Proc. Natl. Acad. Sci. USA Vol. 86, pp. 8607-8609, November 1989 Mathematics

The cyclotomic trace and the K-theoretic analogue of Novikov's conjecture (algebraic K-theory/Waldhausen's A-theory/Hochschild )

M. BOKSTEDTt, W.-C. HSIANGt, AND 1. MADSEN§ tFakultdt fOr Mathematik, Bielefeld University, D-4800, Bielefeld, Federal Republic of Germany; tDepartment of Mathematics, Princeton University. Princeton, NJ 08544; and §Matematisk Institut, Aarhus Universitet, DK-8000 Aarhus C, Denmark Communicated by William Browder, July 31, 1989 (receivedfor review May 1, 1989)

ABSTRACT A trace construction, the cyclotomic trace, is h-cobordism (W"+1; N", N') and a map f:W"' --+M" such given. It associates to algebraic K-theory of a group ring, or that fJNi = fi. The set of equivalence classes is denoted better to Waldhausen's A-theory, equivariant stable yfh(Mn). Let us next generalize 5fh(M11) to yh(M'i x Dk, a) by classes of the free-loop space of its classifying space. The considering homotopy equivalences f:(N"+k, dN"+k) (M" cyclotomic trace detects the Borel classes in algebraic K-theory x Dk, M" x aDk) with f aN"+k a homeomorphism subject to of the integers. It is used to prove, for a wide class of groups, an equivalence relation from h-cobordism (relative to the that the K-theory assembly map is rationally injective. This is homeomorphism along the boundary). In fact, it follows from the K-theoretic analogue of Novikov's conjecture. the general machinery that there is a spectrum Sh/(M") such that the structure set and its generalizations become the homotopy groups of Sh(Mn). On the other hand, we have the Section 1. Introduction surgery spectrum L(7r1M, w), where w: 71M -* {±1} is the orientation map of r1M. For simplicity, let us consider the When Novikov began his celebrated work on topological orientable case only and let us set L(krr1M) = L(TI1M, 1). We invariance of Pontrjagin classes (1), he made the following also have the homology spectrum h(M', L(1)) with values in conjecture. L(1) = L(1, 1)-i.e., the fl-spectrum associated with M' A CONJECTURE 1.1 (Novikov's original conjecture). Let f: L(1). There is a natural map, the assembly map (4, 5): M' M' be a homotopy equivalence, and let x1, . ., E- H1(Mn, c>). Then, a:h(M', L(1)) -L(7rM), [1.2] L*(Mn) U f*(Xl U ... U xs)rMnj and we have the fibration $Sh(Mn) -->h(Mn, L(1)) ->L(7rM). [1.31 = U (x1 U ... U E Q, L*(Mn) xs)[M2I The map from §h(M") to h(M", L(1)) systematically measures the difference (and the higher order differences) of the where L*(Mn) is the total L-genus of Mi (i = 1, 2). characteristic classes of homotopy equivalent . If x1, , x, are linearly dependent, both sides of the CONJECTURE 1.3 (Novikov's conjecture on higher signa- equation are zero and there is nothing to prove. If x1, .. . tures reformulated). The assembly map x, are linearly independent, there exists a map g:M2' -* T'(l ' s) such that we can choose a linearly independent subset a:h(Mn, L(1)) 0 Q -* L(ir1M) 0 Q .. , t, of H'(T'; Q) with xi = g*ti. So, this leads to the = = is split monomorphism. Here 0) C means rationalizing the following construction: Let K K(r, 1) B7r and let t E spectra. HS(K; Q) be a fixed rational cohomology class. For a map Let us now relate the conjecture to K-theory. Let h(BIr, g:M"- K, where M" is a closed , we set K(Z)) be the homology spectrum of Bir with values in the spectrum K(Z) = BGL(Z)+ and let K(Z[v]) be the K-theory L(g, t)(Mn) = (L*(Mn) U g*(t))[Mn] E (U [1.11 spectrum ofthe group ring Z[7] . There is an assembly map (6): and call it Novikov's higher signature of M" associated to (g, -* [1.41 t). It depends only on the