# nLab Peterson space

## Definition

Given a finitely generated abelian group $A$ and $n\ge 3$, the $n$th Peterson space $P^n(A)$ of $A$ is the simply connected space whose reduced cohomology groups? vanish in dimension $k\ne n$ and the $n$th cohomology group is isomorphic to $A$.

## Existence and uniqueness

The Peterson space exists and is unique up to a weak homotopy equivalence given the indicated conditions on $A$ and $n$.

There are counterexamples both to existence and uniqueness without these conditions.

For example, the Peterson space does not exist if $A$ is the abelian group of rationals.

## Functoriality

If $n\ge 4$, then $P_n$ is a functor from abelian groups without 2-torsion to the homotopy category of pointed spaces.

In fact, for all $n\ge 4$ the map

$Hom(P^n(B),P^n(A))\to Hom(A,B)$

is an isomorphism if $A$ has no 2-torsion.

## Corepresentation of homotopy groups with coefficients

For all $n\ge2$, we have a canonical isomorphism

$\pi_n(X,A)\cong [P^n(A),X],$

where the left side denotes homotopy groups with coefficients and the right side denotes morphisms in the pointed homotopy category.

## Relation to Moore spaces

Moore spaces$M_n(A)$ are defined similarly to Peterson spaces,

We have natural weak equivalences

$P^n(A) \simeq M_n(Hom(A,\mathbf{Z}))$

if $A$ is a finitely generated free abelian group and

$P^n(A) \simeq M_{n-1}(Hom(A,\mathbf{Q}/\mathbf{Z}))$

if $A$ is a finite abelian group.

## Examples

If $A=\mathbf{Z}$, then $P^n(A)=S^n$, so $\pi_n(X,\mathbf{Z})=\pi_n(X)$.

If $A=\mathbf{Z}/k\mathbf{Z}$, then $P^n(A)$ is obtained by attaching an $n$-cell to an $(n-1)$-sphere along a map of degree $k$. Thus, $\pi_n(X,\mathbf{Z}/k\mathbf{Z})$ is defined for all $n\ge 2$.