Freudenthal suspension theorem
- Freudenthal suspension theorem
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. 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.
tatement of the theorem
Let "X" be an "n"-connected pointed space (a pointed CW-complex or pointed simplicial set). The map
:"X" → Ω("X" ∧ "S"1)
induces a map
:π"k"("X") → π"k"(Ω("X" ∧ "S"1))
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" ≤ 2"n" and an epimorphism if "k" = 2"n" + 1.
A basic result on loop spaces gives the relation
:π"k"(Ω("X" ∧ "S"1)) ≅ π"k"+1("X" ∧ "S"1)
so the theorem could otherwise be stated in terms of the map
:π"k"("X") → π"k"+1("X" ∧ "S"1),
with the small caveat that in this case one must be careful with the indexing.
Corollary 1
Let "S""n" denote the "n"-sphere and note that it is ("n" − 1)-connected so that the groups π"n"+"k"("S""n") stabilize for
:"n" ≥ "k" + 2
by the Freudenthal theorem. These groups represent the "k"th stable homotopy group of spheres.
Corollary 2
More generally, for fixed "k" ≥ "1", "k" ≤ 2"n" for sufficiently large "n", so that any "n"-connected space "X" will have corresponding stabilized homotopy groups. These groups are actually the homotopy groups of an object corresponding to "X" in the stable homotopy category.
References
*P.G. Goerss and J.F. Jardine, "Simplicial Homotopy Theory", Progress in Mathematics Vol. 174, Birkhäuser Basel-Boston-Berlin (1999).
Wikimedia Foundation.
2010.
Look at other dictionaries:
Freudenthal — is a surname and may refer to:* Axel Olof Freudenthal * Dave Freudenthal, U.S. politician * Hans Freudenthal, mathematician * Heinrich Freudenthal, founder of Deutsche Pentosin Werke GmbH * Jacob Freudenthal (1839 1907), German philosopher *… … Wikipedia
Suspension (topology) — In topology, the suspension SX of a topological space X is the quotient space::SX = (X imes I)/{(x 1,0)sim(x 2,0)mbox{ and }(x 1,1)sim(x 2,1) mbox{ for all } x 1,x 2 in X}of the product of X with the unit interval I = [0, 1] . Intuitively, we… … Wikipedia
Hans Freudenthal — (September 17, 1905 – October 13, 1990) was a Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education. Freudenthal was born in Luckenwalde … Wikipedia
Stable homotopy theory — In mathematics, stable homotopy theory is that part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the… … Wikipedia
List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… … Wikipedia
Groupes d'homotopie des sphères — Enroulement d une sphère à deux dimensions autour d une autre sphère En mathématiques, et plus spécifiquement en topologie algébrique, les groupes d homotopie des sphères sont des invariants qui décrivent, en termes algébriques, comment des… … Wikipédia en Français
List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… … Wikipedia
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… … Wikipedia
Triangulation (topology) — In mathematics, topology generalizes the notion of triangulation in a natural way as follows: A triangulation of a topological space X is a simplicial complex K , homeomorphic to X , together with a homeomorphism h : K o X . Triangulation is… … Wikipedia
