nLab Pi-algebra

Idea

A $\Pi$-algebra is an algebraic model for the homotopy groups $\pi_*X$ of a pointed topological space, $X$, 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 $\Pi$ of homotopy operations

The category $\Pi$ of homotopy operations has

• as objects - pointed CW-complexes with the homotopy type of a finite wedge product of spheres of dimensions $\geq 1$;

• as morphisms - homotopy classes of (pointed) continuous functions between them.

Properties

• $\Pi$ is a pointed category and has finite coproducts (given by the finite wedges), but not products.

• There is a functor, smash product $i : \Pi\times \Pi \to \Pi$, which sends an object $(U,V)$ to $U\wedge V = (U\times V)/((U\times *)\vee(*\times V))$, which preserves coproducts in each variable.

This category $\Pi^{op}$ is a finite product theory, in the sense of algebraic theories whose models are:

$\Pi$-algebras

Let $Set_*$ denote the category of pointed sets.

Definition

A $\Pi$-algebra is a functor $A: \Pi^{op}\to Set_*$, which sends coproducts to products.

A morphism of $\Pi$-algebras is a natural transformation between the corresponding functors.

Properties

• A $\Pi$-algebra $A$ satisfies $A* = *$.

• The values of a $\Pi$-algebra $A$ are determined by the values $A_n = A(S^n)$, that it takes on the spheres, $S^n$, $n\geq 1$.

• A $\Pi$-algebra can be considered to be a graded group $\{A_n\}_{n=1}^\infty$ with $A_n$ abelian for $n\gt 1$, together with

  • ‘Whitehead product’ homomorphisms :

$$[-,-] : A_p\otimes A_q \to A_{p+q-1}$$

for $p,q \geq 1$ (the case where they are equal to 1 needs special mention, see below.)

• ‘composition operations’, $-\cdot \alpha : A_p\to A_r$ for $\alpha \in \pi_r(S^p)$, $1\lt p\lt r$,

which satisfy the identities that hold for the Whitehead products and composition operations on the higher homotopy groups of a pointed space, and

• a left action of $A_1$ on the $A_n$, $n\gt 1$, which commutes with these operations.

The Whitehead products include

• $[\xi,a] = {}^\xi a - a$, where ${}^\xi a$ is the result of the $A_1$-action of $\xi \in A_1$ on $a\in A_r$, $r\gt 1$; similarly for a right action;

• the commutators $[a,b] = a b a^{-1} b^{-1}$, for $a,b \in A_1$.

The homotopy $\Pi$-algebra of a pointed topological space.

For a pointed space $X$, and $U \in \Pi$, define a $\Pi$-algebra $\pi_* X$ by $\pi_* X(U) = [U,X]_*$, the set of pointed homotopy classes of pointed maps from $U$ to $X$.

This is a $\Pi$-algebra called the homotopy $\Pi$-algebra of $X$.

The realisability problem

Suppose $A: \Pi \to sets_*$ is an abstract $\Pi$-algebra, the realisability problem for $A$ is to construct, if possible, a pointed space $X$, such that $A\simeq \pi_* X$. The space $X$ is called a realisation of $A$.

Things can be complicated!

1. The homotopy type of $X$ is not always determined by $A$ (hence ‘a’ rather than ‘the“ realisation) , so that raises the additional problem of classifying the realisations.

2. Not all $\Pi$-algebras can be realised, in fact

Theorem (Blanc 1995)

Given a $\Pi$-algebra, $A$, there is a sequence of higher homotopy operation?s depending only on maps between wedges of spheres, such that $A$ is realisable if and only if the operations vanish coherently.

Example (Blanc 1995)

For $p\neq 2$, a prime and $r\geq 4(p-1)$, $\pi_*S^r \otimes \mathbb{Z}/p$ cannot be realised (and if $p = 2$, one uses $r\geq 6$).

(There is a problem, discussed in Blanc’s 1995 paper, that the composition operations need not be homomorphisms, so tensoring with $\mathbb{Z}/p$ has to be interpreted carefully.)

Simply connected $\Pi$-algebras

A $\Pi$-algebra, $A$, is said to be simply connected if $A_1= 0$.

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 $A$ into a graded Lie ring (with a shift of indices).

Truncated $\Pi$-algebras

The beginnings of a classification theory for $n$-truncated $\Pi$-algebras can be found in Frankland’s thesis (link given below).

References

• C.R. Stover, A Van Kampen spectral sequence for higher homotopy groups, Topology 29 (1990) 9 - 26.

David Blanc has written a lot on the theory of these objects. An example is

• David Blanc, Loop spaces and homotopy operations, Fund. Math. 154 (1997) 75 - 95.

The realisability problem is discussed in

• David Blanc, Higher homotopy operations and the realizability of homotopy groups, Proc. London Math. Soc. (3) 70 (1995) 214 -240,

and further in

• David Blanc, Algebraic invariants for homotopy types, Math. Proc. Camb. Phil. Soc. 127 (3)(1999) 497 - 523. (preprint version on the ArXiv.)

There are more recent results on the realisability problem in Martin Frankland‘s thesis.