A -algebra is an algebraic model for the homotopy groups of a pointed topological space, , together with the action of the primary homotopy operations on them, in the same sense that algebras over the Steenrod algebra are models for the cohomology of a space.
Constructions of this type exist in many pointed model categories. It suffices to have a collection of spherical objects.
The category of homotopy operations has
is a pointed category and has finite coproducts (given by the finite wedges), but not products.
This category is a finite product theory, in the sense of algebraic theories whose models are:
Let denote the category of pointed sets.
A morphism of -algebras is a natural transformation between the corresponding functors.
A -algebra satisfies .
The values of a -algebra are determined by the values , that it takes on the spheres, , .
A -algebra can be considered to be a graded group with abelian for , together with
for (the case where they are equal to 1 needs special mention, see below.)
which satisfy the identities that hold for the Whitehead products and composition operations on the higher homotopy groups os a pointed space, and
The Whitehead products include
, where is the result of the -action of on , ; similarly for a right action;
the commutators , for .
For a pointed space , and , define a -algebra by , the set of pointed homotopy classes of pointed maps from to .
This is a -algebra called the homotopy -algebra of .
Suppose is an abstract -algebra, the realisability problem for is to construct, if possible, a pointed space , such that . The space is called a realisation of .
Things can be complicated!
The homotopy type of is not be determined by (hence ‘a’ rather than ‘the“ realisation) , so that raises the additional problem of classifying the realisations.
Not all -algebras can be realised, in fact
Given a -algebra, , there is a sequence of higher homotopy operation?s depending only on maps between wedges of spheres, such that is realisable if and only if the operations vanish coherently.
For , a prime and , cannot be realised (and if , one uses ).
(There is a problem, discussed in Blanc’s 1995 paper, that the composition operations need not be homomorphisms, so tensoring with has to be interpreted carefully.)
A -algebra, , is said to be simply connected if .
In this case the universal identities for the primary homotopy operations can be described more easily (see Blanc 1993). These include the structural information that the Whitehead products make into a graded Lie ring (with a shift of indices).
The beginnings of a classification theory for -truncated -algebras can be found in Frankland’s thesis (link given below).
David Blanc has written a lot on these objects. An example is
The realisability problem is discussed in
and further in
There are more recent results on the realisability problem in Martin Frankland’s thesis.