homotopy class of g and of t. If K a:h(B7r, K(Z)) K(Z[LT]) = point, then it is Hirzebruch's signature. Conjecture 1.1 has essentially gotten from tensoring (1 x 1) matrices of Zir and the following generalization. (n x n) matrices of GL(n, Z). CONJECTURE 1.2 (Novikov's conjecture on higher signa- The following conjecture, presented in ref. 7, is the K- tures). Let f: Mn -* Mn be a homotopy equivalence of closed theoretic analogue of Conjecture 1.3. manifolds and let g: Mn -* K = Bir be a map. Then L(g, t) CONJECTURE 1.4. The assembly map ofmap 1.4 is alwvays (Mn) = L(gf, t)(Mn). In other words, the higher signatures are a rational direct factor. homotopy invariants. In this paper, we announce the following theorem. Let us now reformulate the above conjectures from the THEOREM A. If Hi(BF, Z) is finitely generated for i -' 0O point of view of (2, 3). Consider the structure then Conjecture 1.4 is true. set fh(M,). We first introduce the set of homotopy equiva- In fact, we shall prove the corresponding conjecture for lences for maps f:N" -* M" of closed manifolds into M". We Waldhausen's K-theory of BF, A(BF), which is rationally say that fi:N' -* M" (i = 1, 2) are equivalent if there is an equivalent to Quillen's K-theory spectrum K(Z[ir]) and de- duce Theorem A from it. The publication costs of this article were defrayed in part by page charge We are most indebted to Gunnar Carlsson for suggesting payment. This article must therefore be hereby marked "advertisement" Sould's paper (8) to us. Our approach is motivated by Sould's in accordance with 18 U.S.C. §1734 solely to indicate this fact. construction of Borel's generators of K(Z), by using Chern

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characters in etale cohomology, even though the precise (SdmNcYGLk(R))c .. is factored through the "homotopy in- relationship is not yet clear between Soulk's work and ours. verse limit" holim (SdmNCYGLk(R))c and we have the cy- It should be a subject for future studies. We also use the clotomic trace map machinery developed for solving Segal's and Sullivan's con- jectures and we hope that the methods employed in this paper Trc:BGLk(R) -- holim (SdmTHH(R))Ct. [2.6] will be useful for other problems. When we use Segal's F-structure (12) or permutative cate- Section 2. Cyclotomic Trace Map gories (13), the cyclotomic trace map turns out to induce an infinite loop space map or a map to spectra with domain K(R). For a topological space Y, the unreduced suspension spec- We observe that the Cm action on SdmTHH(R) is equi- trum fl'S'(Y+) is denoted by Q(Y). If X is connected, we variantly homotopic to Qc..(ABF), where QC..t denotes the consider the "homotopy ring" R = Q(flX), where fIX unreduced representation sphere spectrum of C,,. The space denotes the loop space of X. Waldhausen defines the alge- offixed points of Qc..(ABF) is known by a result oftom Dieck braic K-theory A(X) of X as Quillen's "+" construction (14) to be BGL(R)+ x Z of BGL(R) = lim BGLA(R), where GLk(R) denotes the homotopy units of Mk(R) (9). Generalizing Den- [Qc .. (ABn)]c ..= H' Q(ECtn1k x c.