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Hermitian algebraic K -theory and dihedral homology
Song, Yongjin, Ph.D.
The Ohio State University, 1990
UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 HERMITIAN ALGEBRAIC A-THEORY
AND DIHEDRAL HOMOLOGY
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy in the Graduate
School of the Ohio State University
By
Yongjin Song, B.S., M.S.
The Ohio State University
1990
Dissertation Committee: Approved by
Zbigniew Fiedorowicz
Dan Burghelea
Ruth Chamey Adviser Department of Mathematics ACKNOWLEDGEMENT
I like to thank Prof. Z. Fiedorowicz for his guidence and insight throughout the research. He is a great mathematician and has guided me for years with great kindness.
I express my sincere appreciation to Drs. C. Ogle and W. Gajda. Throughout the seminars with them I learned much from their insight and knowledge. I also thank the members of my advisory committee, Drs. D Burghelea and R. Charney, for their reviewing and comments. I Thank P. J. Prieto-Cox for helping me resolve some problems in typing this thesis. I express sincere appreciation to my parents for their support and sacrifice. To my wife Cheunsoon and my two children Jean Young and
Wonho, I offer sincere thanks for their encouragement VITA
August 7, 1958 ...... Bom - Seoul, Korea
1981 ...... B.S., Seoul National University,
Seoul, Korea
1981-1984 ...... Instructor, Dept, of Mathematics,
Korea Naval Academy
1985 ...... M.S. in Mathematics,
The Ohio State University
1984-Present ...... Teaching Associate, Dept, of
Mathematics, The Ohio State
University
FIELD OF STUDY
Major Field: Mathematics
Studies in Topology. TABLE OF CONTENTS
ACKNOWLEDGMENT...... ii
VITA ...... iii
TABLE OF CONTENTS ...... iv
INTRODUCTION ...... 1
CHAPTER
I. PRELIMINARIES
§ 1.1. Simplicial Objects, Cyclic Objects and Dihedral Objects 7
§ 1.2. Cyclic Homology and Dihedral Homology ...... 15
§ 1.3. Algebraic Af-theory of Simplicial R in g s ...... 20
II. VOLODIN HERMITIAN AT-THEORY FOR DISCRETE RINGS
§ 2.1. Karoubi's Hermitian algebraic Af-theory ...... 22
§ 2.2. Volodin Hermitian Algebraic Af-theory ...... 27
III. HERMITIAN VOLODIN CONSTRUCTION FOR SIMPLICIAL RINGS
§ 3.1. Definitions of Hermitian Af-theory for simplicial r in g s ...... 38
§ 3.2. Hermitian Volodin Construction for simplicial r in g s ...... 40
IV. THE MAIN THEOREM
§ 4.1. A Map from Hermitian Af-theory to Dihedral Homology ...... 47
§ 4.2. The Main Theorem ...... 52
§ 4.3. Reduction to a Special Case ...... 53 V. CALCULATIONS IN THE SPECIAL CASE
§ 5.1. Notations ...... 61
§ 5.2. Dihedral Homology Calculation ...... 62
§ 5.3. A Theorem for Homology Calculation ...... 65
§ 5.4. Calculation of £L/J+1(/) ® Q ...... 73
REFERENCES ...... 80
v INTRODUCTION
Waldhausen ([Wl]) extended Quillen's higher algebraic Af-theory to simplicial A A rings. For a simplicial ring /?*, Af,(/?*) := 7T,- BGL(/?*) for j ^ 1, where GL(/?„), which is a grouplike simplicial monoid, is a collection of connected components of all A matrices so that 7TqGL(R^) = GL(7^R+). Burghelea and Fiedorowicz defined
A hermitian algebraic Af-theory of a simplicial hermitian ring /?*: p ;(/?*) := KiBp{R^) ,
A where £0(R+) is a certain simplicial category (instead of a simplicial monoid of matrices). Given a space X, following Waldhausen, we define the hermitian algebraic Af-theory of X as p (X ) = ^(Z[G„X]), where G Jt is the Kan's free simplicial group of loop space and Z[GJC] is its group ring with the involution induced by the orientation co e H l(X; Z/2). As we define, following Waldhausen, the Af-theory
Whitehead space Wh^(X) we define the hermitian Af-theory Whitehead space Wh^(X) from the fibration sequence jLM A„x+ -*• *£,(*) ->• W h^m .
