Forum Math. 1 0 ( 1 9 9 8 ) , 5 3 3 — 6 0 3 Forum Mathematicu m © d e G r u y t e r 1 9 9 8 Duality i n Wa l d h a u s e n Ca t e g o r i e s Michael We i s s and Bruce Williams (Communicate d b y An d r e w Ra n i c k i ) A b s t r a c t . W e d e v e l o p a th e o r y o f Spanier-Whitehea d du a l i t y i n ca t e g o r i e s wi t h cofibration s a n d we a k equivalence s (Waldhause n categories , fo r short). Th i s in c l u d e s L-theory, the i n v o ¬ l u t i o n o n K - t h e o r y introduced b y [V o ] i n a sp e c i a l ca s e , an d a map Ξ re l a t i n g L-theory to t h e Ta t e sp e c t r u m o f $ ac t i n g o n K-theory. Th e map Ξ i s a distillatio n o f th e lo n g ex a c t Rothenberg se q u e n c e s [Sha], [Ra1], [Ra2], in c l u d i n g an a l o g s in v o l v i n g hi g h e r K-groups. It g o e s b a c k to [WW2] i n s p e c i a l c a s e s . A m o n g the e x a m p l e s c o v e r e d he r e , but not i n [WW2], a r e ca t e g o r i e s o f re t r a c t i v e sp a c e s wh e r e th e notion o f we a k equivalenc e in v o l v e s control. 1 9 9 1 Mathematic s Su b j e c t Classification : 19 D 1 0 , 57 N 6 5 , 5 7 R 6 5 . 0 . Introductio n For an y ri n g R, Qu i l l e n has de f i n e d a n algebraic K-theory Ω - s p e c t r u m , $. Hi s construction i s i n terms o f the category o f finitely ge n e r a t e d pr o j e c t i v e mo d u l e s o v e r R, but i t ca n b e a p p l i e d to an y ex a c t category. I n order to st u d y concordances o f m a n i f o l d s Wa l d h a u s e n generalize d Qu i l l e n ’ s construction to ap p l y to what W a l d ¬ h a u s e n ca l l s ca t e g o r i e s wi t h co f i b r a t i o n s an d we a k eq u i v a l e n c e s . (F o l l o w i n g Thomason w e wi l l ca l l th e m Wa l d h a u s e n categories. ) A n ex a m p l e i s gi v e n b y the c a t e g o r y o f ba s e d compact CW - s p a c e s . More ge n e r a l l y , the ca t e g o r y o f retractive r e l a t i v e CW - s p a c e s $ o v e r a fixed s p a c e X, wi t h compact quotient Y/X, i s an e x a m p l e o f a Wa l d h a u s e n ca t e g o r y ; it s K-theory sp e c t r u m i s kn o w n a s $. W a l d h a u s e n su p p l i e d se v e r a l po w e r f u l tools al o n g wi t h hi s construction o f $ f o r a Wa l d h a u s e n ca t e g o r y $, su c h a s the additivity theorem, the approximation theorem, and the ge n e r i c fibration theorem. Ev e n i f on e i s primarily in t e r e s t e d i n t h e algebraic K-theory o f ri n g s th e s e tools ha v e important applications [Sta], 1 Bo t h a u t h o r s s u p p o r t e d i n p a r t b y N S F g r a n t . 5 3 4 M . W e i s s , B . W i l l i a m s [Tho2]. Also i n [CPed] ex c i s i o n in controlled algebraic K-theory o f rings i s proved using Waldhausen’s machinery. Suppose that R i s a ring with involution (involutory anti-automorphism). The problem o f cl a s s i f y i n g manifolds up to homeomorphism or diffeomorphism lead W a l l to de f i n e algebraic L-groups Ln(R) in terms o f quadratic forms on finitely generated projective (or fr e e , or based fr e e ) modules over R. The L-groups turned out to be the homotopy groups o f a spectrum [Q]. Ranicki [Ra3] associates such L-theory spectra to any additive category with chain duality. (There i s a quadratic L-theory spectrum made using quadratic forms, and a symmetric L-theory spectrum made using symmetric forms.) Specifically , by using additive categories o f “modules parametrized b y a simplicial complex” Ranicki gave an algebraic description o f Quinn’s L-theory assembly map which i s us e d to cl a s s i f y manifolds up to ho m e o ¬ morphism. Also Ranicki’s chain duality setup has been used to construct controlled versions o f L-theory fo r rings [FP]. The assembly map i n L-theory can then also b e id e n t i f i e d with a forget control map. This has been used to prove many cases o f the Novikov conjecture; se e [FRR]. I f one wants to study the spaces o f homeomorphisms or diffeomorphisms o f a manifold, then the L-theory o f rings or ev e n additive categories i s not adequate. One i s fo r c e d to consider L-theory and K-theory o f certain Waldhausen categories equipped with duality, and a certain map Ξ relating the L-theory to the K-theory. For this reason w e need a theory o f duality in Waldhausen categories which, unlike Ranicki’s chain duality theory, allows “nonlinear” cases; and w e need to understand, i n this generality, L-theory, K-theory, and the Ξ-map. Waldhausen categories o f retractive spaces where the notion o f weak equivalence involves control are important i n applications to geometry and should therefore b e the central examples o f such a theory. T o explain what Ξ i s about, w e note that many o f the classical invariants fo r symmetric forms over a ring R take values i n groups constructed fr o m Ki (R) where i = 0,1, or 2. This suggests that there should b e a connection between the symmetric L-theory spectrum $ and the algebraic K-theory spectrum $. S i n c e the classical constructions o f L-theory are modelled on the pre-Quillen definitions o f “ l o w dimensional K-theory”, the connection i s hidden. In [WW2] w e established the connection b y constructing Ξ, a natural map fr o m $ to the Tate spectrum f o r $ acting on $. A nonlinear version o f this does already appear i n [WW2], but i t i s ve r y limited and there i s no mention o f “control”. But i t was certainly motivated b y our study o f spaces o f homeomorphisms. In this paper w e introduce the notion o f a Spanier-Whitehead product $ on a Waldhausen category $. This i s a functor $ f r o m $ to pointed spaces. The space $ should be thought o f as the space o f pairings between C and D. I f $ s a t i s f i e s certain axioms listed i n §2, reminiscent o f Spanier-White- D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 3 5 h e a d duality, then fo r ev e r y C i n $ there ex i s t s an essentiall y unique ob j e c t C i n $ wh i c h co m e s w i t h a nondegenerate pairing $.