(ABr)cFV/k). [2.7] nis' trace map, one of us (M.B.) introduced the topological klm Hochschild spectrum THH(R) and the trace map from A(X) We project to the component corresponding to k = 1 (in fact, to THH(R). Combining this with a certain power map, similar the image of the cyclotomic trace is contained diagonally in to one considered by Goodwillie in a letter to Waldhausen the product). There results an infinite loop map (private communication), we produce a map which shall be called the cyclotomic trace map (a name suggested to us by Trc:A(Br)-*holim Q(ECn xcABr). [2.8] Soule). Let us indicate the construction. Let (NCYGLk(R))l1l be the component of the cyclic space NCYGLk(R) of the Section 3. Reducing the Problem to the Case of A(*) group-like monoid 1 E GLA(R) containing GLA(R). We have If X = Br for a discrete group F, then we have a component a map i:BGLk(R) -- NcYGLk(R)Ill and a natural map essen- tially induced by inclusions: decomposition AX = HI At,[, where [g] runs through the conjugacy classes ofg E F and Al,, consists ofthe loops freely homotopic to g. Correspondingly, we have NCY(F) = LI Br,,,, j:BGLk, (R) A NCYGLk(R)[lI -* NCYGLk(R) where rFl is the centralizer of [g] and BrFl is C,,,- THH(M (R)). equivariantly homotopic to Ajgj. Moreover, the inclusion of [2.11 Br intoA[j1 as the constant loops is a (weakly equivariant but Recall that NCYGLA(R) and THH(Mk(R)) are 51 spaces (since not equivariant) homotopy equivalence, so they are cyclic spaces in the sense of Connes (10), and hence they are Cm spaces for the cyclic subgroup of order m, Cm C ES' X siBF[,] - BS' x SiA[l] - CP' x Br. [3.1] 51. Inspired by Segal (11), we consider the m-fold edge-wise Projecting from NCY(F)+ to BF[11+ and composing with the subdivision SdmX. It is a simplicial space whose set of (k - above 1)-simplices is equal to the (mk - 1)-simplices ofX, and there homotopy equivalence, the maps displayed in 3.1 is a homeomorphism induce- an infinite loop map Trc:A(BF) -> Q(l(CPX x BrA.+) [3.2] D,,:ISd,,,Xl ---> IXI. [2.2] The profinite completion is a product of its p-completions so If X is a cyclic space X, the Cm C S' action on JXI from the Trc = fITrcp, and we can consider each p-completion sepa- cyclic structure ofX is compatible with the obvious simplicial rately. Cm action on SdmX. Furthermore, there is a power map For a given prime p, we need the following condition on the Am:NcY(r) > (Sdm Ncy(f))c .. integral homology groups of BF: Condition The map (gI. g,) (go , **,,gn * go,...*, *g), [2.3] (Cp): which after composing with Dm gives a map into INcY(F)Ic-. 0 Cl lim For a group-like monoid F with 1, we have a natural map Hi(BF) Hi(BF) CP,.]0 k:INcY(F)I - ABF, [2.41 is injective for every i. This is satisfied for example if Hi(BF, Z) is finitely gener- where ABr is the free-loop space of the classifying space BF ated for i _ 0. such that k is an equivariant homotopy equivalence for every THEOREM B. If Trc :A(*) -- Q(Y(CP+))Ais rationally Cm C S1 (but not S 1 itself) and Am corresponds to the obvious injective, then the K-theoretic analogue of Novikov's Con- power map on the free-loop space. Combining with the jecture, Conjecture 1.4, is true for a discrete group F satis- equivariant Morita equivalence ofTHH(Mk(R)) and THH(R), fying condition (Cp). we have the mth cyclotomic trace map: Section 4. Study of on Trc(n):BGLk(R) ANCYGLk(R)[jI -* NCYGLk(R) Trcp A(*) We now study Trcp on A(*). Consider the following induction diagram ' SdmNcYGLk(R)c,.. smash SdTHH(Mk(R))Cfl BGL,(Q(Cin)) C (SdmTHH(Q(Ctn)))C.. (QC..(LAi ))c.. Mna Sd ,THH(R)- D,, THH(R)cm [2.5] I Ind,,, I Ind,,, Here we have dropped the bars to indicate topological Trc"") realization. It turns out that the map from BGLk(R) to BGLin(Q(j)) 30 (SdmTHH(Q(1)))C"' ---* (QC...