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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 4th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated 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]]. ## Related concepts * [[Moore space]] * [[Eilenberg–MacLane space]] * [[homotopy groups with coefficients]] ## References * [[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](http://dx.doi.org/10.1007/s11784-010-0020-1). <a href="https://ncatlab.org/nlab/revision/Peterson+space/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Peterson+space">current</a>

   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.

   Related concepts

   • Moore space

   • Eilenberg–MacLane space

   • homotopy groups with coefficients

   References

   • 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.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 4th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: * [[Franklin P. Peterson]], _Generalized Cohomotopy Groups_. American Journal of Mathematics 78:2 (1956), 259–281. [doi:10.2307/2372515](http://dx.doi.org/10.2307/2372515) <a href="https://ncatlab.org/nlab/revision/Peterson+space/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Peterson+space">current</a>

   Added:

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

   v1, current

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 4th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: ## 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]]. <a href="https://ncatlab.org/nlab/revision/Peterson+space/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Peterson+space">current</a>

   Added:

   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.

   v1, current

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 4th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexFunctoriality and examples. <a href="https://ncatlab.org/nlab/revision/Peterson+space/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Peterson+space">current</a>

   Functoriality and examples.

   v1, current