DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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The standard circle loops $ω_{n} (t) = [n t]$ for $n \in Z$

Definition

Let $I = [0, 1]$ . For every integer $n$ , define ${\tilde{ω}}_{n} (t) = n t$ and $ω_{n} = p \circ {\tilde{ω}}_{n}$ .

The function $t \mapsto n t$ is continuous by Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, and $p$ is continuous because it is the quotient projection of The circle as $S^{1} = R / Z$ with basepoint $[0]$ . Hence $ω_{n}$ is continuous by Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous. Moreover,

$ω_{n} (0) = [0], ω_{n} (1) = [n] = [0] .$

Thus $ω_{n}$ is a based loop at $[0]$ in the sense of Based loops and the fundamental group. This definition includes $n = 0$ , when the loop is constant, and all negative integers.

Depends on

Used by

A based circle loop is nullhomotopic exactly when its degree is zero Corollary
A complete manifold with zero global injectivity radius Counterexample
Based circle loops with the same endpoints need not be path-homotopic Counterexample
A loop that traverses the circle once and then pauses is homotopic to the standard loop Example
The geometric loops t↦(cos 2π nt,sin 2π nt) have degree n Example
FALSE: every continuous self-map of the circle is nullhomotopic False statement
deg(ωₙ)=n for every integer n Proposition

Dependency tree · two levels

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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)

The standard circle loops ωn(t)=[nt]ωn​(t)=[nt] for n∈Zn∈Z