Burghelea and Fiedorowicz showed in [B-F2] that in some range Wh^ (M) computes the rational homotopy groups of H(M, 3Af)/Homeo(M, 3Af) for an even dimensional compact topological manifold M.
As A. Connes introduced cyclic homology in 1982, it was immediately noticed that cyclic homology of a ring R is closely related to algebraic Af-theory of R. Since then many mathematicians ( Burghelea, Hsiang, Staffeldt etc.) have obtained interesting results in this direction. Goodwillie in [G3] proved the following theorem which seems
1 to be the most general result. Theorem (Goodwillie, [G3]): Let / : R —► S be a map of simplicial rings such that the induced map tc^R —► tc^S is a suijection with nilpotent kernel. Then Afn(/)® Q = //Cn.j(/) ® Q .
This theorem may be approached from the following point of view: Given a ring R, Af„(/?) is isomorphic to the primitive part of the Hopf algebra H+{GUR)\ Q). Loday and Quillen showed in [L-Q] that HC+_X(R) ® Q = Prim Hj^gl (R ® Q); Q). While the relation between the homology of the general linear groups and the homology of their Lie algebras is quite complicated in general, there is a much closer relationship between the homology of their nilpotent subgroups and their Lie algebras. This is the basis of Goodwillie's proof. Loday and Procesi showed in [L-P] that ^ H D ^ R ) ® Q
= Prim H^(£(S)(R ® Q); Q), so it seemed natural to conjecture the analogous relation between hermitian algebraic Af-theory and dihedral homology. Krasauskas and
Solov'ev ([K-S]) proved that for a simply connected based topological space X, there is a natural isomorphism gL*(X) ® Q = ^ ( Z ) ® Q ® HD *(C„(£2Y; Q)). This is the analogous result to Burghelea's ([Bu]) which is in the case of linear algebraic A'-theory of spaces. The main result of this paper is the following:
Main Theorem. Suppose /: R — ► S be a map of simplicial hermitian rings such that the induced map n ji —► 7TqS is a suijection with nilpotent kernel. Then /,„(/) ® Q = HC~ ,
This is analogous to Goodwillie's ([G3]). Despite the additional complexity present in hermitian algebraic AT-theory the general line of the proof of the Main
Theorem is very similar to Goodwillie's. In the proof of Main Theorem we reduce the theorem to the special case of a split square zero ideal in a discrete ring. The hermitian
Volodin space of simplicial rings plays a key role in this reduction. This paper consists of four parts. First, the construction of hermitian Volodin space of discrete rings. Secondly, the construction of hermitian Volodin space of simplicial rings. Thirdly, the reduction of the Main Theorem to the special case.
Finally, the proof of Main Theorem for the special case.
In chapter II, we define Karoubi's hermitian Af-theory and construct the hermitian Volodin space for discrete rings. Let R be a hermitian ring (i.e., with an involution t: R —>R). The e-hermitian orthogonal group f l n n(R) is defined to be the group of automorphisms on (Rn) ® (Rn)* preserving the hyperbolic form Bp defined by Bp ((p ,f ), (q, gj) = f(q ) + e g (p) . p ^ i R ) denotes the subgroup of e- hermitian elementary matrices. Since hermitian elementary matrices satisfy hermitian Steinberg relations ([Ba], Lemma 3.16), eEn^l(R) is perfect. The commutator subgroup of /)(R) = ]ir^Onjl(R) is pE(R) = lit^ P„„(R) ([Ba], p. 31). The z-th hermitian Af-theory P fR ) is defined by P fR ) = ni B p (R )+, where + means the plus construction with respect to the perfect normal subgroup p(R). The e-hermitian triangular subgroup p ^ n(R) is defined to be T^(/?) fl Onjl(R), where
