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
Has Statement Form
In Branch
Source Wolfram MathWorld: Freudenthal Suspension
Theorem
Proved By
Source Freudenthal suspension theorem, Wikipedia
Sources
1. Freudenthal suspension theorem, Wikipedia
Statement of the theorem
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.
Proved By: Hans Freudenthal, Lead
It was proved in 1937 by Hans Freudenthal.
In Group: Algebraic Topology, Lead paragraph
In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory.
View the SourceWolfram MathWorld: Freudenthal Suspension
Theorem
Wolfram MathWorldIn Branch: Algebraic Topology, Subject classificationsQuote, In Branch: Algebraic Topology, Subject classifications
Topology Algebraic Topology
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