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Homotopy relative to a fixed subspace, and path homotopy relative to endpoints, are equivalence relations

Statement

For fixed spaces $X, Y$ and a fixed subspace $A \subseteq X$ , the relation $≃_{A}$ is an equivalence relation on the set of continuous maps $X \to Y$ that have a prescribed restriction to $A$ . In particular ordinary homotopy is an equivalence relation on the continuous maps $X \to Y$ .

For fixed endpoints $y_{0}, y_{1} \in Y$ , path homotopy relative to the endpoints is an equivalence relation on the set of paths from $y_{0}$ to $y_{1}$ .

Facts & Assumptions

Given: Spaces $X, Y$ , a subspace $A \subseteq X$ , and the relations defined in Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints.

[L1]

Homotopy rel $A$ is reflexive and symmetric (Homotopy relative to a subspace is reflexive and symmetric).

[L2]

Homotopy rel $A$ is transitive by the two-piece reparametrisation construction (Two homotopies relative to the same subspace concatenate after piecewise-linear reparametrisation).

[L3]

A relation is an equivalence relation exactly when it is reflexive, symmetric and transitive (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ).

Proof

technique · direct

1.1

By [L1] and [L2], $≃_{A}$ is reflexive, symmetric and transitive, so it is an equivalence relation by [L3]. Taking $A = \emptyset$ gives ordinary homotopy.

L1L2L3

2.1

Paths from $y_{0}$ to $y_{1}$ are continuous maps $I \to Y$ with one prescribed restriction to the subspace ${0, 1}$ , and their path homotopies are exactly homotopies rel ${0, 1}$ . Hence step 1.1 applies to them.

step 1.1given∎

Depends on

Used by

Dependency tree · two levels

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Sources

A. Hatcher, Algebraic Topology, Section 0 (standard reference, not scraped)
Homotopy lecture notes (University of Padua) (standard reference, not scraped)