T%n(R) = {M e G L^R ) \ Mtj = unless i Volodin space X(R) = UBTa(R). i) p (R ) is acyclic ii) p (R ) is simple for z > 2 iii) TTj fX(R) = p (R). Hence Suslin's argument (cf. [Su]) works in the proof of the desired property of p ( R ) : p (R ) —► B fi(R ) —► B p (R )+ is a homotopy fibration sequence. In chapter HI, we construct the hermitian Volodin space for a simplicial (hermitian) ring /?*. Burghelea and Fiedorowicz ([B-F2]) succesfully defined hermitian Af-theory A A of simplicial rings. Pi(R*) = tt,- Bp>(R+) , I St 1, where B£0(R*) is the classifying 4 a '■*' r space of the simplicial catergory /)(R*) whose set of object is Syn^ n(R*)- A morphism M : A — ► B is a matrix M e GL^iR*) with M AM = B. There is a ~ F /s fibration sequence Syrrr(/?*)—► B^O(R„) — ► BGL(R*). We define the hermitian Volodin space = litwhere ^ ( F . ) := J n . n ^ is a certain subcategory of J9(F„). B J ^ J R * ) is identified with BJT^CR*) which is t t flJfX t n tfl the classifying space of the simplicial group £T^n(/?*). Hence ^R * ) is identified with ,X(R+) = liry U B ,T^n(Rt). We show that 7rleXX/?*) = (tTqF*) and ^ ( F , ) is acyclic. Our goal in this chapter is to show that F*) is a homotopy fiber of the map A A | BfliR*) — ► Bfl{R+) . Simplicial ring case is a little more difficult than discrete ring case, since ^(F*) is not simple for i ^ 2, so we cannot use Suslin's argument directly. A We resolved this problem by replacing F , by the 7-adic completion Fm = limF*//* , where F* is the free resolution of F* and /* = ker (F* —► ^F *). One of the key properties of the hermitian Volodin space J((R*) is that it is defined "dimensionwise." In chapter IV, we state the Main Theorem and reduce it to a special case. We construct two maps Oq(/) : ® Q —> H C*(f®Q) and Pq V) - Z /C r^ ® Q —► H C+ e£.,(«)®Q -----^ H + ( R »Q) In the construction of the map T, we use the following forgetful map F:C*BeO(F*) >C.BGL(R>) 5 (Mq—> M l — > - —4 Mn) I * (* —> * —> - -4 *) A _ where Ai e GL(R), e 5ym (/?). The proof that /Jq(/) is an isomorphism is relatively easy, since under the hypothesis of Main Theorem H P Jf ® Q) vanishes ([G3], Lemma 1.3.3). We show that proving that 0Cq(J) is an isomorphism under the hypothesis of Main Theorem is equivalent to proving it under the following hypothesis: / : R —► 5 is a suijective map of discrete hermitian rings such that the extension is split and kerf = I is a free 5-bimodule. The hermitian Volodin space plays an important role in this reduction. In chapter V, we prove the Main Theorem in this special case. This proof involves a lot of combinatorial calculations. We first calculate ^ n+ff) ® Q and HC~(f) ® Q separately and show that they are actually isomorphic via Potf)'1* 0Cq(J). It is relatively easy to show that HCT^if) ® Q = (pn\ I ® Q), where W^(pn; I ® Q) is the Hochschild homology //*(5®n; D(pn)), D(pn) denotes 7®” equipped with the 5®n-bimodule structure determined by the cyclic permutation pn. w*(Pn> / ® Q) is the coinvariants of the action of the centralizer of pn and W^iPn', /® Q) is its Z/2-invariants. For the hermitian Af-theory calculation we made Theorem 5.3.9, which is analogous to [G3], Theorem IV.3.2. We prove this theorem through a Lie algebra homology calculation with a help of classical invariant theory (cf. [L-Q], [L-P], [P]). We need some combinatorial computations in this proof. This theorem provides us with two facts which are essential in the hermitian Af-theory calculation: i) The quotient map A r£0(B )—► A r£0 ( B ) ^ ^ induces an isomorphism on H+(^G(S); -), where B is a free 5-bimodule, ii) Ur0^ is isomorphic to ^ VQ(p). We have