n1sphere

n1sphere is a notation used for the pointed version of the standard n-sphere. In this context, n1sphere, typically denoted S^n_1, refers to the n-sphere equipped with a distinguished basepoint, making it a pointed topological space. This structure is convenient in the study of homotopy theory, reduced homology, and related constructions where a distinguished point is required.

Definition and structure. Let S^n_1 denote the pair (S^n, *), where * is a fixed point in the n-sphere chosen as the basepoint. A map f between pointed spaces f: (X, x0) → (Y, y0) is basepoint-preserving if f(x0) = y0. Thus, a map from S^n_1 to another pointed space preserves the distinguished point.

Construction and models. S^n_1 can be given a standard CW structure with a single 0-cell at the basepoint and a single n-cell attached, yielding the familiar CW presentation of the n-sphere but explicitly as a pointed space. This viewpoint underpins many constructions in algebraic topology, where the basepoint governs how spaces are joined or mapped.

Key properties. As a pointed space, S^n_1 supports the usual homotopy-theoretic notions with basepoint-preserving maps. Its (reduced) homology agrees with that of the unpointed sphere in positive degrees: tilde H_k(S^n_1) ≅ tilde H_k(S^n). The basepoint enables operations such as the smash product X ∧ S^n_1 and the formation of the sphere spectrum via suspensions.

Variants and usage. The notation S^n_1 is not universal; many texts simply write S^n with a basepoint or use S^n_* or S^n_*. The concept remains central in pointed topology and stable homotopy theory, where pointed spheres serve as fundamental building blocks.

See also. S^n, pointed topological space, reduced homology, smash product, sphere spectrum.