# nLab generalized (Eilenberg-Steenrod) homotopy

The ordinary homotopy groups of a space $X$ are

${\pi }_{n}\left(X\right)={\pi }_{n}^{{S}^{0}}\left(X\right)=\left[{\Sigma }^{n}{S}^{0},X\right]=\left[{S}^{n},X\right],$\pi_n(X) = \pi^{S^0}_n(X) = [\Sigma^n S^0, X] = [S^n, X],

where ${S}^{0}$ is the 0-sphere. We can choose another based space, say $A$. Thus,

${\pi }_{n}^{A}\left(X\right)=\left[{\Sigma }^{n}A,X\right],$\pi^{A}_n(X) = [\Sigma^n A, X],

are the generalized homotopy groups of $X$ with (co)-coefficients in $A$.

But should this page, mentioning Eilenberg-Steenrod, be about generalized stable homotopy? I.e., should we focus on ${\Sigma }^{n}A$ as a spectrum? Mind you, in spectrum it requires ${E}_{n}\cong \Omega {E}_{n+1}$, where $\Omega$ denotes the based loop space. Don’t we want the requirement ${E}_{n+1}\cong \Sigma {E}_{n}$? Need to check whether adjunction means this makes no difference.

Tim: To my mind, there should be a spectrum based generalised stable cohomotopy of $X$ as well perhaps, but the paradigm we have been using has been that it is the spaces that are the first importance here so I would stick with homotopy as $\left[{\Sigma }^{n}A,X\right]$ but also would ask about not using pointed spaces. The free case is possibly more fun and useful.

## Reference

• Hans Baues, Algebraic Homotopy, Cambridge University Press, 1989, p. 117

Revised on October 6, 2010 14:20:02 by Tim Porter (95.147.237.88)