the following diagram of fibration sequence I I B fiiS ) - > p (S ) Note that Hn(Fx\ Q) = A n£Q(/ ® Q). £LW+1(/) ® Q = 7tn(F2) ® Q is isomorphic to the primitive part of Hn(F2, Q). We show that //n(F2; Q) = Hn{Fx\ Q) = (A”eO (/ ® We analyze the coproduct structure of the Hopf algebra //n(F2; Q) = (y\”£0 ( / ® Q ))^ ( 5 ) and figure out the primitive part, which is isomorphic to W0(pn)N(p ) = Wq (p„; / ® Q) (Lemma 5.4.8). Hence we have p n+l(f) ® Q = ^ ( p n; f® Q). Finally we have that £L„+1(/) ® Q and HC~(f) ® Q are isomorphic via Pq(J)'1° (Xq(/) by the commutativity of diagram 5.4.10. CH APTER I PRELIMINARIES § 1.1 Simplicial objects, cyclic objects and dihedral objects. Definition 1.1.1 Let A be a category whose objects are [n] = {0, 1, — , n), n £ 0, and a morphism [m\ — > [ri\ is a nondecreasing function. A is called the simplicial category. A simplicial object is a contravariant functor from A to a category C which is usually a category of sets, topological spaces, abelian groups or rings etc. We denote a simplicial object X : Aop—► S by Xn denotes X([n]), which is called the set of ♦ n-simplices. g : Xn —► Xm denotes X(g : [m] —► [«]). Asimplicial map between two simplicial objects /:X „ —► Ym is a natural transformation from X+ to Y\. There is an alternative definition of simplicial object A simplicial object is a collection of objects {^n}n^o together with functions di : Xn —► X n l (0 < i < n) and si : Xn — ► Xn+l (0<,i<, n) satisfying the following relations didj = dj-idi if i siSj = Sj■J/.1 if i> j ' sj - i d i i f * < J ii identity if /= j, j + ja. •J? „ S jd i_l if i > j + 1 We call d- s face maps and i,-'s degeneracy maps. A simplicial map f:X ^ — > Y+ is a collection of maps f n: Xn—► Yn ( n ^ O ) which commute with face and degeneracy maps. 7 8 For a simplicial object X, a simplex x e X is called degenerate if x = 5,(x') for some x 'e X and i. Otherwise it is called nondegenerate. Definition 1.1.3 Let X t be a simplicial object. The geometric realization |x ,J of X is defined to be the quotient of x A" by the following relations: Let x e Xn and A : [ml — > [w] in A. Then (A* x, (tQ, — , tm)) « (x, A„(r0, - , rm)). We may think of [m] as vertices of Am, so A* : Am — > A” is a linear extension of A on baiycentric coordinates. IX+1 is clearly a CW-complex. We may regard a simplicial object X* as a topological space I Xm I together with a decomposition into simplices. 1.1.4 Classifying space of a category. Let & be a small category. Then we can form a simplicial set B m C , which is called bar construction (or nerve) of G: Let BqC = obj(2T). For n ^ 1, n-simplices Bn G is a set of all possible chains of morphisms of the form A0 A j ••• —£• An , At e obj(S), ctj e m or(C). The /'-th face map di deletes the t-th object and composes maps if necessary. The i-th id degeneracy map si replaces Ai by A i —► Ai . The classifying space B G of a category G is defined by B C = I Bm G \. B& may also be regarded as the space of commutative diagrams in £. In particular, for a group G, we may regard G as a g category with a single object * and morphisms * —► * for all g e G. Then we have BG =K(G, 1). 1.1.5 Multisimplicial object. Acontravariant functor X : Aop x Aop—► Miscalled a bisimplicial object in a category C. Given a bisimplicial set X, its geometric 9 realization is defined by that of its diagonal object Diag(X). There are natural homeomorphisms I Diag(X) | = | m \—> X mm I =\m h~> I n f—> X mn | | = | n I m h-> X mn I I (cf. [Q3], p. 10). More generally, a k-multisimplicial object is a contravariant functor from the product of ^-copies of A to U. Given a fc-multisimplicial object X, its geometric realization is also defined by that of its diagonal simplicial object Diag(X). Of course the several geometric realizations are naturally homeomorphic as above. 