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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 4th 2021

Created with the following content:

## 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.

## 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.

## References

• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 4th 2021

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 4th 2021

## 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.

• CommentRowNumber4.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 4th 2021

Functoriality and examples.