DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Suspension and sphere prespectra

Definition

For a well-pointed based CGWH space $X$ , its suspension prespectrum is

$(Σ^{\infty} X)_{n} = S^{n} \land X,$

where $S^{n} = (S^{1})^{\land n}$ and $S^{0}$ is the smash unit. Its structure map

$S^{1} \land (S^{n} \land X) ⟶ S^{n + 1} \land X$

is the canonical associativity homeomorphism, with the new $S^{1}$ coordinate placed first. Coherence makes the iterated construction unambiguous.

The sphere prespectrum is

$S = Σ^{\infty} S^{0}, S_{n} = S^{n} .$

These are prespectra. The definition does not assert that an arbitrary suspension prespectrum is an Omega-spectrum.

Depends on

Sequential prespectra, spectra, and adjoint structure maps

Used by

An unstable homotopy class need not yet be stable Counterexample
Stable stems of the sphere Definition
Suspension prespectra of spheres are shifts Example

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. P. May, A Concise Course in Algebraic Topology (standard reference, not scraped)
Allen Hatcher, Spectral Sequences in Algebraic Topology, Chapter 2 (standard reference, not scraped)