Given a finitely generated abelian group and , the th Peterson space of is the simply connected space whose reduced cohomology groups? vanish in dimension and the th cohomology group is isomorphic to .
The Peterson space exists and is unique up to a weak homotopy equivalence given the indicated conditions on and .
There are counterexamples both to existence and uniqueness without these conditions.
For example, the Peterson space does not exist if is the abelian group of rationals.
If , then is a functor from abelian groups without 2-torsion to the homotopy category of pointed spaces.
In fact, for all the map
is an isomorphism if has no 2-torsion.
For all , 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 are defined similarly to Peterson spaces,
using homology instead of cohomology.
We have natural weak equivalences
if is a finitely generated free abelian group and
if is a finite abelian group.
If , then , so .
If , then is obtained by attaching an -cell to an -sphere along a map of degree . Thus, is defined for all .
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 5, 2021 at 02:10:50. See the history of this page for a list of all contributions to it.