1.1.6 Two sided bar construction ([M2], [F], [B-Fl]). Let 2? be a Top-like category, e„ category of sets, simplicial sets or topological spaces etc. A monad is an endofunctor F : C — > 2T together with natural transformations f i : FF —>■ F and i]: 1 — > F satisfying associativity and unicity ([M2], p. 10). An object X in £ is called an F-algebra if there exists a map ev : FX —► X compatible with fj. and 77. An F-functor is a functor G together with a natural transformation A: GF —► G compatible with fi and Tf. In particular, F itself is an F-functor together with fj.: FF — ► F. Given a monad F, an F-algebra X and an F-functor G, the two sided bar construction B^(G, F, X) is defined to be a simplicial object in G whose n-simplices are GF^X. The face maps are induced by A: GF —► G , )J. : FF —► F and ev : FX —► X . The degeneracy maps are induced by 7]: 1 —► F . The two sided bar construction has the following property: i) X — ► B+(F, F, X) and £*(F, F, X) —► X ii) GX —► F*(G, F, FX) and 5*(G, F, FX) —> X are inverse homotopy equivalences. 10 Let be a simplicial monoid Then | A/„ | is a topological monoid. is called grouplike if is a group. If is grouplike then we have | | ~ Q | BM^ | (See for example [G3], 1.1.8). Given a grouplike simplicial monoid , we can construct a simplicial (free) group which is homotopy equivalent to by a two sided bar construction. The two maps Mm <----- B^(J, J, ------► B^(G, J, are equivalences, where J is the James (free monoid) construction and G is (based) free group construction (cf. [B-F2], [F]). Replacing a grouplike simplicial monoid Mm by a simplicial free group will play an important role in the proof of the Main Theorem. We can also replace a simplicial ring R* by a simplicial free ring BJJ, T, R J which is homotopy equivalent to /?*, where T is tensor algebra construction on J?*. 1.1.7 Let A™ = {f: [n] —► [m] I /preserves the order } = hom^([n\, [m]). Then A* is a simplicial set. It has clearly one nondegenerate m-simplex and I ATI = A m. A* is called standard m-simplex. We may regard a p-simplex of A£* as a (p+l)-tuple (a0, av •••, ap) of integers such that 0 £aQ <, — £ a m. We have I Ai I = A 1 = I = [0,1]. The set {un : pi] — ► 1 1 un(i) = 0 for all j} realizes 0 e [0,1] and (vn : pi] —► 1 1 vn(i) = 1 for all /} realizes 1 e [0,1]. Let f, g : X* —► Y+ be two simplicial maps. Then / and g are said to be homotopic if there exists a simplicial map (which is called simplicial homotopy) F : A i x X* —► L* such that F I QxX = / and F \ lyX = g . 1.1.8 Given a simplicial abelian group A*, we can define a chain complex Chm(AJ n by Chn{A^)=An, with boundary map d= . We also define a normalized 1=0 11 chain complex Nm(A J of C7t„(X*) either by Nn(Am) = An / (degenerate n-simplices) n or NfliAJ =j\ker{d^: An —► An_j). The functor is an equivalence from simplicial abelian groups to chain complexes ([D-P], 3.6). We also have H ^C hiAj) = Hj(N(A+)), which is naturally isomorphic to tt,- (AJ (cf. [La], VII.5.2). A simplicial space is called proper if all degeneracy maps arc cofibrations. Lemma 1.1.9 ([Se], Appendix A) Let f c X^ —► be a map of proper simplicial spaces such that each f n : X n —> Yn is a homotopy equivalence (resp. homology equivalence). Then |/J : | x j —>• I Y\ I is a homotopy equivalence (resp. homology equivalence) Lemma 1.1.10 ([W2], Lemma 5.2) Let X„ — > Y+ —► Z„ is a sequence of proper simplicial spaces such that the composed map X„ —► Z„ is constant. Suppose that for all n > 0 Xn —► Yn —► Zn is a homotopy fibration sequence and Zn is connected. Then Ix* | —► | T* | — > |z* | is also a homotopy fibration sequence. Taking the levelwise path fibration, we have the follosing consequence: Lemma 1.1.11 Let X^ be a pointed proper simplicial space such that Xn is connected for all n. Then we have I OX* I = Q | X„ I. Definition 1.1.12 A cyclic object is a simplicial object X* together with a cyclic /l+l operator tn: Xn —► X n satisfying r =id and 12 ^ + 1^,-1 if 1 * * * " - 1 fn + 1 ^n - 1 if / = 0 A cyclic set can be thought of as a functor A01* —► Sets , where A is Connes cyclic category (cf [B-Fl], [C]). The simplicial category A is a subcategory of A and any morphism [m\ —► [n] in A can be uniquely expressed as a • < « ’ where a : [m] —► [/i] in A and e Z/m+1 = Aut [m]. Suppose X is an Sl-space with S ^ X —► X, (z, x) I—► zjc. Then we can endow the total singular complex Sing„(X) with a structure of cyclic set. The cyclic operator tn is given as follows : Let o : An —► X be a singular n-complex. Then define tn o: A” —► X by tno (n0, — , un) = e2ntu°(7 (uv •••, un, u0). Conversely, if X* is a cyclic set, then |x j is an S^space. ([B-Fl], Proposition 1.4) Suppose X is a based space and R a ring. Let C„(QX; R) be the algebra of singular chains on the Moore loop space QX. Burghelea-Fiedorowicz ([B-Fl]) and Goodwillie ([G1]) showed that the cyclic homology //C„(C„(QX; R)) := HC^ipX; R) is naturally isomorphic to the S'1-equivariant homology of free loop space Xs H ^ (X s l; R) := H ^ E S 1 x i X5*). Definition 1.1.13 Let Dn = ( x, y | xn = y 2 = 1, y ’^ y = x '1 ) be the dihedral group and Qn = (x ,y \ x? = y 2 , y’xx y = x~l ) be the quaternion group. A dihedral (resp. quaternionic) object is a simplicial object X„ together with the action of dihedral group £>n+ 1(resp. quaternionic group Qn+i) on Xn satisfying d ^ x - x diA , ds x = x j for 1 < / < n diy=ydn.i , siy=ysn,i for 0 ^ i < n with dt : Xn —> X nA , : X n —> X„+1 . 13 Remark 1.1.14 As simplicial object and cyclic object, a dihedral object can be regarded as a contravariant functor AD°V —► 2T, where AD is the dihedral category (See [F-L], Definition 1.1). The objects of AD are same as the simplicial category A f A and each morphism is uniquely expressed by [m] [m] — > [n], where A e HomAQm], [n]) and / e Dm+l = Aut([m]). Following Fiedorowicz-Loday ([F-L]), simplicial, cyclic, dihedral and quaternionic objects are special cases of G*-objects, where G* is a crossed simplicial group. When G„ = C„ = {C,„}/j20= {Z//i+l}^0, a G*-object is a cyclic object and when G* = a G*-object is a dihedral object and so on. Aboughazi-Fiedorowicz-Loday gave a complete classification theorem of crossed simplicial groups (See [F-L], Theorem 3.6). Example 1.1.15 Let A be an associative algebra with unity over a commutative ring k. Let ZnA = A®n+1, the tensor product over k. Then Z^A is a cyclic ^-module with face maps, degeneracy maps and cyclic operators as follows: ( a 0, — , a ta i+l, ••• , a n) if r < i <> n - 1 (i) dt (aQ, - ,a n) = - (a na 0, a v - , a n. x) if i — n (it) (fl0» (a0, ”•, flj’, 1, , flfl) , 0 ^ i n (iii) tn (a0, - , an) = (an, aQ, - , an.x) Suppose that A is equipped with an involution t : A —>• A , a I—► a which is trivial on k and aB = 5 a . Then Z*A is a dihedral (hence a quaternionic) ^-module equipped with x (a0, - , an) = (-1 f ( a n, a0, - , anA) y (a0, - , an) = (_i)', 1.1.16 Cyclic bar construction. Let £ be a small category. Then