We associate to every exact category with duality (C, D, η) a real symmetric spectrum KR(C, D, η). We call KR(C, D, η) the real algebraic K-theory spectrum of (C, D, η) and call its equivariant homotopy groups KRp,q(C, D, η) the real algebraic K-groups of (C, D, η). We show that the groups KRp,0(C, D, η) are canonically isomorphic to Schlichting's higher Grothendieck-Witt groups GWp(C, D, η), which generalize Karoubi's hermitian K-groups, and that the homotopy groups of the underlying non-equivariant symmetric spectrum of KR(C, D, η) are canonically isomorphic to Quillen's K-groups Kp(C). The real algebraic K-theory spectrum thus is an algebraic analogue of Atiyah's real K-theory spectrum.
Our main results concerning this construction are real versions of the group-completion theorem and the additivity theory, stated in the introduction as Theorem A and Theorem B, respectively. We stress that all theorems in the book, including Theorem A and Theorem B, are valid also if 2 is not invertible in the hom-abelian groups of C.
A word of warning is in order. The book is in preliminary form and, at present, only contains a full proof of Theorem A. There are also some inconsistencies stemming from the two possible choices of how to view a partially ordered set as a category.
Lars Hesselholt
<larsh@math.nagoya-u.ac.jp>