CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.

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A group acting freely on a positive even sphere has at most two elements

Statement

If a group $Γ$ acts freely by homeomorphisms on $S^{2 m}$ with $m \geq 1$ , then $∣ Γ ∣ \leq 2$ . If it is nontrivial, it is isomorphic to $Z / 2 Z$ . The antipodal action realizes the nontrivial case.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

For $n \geq 1$ , any continuous fixed-point-free map $f : S^{n} \to S^{n}$ has degree $(- 1)^{n + 1}$ . Consequently, a self-map of any other degree has a fixed point. (A fixed point free sphere map has antipodal degree)

[F2]

For $n \geq 1$ and continuous sphere self-maps $f, g$ , homotopic maps have the same degree and $\deg (g \circ f) = \deg (g) \deg (f)$ . Every homotopy equivalence $S^{n} \to S^{n}$ has degree $1$ or $- 1$ . (Degree is homotopy invariant and multiplicative under composition)

[F3]

On $S^{n} \subset R^{n + 1}$ for $n \geq 1$ , the identity, a constant map, any single coordinate reflection, and the antipodal map have degrees $1$ , $0$ , $- 1$ , and $(- 1)^{n + 1}$ respectively. (Degree of identity constant reflection and antipodal sphere maps)

Proof

1.1F2

Let $a_{g}$ be the homeomorphism associated to $g \in Γ$ . F2 makes $d (g) = \deg (a_{g})$ a homomorphism $Γ \to {1, - 1}$ , since $a_{g h} = a_{g} \circ a_{h}$ and a homeomorphism is a homotopy equivalence.

2.1F1step 1.1algebra

For every $g \neq e$ , freeness says $a_{g}$ has no fixed point. F1, in positive even dimension, gives $d (g) = - 1$ . Thus $\ker d = {e}$ , so $d$ is injective and $∣ Γ ∣ \leq 2$ , including the trivial group. A nontrivial subgroup of ${1, - 1}$ is the entire two-element group.

3.1F3algebra∎

The antipodal involution squares to the identity and has no fixed point on a unit sphere: $x = - x$ would imply $x = 0$ . Together with the identity it therefore gives a free two-element action, of nonidentity degree $- 1$ by F3.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Hatcher, Algebraic Topology, Proposition 2.29, p.135 (standard reference, not scraped)