(1))C"'. R.11 Downloaded by guest on September 25, 2021 Mathematics: B6kstedt et al. Proc. Natl. Acad. Sci. USA 86 (1989) 8609 We have the following facts: image of the S1-transfer from >CP+ to Q'(BCp,,) associated PROPOSITION 4.1. Indm:(Qcm-Agl)cm (QCm(1))Cm is trivial with the S1-fibration BCp,, -* CPX. The classes t, and v4,,1 for g # 1. denote the generators ofH1(S1; Z) and H4m(BCp,,; Zip"). The PROPOSITION 4.2. AtL, is a model of Bcm(Cm), the Cm- fact that the p-adic L-function appears in map 4.4 is based in equivariant BCm, and we can apply the decomposition 2.7 to part on joint work of S. Bentzen and I.M. Taking the inverse (QCmAjj)Cm. In this case Indm:(QCmAtii)Cm (QCm(M))Cm can limit, we have that be computed as the usual equivariant Cm-transfer. PROPOSITION 4.3. The Indm functions are compatible with Trc :A(*) -* Q(XCPo+), A respect to the homotopy inverse limit. From these, we reduce our problem to considering Trc(m) is a rational injection on homology ifp is a regular prime. This (with m = p", p - 5 orp = 2, 3, and n _ 2) on the appropriate proves Theorem A. "subgroup" of GL1(Q(Cp,,)). First, however, we replace Q by the mapping space Q' of We thank R. Cohen, T. Goodwillie, J. Jones, and F.Waldhausen the localized spheres S"[1/q] where q is any integer which for valuable philosophical as well as practical suggestions. In fact, a generates the units modulo p2. This leads to a A'(X). result similar to Theorem B was first announced by R. Cohen and J. = = 7' = Jones in private communication. W.-C.H. was supported by a Rationally, A(*) K(7Z) and A'(*) K(Z') with Z[1/q]. National Science Foundation grant, and I.M. was supported by the By Quillen's localization theorem (or by ref. 15) A(*) and Danish Science Council. A'(*) differ rationally only in dimension one. Let u,, be the unit of Z'[Cp,,] given by u,, = 11qT(112(Tq 1. Novikov, S. P. (1970) in Essays on Topology and Related - 1)/T - 1), where T is a generator of Cp, and consider the Topics: Memoires dedies a G. de Rham, (Springer, New York), map pp. 147-155. 2. Farrell, T. & Hsiang, W. C. (1981) Ann. Math. 113, 199-209. 5' x BCp,, - BGL,(Q'(Cp,,)), [4.2] 3. Wall, C. T. C. (1970) Surgery on Compact Manifolds (Aca- demic, London). where S' corresponds to the infinite cyclic subgroup gener- 4. Quinn, F. (1969) Ph.D. thesis (Princeton University, Princeton, ated by u,,. Since the u,, are compatible under norm maps, the NJ). inclusion in map 4.2 composed with the maps BGL,(Q'(Cp,,)) 5. Ranicki, A. (1979) Algebraic Topology, Aarhus, 1978: Proceed- C BGLp,,(Q') -> A'(*) are compatible for various n and give ings, Lecture Notes in Mathematics (Springer, New York), no. a map from 51 x holim to A'(*). Moreover, 763, pp. 275-316. BCp,, 6. Loday, J.-L. (1976) Ann. Sci. Ec. Norm. Super. Ser. 4 9, (holim BCp,,) = (CPX7t 309-377. So we consider 7. Hsiang, W. C. (1983) Geometric Applications of Algebraic K-Theory (Int. Congress of Mathematicians, Warsaw). ) X n Cjn 8. Soule, C. (1981) in Algebraic K-Theory, Evanston, 1980: Pro- Trc(P S1 BZIP __,(Q (A[,,))Cln Trf) (QCp~n(j))Cj,n. [4.3] ceedings, Lecture Notes in Mathematics (Springer, New York), no. 854, pp. 372-401. Taking advantage of Propositions 4.1 and 4.2, one finds that 9. Waldhausen, F. (1978) Proc. Symp. Pure Math. 32, 35-60. the map 10. Connes, A. (1983) C.R. Acad. Sci. Paris Ser. I Math. 296, 953-958. (Trc(P ))*:H4mn+1(S1 x BCp,,; Zip") 11. Segal, G. (1973) Invent. Math. 21, 213-221. 12. Segal, G. (1974) Topology 13, 293-312. 13. May, J. P. (1977) E& Ring Spaces and E& Ring Spectra, Lecture -> H4,n+l(Q'(BCp,,); Zlpn) [4.4] Notes in Mathematics (Springer, New York), no. 577. 14. Dieck, T. tom (1975) Arch. Math. (Basel) 26, 650-662. sends the element L1 x v4m to a Z/p" summand by multipli- 15. Waldhausen, F. (1982) Algebraic Topology, Aarhus, 1982, cation with (q-2m - 1)Lp(1 + 2m, co-2), where Lp denotes Lecture Notes in Mathematics (Springer, New York), no. 1051, thep-adic L-function. The Z/p" summand in question is in the pp. 173-195. Downloaded by guest on September 25, 2021