K?+ K} N+ KY+ K~ -v 1 i 1 1

Transcription

1 68 RELATIONS BETWEEN HIGHER ALGEBRAIC K-THEORIES D. Anderson~ M. Karoubi, J. Wagoner ~ In this paper we outine the construction of a sequence of natura transformations K?+ K} N+ KY+ K~ -v 1 i 1 1 between the higher agebraic K-theories K@ of [Q] K~ N of [W] 1 ~ 1 KY of [V] K~ -v of [K-V] [G]. The composition 1 K9 + K~ -V is the map constructed by Gersten [G]. Let G be a discrete group. Let NG be the simpicia set in [S] such that the geometric reaization INGI is the cassifying space BG of G. Let AG denote the contractibe simpicia set whose k-simpices are (k+)-tupes (g0,...,gk) which the face degeneracy operators being deetion insertion. The geometric reaization IAGI sha be denoted by EG. The map of simpicia sets AG NG given by - -i (g0'''''gk) (g01g... 'gk-gk ) induces as in [S] the cassifying fibration for G: G EG BG For any coset e.h of a subgroup H of G et Ee.H : IA~.HI where A. H is the subcompex of AG whose k-simpices are (k+)- tupes (g0,g,...,gk) with e.h for 0 ~ i ~ k. ~Partiay supported by NSF GP-34217X 75

2 69 Let {H i } be a coection of subgroups of G. Let A{G,~-H i} denote the bisimpicia set whose (k,~)-simpices are of the form (~0.H0 C... C ~k. Hk; (g0,...,g~)) where gi E ~0-Ho for 0 < i ~ ~. The boundary degeneracy maps in each factor are the usua deetions insertions. Let AV{G,e.H i} be the simpicia space obtained by vertica reaization: in dimension k, AV{G,e.Hi } is the disjoint union the spaces (~0.H0 C... C ~k.hk) E 0.H 0 Then as for any bi-simpica set Idiag A{G,e'Hi] I ~ IAV{G,~-Hi}I We sha et E{G,~.H i} denote Idiag A{G,~-Hi} I. The reader is aerted to the sight notationa difference between this the E(X,X i) of [W]. Let N{H i] be the bi-simpicia set whose (k, )- simpices are of the form (H 0 C... C Hk; (g,...,g )) where gi E H 0 for I < i <. The boundary degeneracy operators in the first factor are deetion insertion, the ones in the second factor are those of NH 0. Let NV{H i} be the vertica reai- zation of N{Hi}; this is a simpicia space which in dimension k is the disjoint union of the spaces We aso have (H0 C... C Hk) BH 0 I diag N{Hi} I = INV{Hi} Let B{Hi} denote Idiag N{Hi} I. 74

3 70 There is a commutative diagram (*) E{G,~'H i} B{H i} > EG ~, BG induced by the foowing diagram of maps on the eve of k-simpices: (~o-ho C... C ~k.hk; (go,...,gk))--~(go,...,gk) (H 0 c c Mk~ (g~ig,.,g{gk)) (g~ig, _ ~...,gk_igk ) Lemma i. The sequence E{G,~.H i} B{H i} + BG is a homotopy fibration. Proof. The simpicia set A{G,~.H i} is the pu back of the maps AG + NG N{H i} NG. Since reaization of simpieia sets com- mutes with pubacks (*) is a cartesian square. In particuar E{G,~.H i} B{H i} is a fibration with discrete fiber G. Since EG is contractibe the "nine-emma" of [T; Appendix, 6] shows E{G,~-H i} is homotopy equivaent to the homotopy fiber of B{H i} BG. We reca briefy the definitions of K~ N K v See [V] 1 i" [W]. Let A be an associative ring with unity A ~ be the stard right A-modue with basis e,e2,...,en,... A semi- stard fag P = {PI C... c Pk} in A ~ is a sequence of free submodues such that for some n > 1 each subspace P. is spanned by a finite subset of {e,...,e n} such that Pk = An. We say 75

4 71 P = {PI C... C pz} refines P, written P ~ P, provided there! is an increasing sequence n<...< n k with Pi = Pn. for 1 ~ i ~ k. If P is a fag with Pk = An et Up C GLn(A) c GL(A) be the sub- group of eements of the form I + N where I is the identity N is an n x n-matrix satisfying N(Pi ) C Pi-I N(P!) : 0. Then et ~(A) = E{(GL(A), ~'Up} for i ~ i et 4 KBN(A) = Wi_~(A). As in (*) we have a cartesian square (t) E{GL(A),~-Up} > EGL(A).i B{Up}?- BGL(A) a homotopy fibration ~(A) B{Up} BGL(A). Let U c GL(A) be the subgroup of upper trianguar matrices with ones on the diagona. Let p denote any finite permutation matrix. Let GLV(A) C EGL(A) be the reaization of the subcompex of AGL(A) whose k-simpices consist of (k+)-tupes (g0,...,gk) -i such that there is some coset ~'pup containing gi for 0 ~ i ~ k. Then in [V] Voodin defines KV(A) = ~i_iglv(a). There is a oartesian square 76

5 72 GLV(A) >EGL(A) L UBpUp -1 ~ BGL(A) P where the space UBpUp -1 denotes the union of the subspaces BpUp -1 P in BGL(A). As in Lemma 1 we have a homotopy fibration GLV(A) > UBpUp -I > BGL(A) P The natura transformation (1) N KY 1 is induced by the correspondence (s 0.UP0 C... C ~k'up k; g0'''''gk ) (g0'''''gk)" It can be shown that the direct sum of matrices induces on ~(A) GLV(A) the structure of a homotopy associative commutative H-space furthermore we have kbass... i <A) = K~N(A) = K~(A) See [V] [W]. kminor.~, K~N(A) K~(A) 2 ~) : = " Theorem A. The fundamenta groups of B{Up} UBpUp -I are perfect P the sequences A GL(A) B {Up}+ BGL.A. +( ] 77

