The Freudenthal Suspension Theorem describes how the homotopy groups of a sufficiently connected space stabilize under repeated suspension, showing that the natural suspension map on homotopy groups becomes an isomorphism once the dimension considered is small enough relative to the space's connectivity. Named for Hans Freudenthal, it is the foundational result that makes stable homotopy theory possible, since it guarantees that homotopy groups eventually stop depending on how many times a space has been suspended.
Facts
StatementThe suspension theorem states that the induced map on homotopy groups is an isomorphism if k <= 2n and an epimorphism if k = 2n + 1. 1 Classification
Statement Form Statement Form Connections
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Freudenthal suspension theorem, Wikipedia
Statement section
The suspension theorem then states that the induced map on homotopy groups is an isomorphism if k <= 2n and an epimorphism if k = 2n + 1.
History section
The theorem was proved in 1937 by Hans Freudenthal, with his original paper published in 1938.
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