nLab
generalized (Eilenberg-Steenrod) homotopy

The ordinary homotopy groups of a space XX are

π n(X)=π n S 0(X)=[Σ nS 0,X]=[S n,X], \pi_n(X) = \pi^{S^0}_n(X) = [\Sigma^n S^0, X] = [S^n, X],

where S 0S^0 is the 0-sphere. We can choose another based space, say AA. Thus,

π n A(X)=[Σ nA,X], \pi^{A}_n(X) = [\Sigma^n A, X],

are the generalized homotopy groups of XX with (co)-coefficients in AA.

But should this page, mentioning Eilenberg-Steenrod, be about generalized stable homotopy? I.e., should we focus on Σ nA\Sigma^n A as a spectrum? Mind you, in spectrum it requires E nΩE n+1E_n \cong \Omega E_{n + 1}, where Ω\Omega denotes the based loop space. Don’t we want the requirement E n+1ΣE nE_{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 XX 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 [Σ nA,X][\Sigma^n A,X] 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)