# Contents

## 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

### Properties

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

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

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

## $\Pi$-algebras

Let ${\mathrm{Set}}_{*}$ denote the category of pointed sets.

###### Definition

A $\Pi$-algebra is a functor $A:{\Pi }^{\mathrm{op}}\to {\mathrm{Set}}_{*}$, 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 ${A}_{n}=A\left({S}^{n}\right)$, that it takes on the spheres, ${S}^{n}$, $n\ge 1$.

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

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

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

* ‘composition operations’, $-\cdot \alpha :{A}_{p}\to {A}_{r}$ for $\alpha \in {\pi }_{r}\left({S}^{p}\right)$, $1,

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

* a left action of ${A}_{1}$ on the ${A}_{n}$, $n>1$, which commutes with these operations.

* $\left[\xi ,a\right]={}^{\xi }a-a$, where ${}^{\xi }a$ is the result of the ${A}_{1}$-action of $\xi \in {A}_{1}$ on $a\in {A}_{r}$, $r>1$; similarly for a right action;

* the commutators $\left[a,b\right]={\mathrm{aba}}^{-1}{b}^{-1}$, 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 ${\pi }_{*}X\left(U\right)=\left[U,X{\right]}_{*}$, 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 $A:\Pi \to {\mathrm{sets}}_{*}$ 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 be 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 $p\ne 2$, a prime and $r\ge 4\left(p-1\right)$, ${\pi }_{*}{S}^{r}\otimes ℤ/p$ cannot be realised (and if $p=2$, one uses $r\ge 6$).

(There is a problem, discussed in Blanc’s 1995 paper, that the composition operations need not be homomorphisms, so tensoring with $ℤ/p$ 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 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.

Revised on February 6, 2013 00:45:08 by Anonymous Coward (66.31.20.175)