we can form a cyclic set BT ( C7) whose n-simplices are circular chains of morphisms A0 A x — —► An -5 ^ i40 , Ai e obj'( S), ccj e mor( 2?). The face maps are composing morphisms and degeneracies are inserting identity morphisms, and Z/n+1 acts by rotating the diagram. We call BT(U) the cyclic bar construction (or cyclic nerve) of C. Suppose that & is a groupoid (every morphism is an isomorphism). Then the subset of BT{ 5) consisting of simplices satisfying an° 7i-f" °t) = 1 ^ isomorphic (as a simplicial set) to the ordinary bar construction BJi U). There is an obvious injection B^( &) — ► BT ( U) defined by ( a p a 2, - , a n) I—► ((^ - a n)'1, a v a2, - , a„). Suppose that 6 is a category with duality D : —> & such that Dop° D = id. Then B°+c( &) is a dihedral object equipped with the Z/2-action as follows: Let R be a ring. Then we can construct a cyclic abelian group Cyc^R; R) = Z^R (See Example 1.1.14) which is called the cyclic bar construction of ring R. We have a chain complex Ch^(Cyc(R\ R)) = C^(R\ R) (See 1.1.8) whose homology group is called the Hochschild homology. More generally, for an R-bimodule B, we can construct a simplicial abelian group Cyc(R\ B) with similar face and degeneracy maps to those of Cyc(R; R). The homology group of the chain complex Ch^(Cyc(R; B )) = C^(R; B) is also called the Hochschild homology and denoted by //*(/?; B). This homology is Morita-invariant (See [G2], p. 403). Let G be a group. Then we also can form a cyclic bar construction BcycG in an obvious way. Let Z BG and ZBcycG be simplicial free abelian groups generated on 15 BG and Bcy°G, respectively. There is a composition of two injective maps of simplicial abelian groups ZjBG —► ZB ^G —► Cyc(Z[G]; Z[G]) which is defined by (gv g2’ - (tei-s*)"1.*!. - ••••«„• § 1.2 Cyclic homology and dihedral homology. Cyclic homology and dihedral homology are associated with cyclic sets and dihedral sets, respectively. In this section we recall Hochschild homology and cyclic homology through [Gl], [G3] and dihedral homology through [L]. We try to follow Goodwillie's notation in [G3] as closely as possible. Let X be a cyclic abelian group, then following [G3], we define four homology groups Hm(X), HC+(X), HPJX) and H~CJJC). (In this paper we use the notation H~C+(X) instead of HC^(X) that Goodwillie used in [G3].) In particular when X = ZA, where A is an algebra over a commutative ring k, then //„(Z4) is the Hochschild homology H+(A) and HC^ZA) is the cyclic homology HCJfi). As in [G3], 1.3, define a double chain complex B+JX) by X q.p -oo < p £ q 0 q < p where Xn =Xn l degenerate simplices. The vertical boundary map _ _ q ' p b : B (.X) — > B _.j(X) is given by b = £ ( - l ) ‘d : . The horizontal boundary map J • 1=0 I A B '• Bp,q(-X) = x n Xn +1 = Bp-l,qW (n = is 8 iven by B = W i . N’ where N = 1 + x + — + V , x = (-1 )ntn . For -°o Then we have H„(X) = Hnt'" (X ) HCn(X) = Hn1J!°°(X) HP„(X) = H „T?’°°(X) H~Cn(X)=H„r~'°(X)- There is a diagram of chain complexes n V ' t 0 —> tJ ’°(X) —► T^'°°(X) —► Tl'°°(X) —► 0 This makes a diagram of long exact sequences — ► H C ^ iX ) -? -> H~C,(X) - > HP,(X) -► HC,_2(X) ( 1.2. 1) I* B H C .