Freudenthal suspension theorem
Encyclopedia
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 group
s and ultimately to stable homotopy theory
. It explains the behavior of simultaneously taking suspensions and increasing the index of the homotopy groups of the space in question. It was proved in 1937 by Hans Freudenthal
.
pointed space
(a pointed CW-complex or pointed simplicial set
). The map
induces a map
on homotopy groups, where Ω denotes the loop functor and ∧ denotes the smash product
. 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.
A basic result on loop spaces gives the relation
so the theorem could otherwise be stated in terms of the map
with the small caveat that in this case one must be careful with the indexing.
by the Freudenthal theorem. These groups represent the kth stable homotopy group of spheres
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s and ultimately to stable homotopy theory
Stable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
. It explains the behavior of simultaneously taking suspensions and increasing the index of the homotopy groups of the space in question. It was proved in 1937 by Hans Freudenthal
Hans Freudenthal
Hans Freudenthal was a Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education....
.
Statement of the theorem
Let X be an n-connectedN-connected
In the mathematical branch of algebraic topology, specifically homotopy theory, n-connectedness is a way to say that a space vanishes or that a map is an isomorphism "up to dimension n, in homotopy".-n-connected space:...
pointed space
Pointed space
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
(a pointed CW-complex or pointed simplicial set
Simplicial set
In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space...
). The map
- X → Ω(X ∧ S1)
induces a map
- πk(X) → πk(Ω(X ∧ S1))
on homotopy groups, where Ω denotes the loop functor and ∧ denotes the smash product
Smash product
In mathematics, the smash product of two pointed spaces X and Y is the quotient of the product space X × Y under the identifications ∼ for all x ∈ X and y ∈ Y. The smash product is usually denoted X ∧ Y...
. The suspension theorem then states that the induced map on homotopy groups is an isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
if k ≤ 2n and an epimorphism
Epimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
if k = 2n + 1.
A basic result on loop spaces gives the relation
- πk(Ω(X ∧ S1)) ≅ πk+1(X ∧ S1)
so the theorem could otherwise be stated in terms of the map
- πk(X) → πk+1(X ∧ S1),
with the small caveat that in this case one must be careful with the indexing.
Corollary 1
Let Sn denote the n-sphere and note that it is (n − 1)-connected so that the groups πn+k(Sn) stabilize for- n ≥ k + 2
by the Freudenthal theorem. These groups represent the kth stable homotopy group of spheres
Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting...
.