Generalized Snaith Splittings

Abstract

A Segal Γ-space A gives a homotopy functor A(X) and a connective homology theory h_*(X;A) = π_*(A(X)). The infinite symmetric product SP^∞(X) and the configuration space C(R^∞;X) ≅ Q(X) are well-known examples of Segal Γ-spaces; the former giving singular homology H_*(X;Z) and the latter stable homotopy theory as their homotopy groups. Here we are concerned with another important example, the Segal Γ-space K leading to connective KO-theory: π_*K(X) = ̃ko(X).<br /> Like the first two examples, such functors A come very often with a filtration A_n(X) which splits after applying another suitable homotopy functor, perhaps even a Segal Γ-space B; in the first two examples one can take B = A and obtain the well-known Dold-Puppe splitting of SP^∞(X) resp. the Snaith splitting of Q(X). Our main result is a splitting of K(X) using the functor B(X₊) ≅ Ω^∞-1(MO∧X₊) representing unoriented cobordism, namely <p align="center"> B(K(X)₊) ≅ B(V ^∞_n=o K_n(X)/K_n-1(X)). </p>