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 $A$.
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.
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
is an isomorphism if $A$ has no 2-torsion.
For all $n\ge2$, we have a canonical isomorphism
where the left side denotes homotopy groups with coefficients and the right side denotes morphisms in the pointed homotopy category.
Moore spaces$M_n(A)$ are defined similarly to Peterson spaces,
using homology instead of cohomology.
We have natural weak equivalences
if $A$ is a finitely generated free abelian group and
if $A$ is a finite abelian group.
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 $n\ge 2$.
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.
Last revised on April 4, 2021 at 22:10:50. See the history of this page for a list of all contributions to it.