Alphabeta MathHow 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.
Stable stems of the sphere
Definition
For , the th stable stem of the sphere is
where is any index for which thereafter. The bonding map is the suspension homomorphism determined by the sphere-prespectrum structure homeomorphism . Tail independence makes the notation independent of .
Depends on
Used by
- Stabilizing a framed point suspends its collapse map Example
- Stabilizing a map between spheres Example
- The zero stable stem is the integers Example
- Freudenthal identifies the eventual suspension system for spheres Lemma
- Stabilizing a framed submanifold suspends its Pontryagin-Thom map Lemma
- The sphere prespectrum groups are the classical stable stems Proposition
- The stable Pontryagin-Thom theorem identifies framed bordism with stable stems Theorem
Dependency tree · two levels
6 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)