DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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Connectivity of a CW pair

Definition

Let $(X, A)$ be a CW pair in the sense of Skeleta, CW subcomplexes, and relative CW complexes, and let $k \geq 0$ be an integer. The pair is $k$ -connected when every path component of $X$ meets $A$ and, for every $a \in A$ and $1 \leq i \leq k$ , the pointed set or group $π_{i} (X, A, a)$ has one element. The positive relative objects are those of Relative homotopy classes and groups, and path components are those of Paths, path-connected spaces and path components. In degree one triviality concerns a pointed set. No relative $π_{0}$ is defined.

Equivalently, for every $0 \leq i \leq k$ , every continuous map $(D^{i}, S^{i - 1}) \to (X, A)$ is homotopic into $A$ while its whole boundary is fixed. For $i = 0$ , use $D^{0} = {*}$ and empty boundary; the clause asks for a path from its image point into $A$ .

No basepoint is chosen per component. In particular $0$ -connectedness only requires that every component meet $A$ , and does not assert that $X$ or $A$ has one component. If $A$ is empty, this definition holds precisely when $X$ is empty. If $X = A$ , it holds for every $k$ .

Facts & Assumptions

[F1]

Relative homotopy classes and groups supplies the all-basepoint positive relative sets and their distinguished constant representatives.

[F2]

Paths, path-connected spaces and path components defines path components by the path equivalence relation. Skeleta, CW subcomplexes, and relative CW complexes supplies the subspace topology on the pair.

[F3]

Relative cubical disk model and compression identifies disk representatives with relative cubical classes, and proves that relative nullity is equivalent to compression into the subspace fixing the entire boundary.

Verification

Given: The pair and integer in the definition.

1.1F1F2F3given

Suppose the component and relative-triviality conditions hold. A map of a zero-disk is a point of $X$ , whose component meets $A$ , so a path to $A$ supplies its compression. For $i \geq 1$ , mark $b = (1, 0, \dots, 0) \in S^{i - 1}$ and set $a = u (b)$ for the specified disk map $u$ . Its relative class based at this actual $a$ is trivial by hypothesis. By [F3] it has a homotopy into $A$ fixing the full boundary, including both endpoints when $i = 1$ . No selected family of basepoints or paths is involved.

1.2F1F2F3given

Conversely suppose all the stated disk-compression conditions hold. The zero-disk condition supplies a path to $A$ for any point of $X$ , so every component meets $A$ . For each $a \in A$ , every positive relative class in degrees $i \leq k$ has a disk representative with marked boundary value $a$ by [F3]. Its stipulated boundary-fixed compression makes the class trivial by the converse in [F3]. Thus all the defining conditions hold.

2.1F1F2step 1.1step 1.2∎

If $A$ is empty and $X$ nonempty, its point disks fail the required path condition; if both are empty there are no such maps or basepoints. When $X = A$ , the constant homotopy of any disk map already ends in $A$ , so the compression condition holds in all degrees. For $k = 0$ there are no positive-degree conditions; for $k = 1$ the positive condition is exactly the pointed degree-one one. Steps 1.1–1.2 prove both formulations equivalent, including their endpoints, without any choice principle.

Depends on

Dependency tree · 2 levels

Sources

Hatcher, Algebraic Topology, printed p346, equivalent definitions of relative connectivity