Pi-algebra in nLab

Contents

Contents

Idea

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

The category Π\Pi of homotopy operations has

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

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

Properties

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

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

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

Π\Pi-algebras

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

Properties

• A Π\Pi-algebra AA satisfies A*=*A* = *.

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

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

  • ‘Whitehead product’ homomorphisms :

[−,−]:A p⊗A q→A p+q−1[-,-] : A_p\otimes A_q \to A_{p+q-1}

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

• ‘composition operations’, −⋅α:A p→A r-\cdot \alpha : A_p\to A_r for α∈π r(S p)\alpha \in \pi_r(S^p), 1<p<r1\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 1A_1 on the A nA_n, n>1n\gt 1, which commutes with these operations.

The Whitehead products include

• [ξ,a]= ξa−a[\xi,a] = {}^\xi a - a, where ξa{}^\xi a is the result of the A 1A_1-action of ξ∈A 1\xi \in A_1 on a∈A ra\in A_r, r>1r\gt 1; similarly for a right action;

• the commutators [a,b]=aba −1b −1[a,b] = a b a^{-1} b^{-1}, for a,b∈A 1a,b \in A_1.

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

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

This is a Π\Pi-algebra called the homotopy Π\Pi-algebra of XX.

The realisability problem

Suppose A:Π→sets *A: \Pi \to sets_* is an abstract Π\Pi-algebra, the realisability problem for AA is to construct, if possible, a pointed space XX, such that A≃π *XA\simeq \pi_* X. The space XX is called a realisation of AA.

Things can be complicated!

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

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

Theorem (Blanc 1995)

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

Example (Blanc 1995)

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

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

Simply connected Π\Pi-algebras

A Π\Pi-algebra, AA, is said to be simply connected if A 1=0A_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 AA into a graded Lie ring (with a shift of indices).

Truncated Π\Pi-algebras

The beginnings of a classification theory for nn-truncated Π\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.