nLab
Peterson space

Definition

Given a finitely generated abelian group AA and n3n\ge 3, the nnth Peterson space P n(A)P^n(A) of AA is the simply connected space whose reduced cohomology groups? vanish in dimension knk\ne n and the nnth cohomology group is isomorphic to AA.

Existence and uniqueness

The Peterson space exists and is unique up to a weak homotopy equivalence given the indicated conditions on AA and nn.

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

For example, the Peterson space does not exist if AA is the abelian group of rationals.

Functoriality

If n4n\ge 4, then P nP_n is a functor from abelian groups without 2-torsion to the homotopy category of pointed spaces.

In fact, for all n4n\ge 4 the map

Hom(P n(B),P n(A))Hom(A,B)Hom(P^n(B),P^n(A))\to Hom(A,B)

is an isomorphism if AA has no 2-torsion.

Corepresentation of homotopy groups with coefficients

For all n2n\ge2, we have a canonical isomorphism

π n(X,A)[P n(A),X],\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 spacesM n(A)M_n(A) are defined similarly to Peterson spaces,

using homology instead of cohomology.

We have natural weak equivalences

P n(A)M n(Hom(A,Z))P^n(A) \simeq M_n(Hom(A,\mathbf{Z}))

if AA is a finitely generated free abelian group and

P n(A)M n1(Hom(A,Q/Z))P^n(A) \simeq M_{n-1}(Hom(A,\mathbf{Q}/\mathbf{Z}))

if AA is a finite abelian group.

Examples

If A=ZA=\mathbf{Z}, then P n(A)=S nP^n(A)=S^n, so π n(X,Z)=π n(X)\pi_n(X,\mathbf{Z})=\pi_n(X).

If A=Z/kZA=\mathbf{Z}/k\mathbf{Z}, then P n(A)P^n(A) is obtained by attaching an nn-cell to an (n1)(n-1)-sphere along a map of degree kk. Thus, π n(X,Z/kZ)\pi_n(X,\mathbf{Z}/k\mathbf{Z}) is defined for all n2n\ge 2.

References

Last revised on April 4, 2021 at 22:10:50. See the history of this page for a list of all contributions to it.