nLab homotopy group with coefficients

Idea

A generalization of the notion of homotopy groups.

Definition

Given a pointed space $X$ and finitely generated abelian group $A$ and $n\ge2$, we set

$$\pi_n(X,A) \;\coloneqq\; [P^n(A),X] \,,$$

where $P^n(A)$ is the $n$th Peterson space of $A$ and $[-,-]$ denotes pointed homotopy classes.

Related concepts

• homotopy groups

• Peterson space

References

• 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.