Stable homotopy theory
Encyclopedia
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 Freudenthal suspension theorem
, which states that for a given CW-complex X the (n+i)th homotopy group
of its ith iterated suspension, πn+i (ΣiX), becomes stable (i.e., isomorphic after further iteration) for large but finite values of i. For instance,
In the two examples above all the maps between homotopy groups are applications of the suspension functor. Thus the first example is a restatement of the Hurewicz theorem
, that πn(Sn) ≅ ℤSn >. In the second example the Hopf map, η, is mapped to Ση which generates π4(S3) ≅ ℤ/2.
One of the most important problems in stable homotopy theory is the computation of stable homotopy groups of spheres. According to Freudenthal's theorem, in the stable range the homotopy groups of spheres depend not on the specific dimensions of the spheres in the domain and target, but on the difference in those dimensions. With this in mind the kth stable stem is
:= lim πn+k (Sn). This is an abelian group for all k. It is a theorem of Serre that these groups are finite if k > 0. In fact, composition makes 
into a graded ring. Nishida's theorem states that all elements of positive grading in this ring are nilpotent. Thus the only prime ideals are the primes in 
≅ ℤ. So the structure of 
is quite complicated.
In the modern treatment of stable homotopy, spaces are typically replaced by spectra
. Following this line of thought, an entire stable homotopy category can be created. This category has many nice properties not found in the (unstable) homotopy category of spaces, following from the fact that the suspension functor becomes invertible. For example, the notion of cofibration sequence and fibration sequence are equivalent.
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...
, stable homotopy theory is that part of homotopy theory (and thus algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the 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...
, which states that for a given CW-complex X the (n+i)th 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...
of its ith iterated suspension, πn+i (ΣiX), becomes stable (i.e., isomorphic after further iteration) for large but finite values of i. For instance,
- ℤ
- ℤ<η> = π3(S2) → π4(S3) ≅ π5(S4) ≅ ...
In the two examples above all the maps between homotopy groups are applications of the suspension functor. Thus the first example is a restatement of the Hurewicz theorem
Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism...
, that πn(Sn) ≅ ℤ
One of the most important problems in stable homotopy theory is the computation of stable homotopy groups of spheres. According to Freudenthal's theorem, in the stable range the homotopy groups of spheres depend not on the specific dimensions of the spheres in the domain and target, but on the difference in those dimensions. With this in mind the kth stable stem is
:= lim πn+k (Sn). This is an abelian group for all k. It is a theorem of Serre that these groups are finite if k > 0. In fact, composition makes into a graded ring. Nishida's theorem states that all elements of positive grading in this ring are nilpotent. Thus the only prime ideals are the primes in ≅ ℤ. So the structure of is quite complicated.In the modern treatment of stable homotopy, spaces are typically replaced by spectra
Spectrum (homotopy theory)
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, any of which gives a context for the same stable homotopy theory....
. Following this line of thought, an entire stable homotopy category can be created. This category has many nice properties not found in the (unstable) homotopy category of spaces, following from the fact that the suspension functor becomes invertible. For example, the notion of cofibration sequence and fibration sequence are equivalent.