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nLab > Latest Changes: Peterson space

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- CommentRowNumber1.
- CommentAuthorDmitri Pavlov
- CommentTimeApr 4th 2021
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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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n\ge 3</annotation></semantics></math>$ , the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ th Peterson space $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>P</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P^n(A)</annotation></semantics></math>$ of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ is the simply connected space whose reduced cohomology groups vanish in dimension $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≠</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k\ne n</annotation></semantics></math>$ and the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ th cohomology group is isomorphic to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ .

Existence and uniqueness

The Peterson space exists and is unique up to a weak homotopy equivalence given the indicated conditions on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ .

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

For example, the Peterson space does not exist if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ is the abelian group of rationals.

Corepresentation of homotopy groups with coefficients

For all $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n\ge2</annotation></semantics></math>$ , we have a canonical isomorphism
πn(X,A)≅[Pn(A),X],

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

Related concepts
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
- CommentRowNumber2.
- CommentAuthorDmitri Pavlov
- CommentTimeApr 4th 2021
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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
- 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M_n(A)</annotation></semantics></math>$ are defined similarly to Peterson spaces, using homology instead of cohomology.

We have natural weak equivalences
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>P</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>M</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P^n(A) \simeq M_n(Hom(A,\mathbf{Z}))</annotation></semantics></math>$
if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ is a finitely generated free abelian group and
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>P</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>M</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Q</mi></mstyle><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P^n(A) \simeq M_{n-1}(Hom(A,\mathbf{Q}/\mathbf{Z}))</annotation></semantics></math>$
if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ is a finite abelian group.

v1, current
- 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

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nLab > Latest Changes: Peterson space

Definition

Existence and uniqueness

Corepresentation of homotopy groups with coefficients

Related concepts

References

Relation to Moore spaces