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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Based circle loops with the same endpoints need not be path-homotopic

Statement refuted

The claim that any two based circle loops with the same initial and terminal points are path-homotopic relative to those endpoints is false.

Facts & Assumptions

Given: The standard loops $ω_{0}$ and $ω_{1}$ in $R / Z$ .

[L1]

A based loop at $x_{0}$ is a path whose values at both endpoints are $x_{0}$ (Based loops and the fundamental group).

[L2]

A based circle loop is nullhomotopic exactly when its degree is zero (A based circle loop is nullhomotopic exactly when its degree is zero).

[L3]

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

[L4]

For every integer $n$ , $ω_{n} (t) = [n t]$ , and $ω_{0}$ is constant (The standard circle loops $ω_{n} (t) = [n t]$ for $n \in Z$ ).

Counterexample

technique · direct

1.1

By [L4], $ω_{0} (0) = ω_{0} (1) = [0]$ and $ω_{1} (0) = [0] = [1] = ω_{1} (1)$ , so both satisfy the two endpoint equalities in [L1]. Their degrees are $0$ and $1$ by [L3].

L1L3L4

2.1

If $ω_{0}$ and $ω_{1}$ were path-homotopic relative to the endpoints, then $ω_{1}$ would be path-homotopic to the constant loop and hence nullhomotopic. But [L2] and [L3] rule this out because $\deg (ω_{1}) = 1 \neq 0$ . Thus equal endpoints do not imply path homotopy.

step 1.1L2L3algebra∎

Depends on

Used by

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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)