Alphabeta Math
RemarkRemark: AI-adaptedProof: Not applicable‡ not proved here
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.

‡ Recorded, not proved here. This statement is included so the library can refer to it honestly, with a citation to the literature. It is not proved anywhere in this library: the track that would prove it has not been developed here yet.

The hairy-ball theorem for even-dimensional spheres

Remark

The triviality of TS1 does not extend to every sphere. For even-dimensional spheres S2m with m≥1, the hairy-ball theorem says that TS2m has no nowhere-zero global section, so those tangent bundles are not trivial. The case S0 is exceptional: its tangent bundle has rank 0 and is trivial. This page does not prove the hairy-ball theorem; the obstruction will be supplied later from degree and Euler-class machinery.

The contrast is already visible in this batch. The sphere's normal line bundle is trivial by the radial field, but the tangent bundle need not be.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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