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,

using homology instead of cohomology.

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 $n\ge 2$.

Related concepts

• Moore space

• Eilenberg–MacLane space

• homotopy groups with coefficients

References

• Franklin P. Peterson, Generalized Cohomotopy Groups. American Journal of Mathematics 78:2 (1956), 259–281. doi:10.2307/2372515

• Joseph A. Neisendorfer, Homotopy groups with coefficients,

  Journal of Fixed Point Theory and Applications 8:2 (2010), 247–338. doi:10.1007/s11784-010-0020-1.