6 73 GLV(A) (UBpUp-I) + BGL(A) + are homotopy fibrations. As a consequence there are maps ~BGL(A) + /x ~ GL(A) 2BGL(A) + > GLV(A) which induce natura homomorphisms (2) Kg(A)m = ~i_i(2bgl(a) +) ~i_i(gal(a)) = K BNi (A) (3) Kg(A)I = ~i-i (2BGL(A)+) ~i-i (GLV(A)) = KY(A)I " In fact (3) is the composition of (i) (2). It wi be shown in + [W'] that H~(B{Up};Z) vanishs for any ring. Hence B{Up} is contractibe Theorem A shows that (2) is an isomorphism. I think the methods of [W'] wi aso show H,(~BpUp-I;z) = 0 therefore (3) is probaby an isomorphism too. Finay we construct the natura transformation K BN K K-V. + A 1 simiar construction works for KY. Reca the definition of K~ -V 1 1 as given in [G]. Let A, = {A n } be the simpicia ring where A n = A[t0,t,...,tn] / t o + t I t n = i the face degeneracy operators ~i:an An_ I si: An A n+ are given by f t~, < i i t, ~ < i ~i(t~) : 0, ~ = i si(t ) = I tz+t +' ~ = i t _, i < Z t + 1, i < 78

7 74 Let GL(A), : {GL(An)} be the corresponding simpicia group. Then as in [G] K~-V(A) = ~i_iigl(a,)i = ~ibgl(a,) where BGL(A,) is the reaization of the diagona in the bisimpicia set NGL(A,) whose (k,~)-simpices are ~-tupes (g,...,g~) with gi E GL(Ak) for i ~ i ~ ~. The vertica face degeneracies are those of NGL(A k) the horizonta face degeneracies come from the simpicia ring A,. Let AGL(A,) be the bisimpicia set whose (k, )-simpices are (~+)-tupes (g0,...,g ) where gi E GL(A k) for 1 < i ~ ~. Let EGL(A,) be the reaization of the diagona. Then there is a homotopy fibration ]GL(A,) I EGL(A,) BGL(A,). There is a simiar fibration for BUp(A,). We want to get a commutative square ike (*) for the simpicia ring A,. For any ring A we ket Up(A) denote Up for the ring A. Let N{Up(A,)} be the bi-simpicia set whose (k,~) simpices are of the form (UP0(A ) C... c UPk(A~) ; (g,...,g~)) where gi E UP0(A ) for i ~ i~. Let A{GL(A,),~Up(A,) } be the bi-simpicia set whose (k, ) simpices are of the form (a 0"UP0(A~) C... C ak-upk(a~) ; (g0,...,g )) where gi E a0.up0(a~) for 0 < i ~ ~. Let B{Up(A,)} = Idiag N{Up(A,)} I 79

8 75 E{GL(A,),~.Up(A,)} : Idiag {GL(A,),~'Up(A,)} Then the foowing is a puback square: E{GL(A,),~.Up(A,)} > EGL(A,) (**) B{Up(A~)} > BGL(A,) Let ~'~(A,) = E{GL(A,),e. Up(A,)} Lemma 2. ~(A,) ~ IGL(A,)[ Proof. As in Lemma i the homotopy fiber of B{Up(A,)} BGL(A,) is ~(A,). Hence it suffices to prove B{Up(A,)} is contractibe. This space has the homotopy type of the reaization of the simpicia space NV{Up(A,)} which in dimension k is the disjoint union of the (P0 ~ "'" ~ Pk ) x BUp(A,). Since A, is an acycic simpicia ring each BUp(A,) is contracti- be; thus up to homotopy type we are ooking at the nerve of the partiay ordered set of semi-stard fags in A ~ which is contract- ibe. Now et A denote the constant simpicia ring whege A = A n for n ~ 0 a the face degeneracy operators are the identity. Then there is a cartesian square ike (**) but in this case we are sti, up to homotopy type, just working with the square (t) when A is not considered as a simpicia ring. The natura incusion of simpicia rings A + A, induces a homotopy commutative diagram of homotopy fibrations 80

9 76 ~L(A) > B{Up(A)} + > BGL(A) + ~"~(A,) ~ B{Up(A,)} > BGL(A,) This then gives the homomorphism (4) K~N(A) : ~i 1 ~(A) > ~i-i IGL(A*)I = K~-V(A) - " Putting (i), (2), the KY 1 sequence of natura transformations anogue of (4) together gives the (s) K? + K} N + + -v whose composition is the map defined by Gersten [G]. When A is eft reguar Quien [Q] has shown this composition to be an isomor- phism. Coroary B. If A is eft reguar K~ N KY 1! direct summ. contain K9 as a References [G] S. Gersten, K-theory of a poynomia extension, preprint from Rice University. [K-V] M. Karoubi 0. Viamayor, Foncteurs K n en agebra et en topoogie, C.R. Acad. Sci. Paris 269(1969), [Q] D. Quien, Higher K-theory for categories with exact sequences, preprint, M.I.T. [S] S. Sega, Categories cohomoogy theories, preprint, Oxford University. [T] E. Thomas, Fieds of tangent k-panes on manifods, Inventiones Math. Vo. 3(1967), p [W] J. Wagoner, Buidings, stratifications, higher K- theories, this voume. [W'] J. Wagoner, Equivaence of agebraic K-theories, to appear. 81