§ Stable homotopy theory

We like stable homotopy groups because of the Freudenthal suspension theorem which tells us that homotopy groups stabilise after many suspensions. The basic idea seems to be something like a tensor-hom adjunction. We have the loop spaces which are like S1→XS^1 \rightarrow X and the suspension which is like S1∧XS^1 \wedge X. The theory begins by considering the tensor-hom-adjunction between these objects as fundamental. So curry stuff around to write things as (S^1, A) -> B and A -> (S^1 -> B), which is Suspension(A) -> B and A -> Loop(B). This gives us the adjunction between suspension and looping.
  • We then try to ask: how can one invert the suspension formally? One tries to do some sort of formal nonsense, by declaring that maps between Σ−nX\Sigma^{-n}Xand Σ−mY\Sigma^{-m} Y , but this doesn't work due to some sort of grading issue.
  • Instead, one repaces a single object XX with a family of objects {Xi}\{ X_i \}called as the spectrum. Then, we can invert the suspension by trying to invert maps between objects of the same index.

§ References