Generalized Snaith Splittings

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). Like the first two examples, such functors A come very often with a filtration An(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 B(K(X)+) ≅ B(V ∞n=o Kn(X)/Kn-1(X)). </p

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..

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