversal to negation, and the constant loop to zero (Degree sends concatenation to addition, reversal to negation, and the constant loop to zero).

[L3]

A group homomorphism $f : G \to G^{'}$ is a function satisfying $f (x y) = f (x) f (y)$ for all $x, y \in G$ (Monoid homomorphism and group homomorphism).

[L4]

$(Z, +, \cdot, 0, 1)$ is a commutative ring with multiplicative identity, so $(Z, +)$ is a group (The integers form a commutative ring).

[L5]

The product of fundamental-group classes is $[α] [β] = [α * β]$ (Based loops and the fundamental group).

Proof

technique · direct

1.1L1L2L5

By [L5], [L1], and the concatenation law in [L2], $Deg ([α] [β]) = Deg ([α * β]) = \deg (α * β) = \deg (α) + \deg (β) = Deg ([α]) + Deg ([β])$ .

2.1step 1.1L2L3L4∎

The target is the additive group of the integers by [L4], and step 1.1 is exactly the product-preservation condition of [L3]. Hence $Deg$ is a group homomorphism. Its identity and inverse laws also agree with the zero and negation formulas of [L2].

Depends on

Used by

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

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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

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