^ X ) Hm(X) where the lower sequence is the Connes-Gysin sequence ([Gl], n.2.2). Recall ([G3], p. 365) that when R is a ring (flat over Z) then we denote HC+(ZR) by HCm(R). When R is a simplicial ring (flat over Z), we may regard Z R as a cyclic simplicial abelian group. Then applying the functor Ch we have a cyclic chain complex Ch (7R ). For each cyclic chain complex X we can define the above four homology groups (cf. [Gl], ID). Each homology group is defined from the total complex of a triple complex i f ’^X) for each choice of - °o < a < fi< These homology groups satisfy desired properties, for example, (1.2.1). When X = Ch (ZR), R a simplicial ring, we call these four homology groups //*(/?), HC+(R), HP+(R) and H CJJR). If / : R —► S is a map of simplicial rings, then from the algebraic mapping cone of the map of cyclic chain complexes Ch (ZR) — > Ch (ZS) (cf.[Gl], p. 200), we construct the relative cyclic homology H C Jf) that fits into a long exact sequence —> HC.(R) HC.(S) —► HCJlf) —> HC^(R) —> X Suppose X is a cyclic vector space over Q, then denote by C*(X) the chain complex C*(X)=Xn / im ( l-(-l)nfn). Recall ([L-Q], Prop. 1.2.) that H ^cliX )) = HC,(X). We recall the Morita invariance (cf. [D-I], [Gl]). Two rings R and S are said to be Morita equivalent if there is an R -5-bimodule P and S-/?-bimodule Q such that P ®5(2 =R as /?-bimodules and Q ® R P = S as S-bimodules. Suppose that R and S are Morita equivalent and B is an /?-bimodule. Then we have an isomorphism on Hochschild homology //*(/?; B) = H^(S\ Q ® R B ®RP) ([D-I], Theorem 3.4.). We say that the Hochschild homology is Morita invariant. In particular given a ring R the matrix ring Mk(R) and R are Morita equivalent. Moreover, if B is an R-bimodule, //*(/?; B) = H JM kiR)\ Mj^B)). Applying the direct limit, the same is true for k = °o. The isomorphism Hm(R) = H+(Mk(R)) is induced by the trace map of cyclic abelian groups Z Mk(R) —► ZR defined by ZpMk(R ) * Z / The same is true for a simplicial ring R (See [G3], p. 368). Cyclic homology is also Morita invariant ([L-Q], Cor. 1.7.). Now we recall (cf. [L]) the definition of dihedral homology. Let k be a commutative ring with 1. Let A be a ^-algebra with 1 equipped with an involution t : A —► A , a I—► a which is trivial on k. Then ZA is a dihedral it-module (See Example 1.1.10) equipped with * (a0, - , an) = (-!)”(an, a0, - , anA) , . t . Sn(n+\)I2, - - - - . y (a0, - , an) = (-1) ' ( a0, an.lt ••• , ax), where x and y are generators of dihedral group Dn+l (or quaternion group Qn+l). Denote the bicomplex Bp q(ZA) by Bp q{A). The horizontal boundary map B : Bpq(A) —► Bp l q(A)(n = q-p) equals More explicitly, B sends n (a0, - , an) to 2 ( - l ) ,#,(l. fl|-, " . an, a0, - , aM) ([L-Q], Prop. 1.11). Suppose i=0 ^ e k. Then Bp q(A) splits into Bp q(A) = Bpq(A)+ ® Bp q(A)~, where ,+_ f Z q-PA+ if P even Z q.pA if p even Bm (A) = and Bpq(A)~= _ . y acts by 1 Z q.pA~ if p odd Z A + if p odd (trivially) on Z^4+ and by -1 on Z„A . Denote 7^’^(/l)+ = 1 1 Bn „ JA)+. Then n n a HP+(A) = Hnr,°°'~m * H C+(A)=Hnr,°‘'0(Af. In the similar way we can define the homology groups H~n(A), HCT(A), HP~(A) and H CT(A). We have a diagram of long exact sequences analogous to (1.2.1): 19 By a hermitian ring we mean a ring with (anti)invoution. We also can define as above the four homology groups IT^R), HCT(R), HP~(R) and H~CT(R) for a simplicial hermitian ring R with ~ e R. Let M be a dihedral ^-module (AD°p-k- module) and N be a codihedral k- module (AD-fc-module). Then we define the tensor product N to be For x e Nn, y e Mm, (ft e H o m ^ n ] , [m]), we have (x ® / ( y ) ) » ( T o r ^ (k, M). The quaternionic homology HQn(M) is similarly defined for a quatemionic /:-module M. Let M be a dihedral (hence a quaternionic) fc-module with 2 G k. Then we have HDn(M) = HQn(M) = HC+n(M) ([L], Theorem 2.5 and Cor. 3.8.). In particular, if M = ZA as above, A a L-algebra, HDn(ZA) = Tor^ik, ZA) is denoted by HDn(A), and we have HDn(A) = HQn(A) s HC^(A) s H„(A*+'/(l-x, 1-y), b) and _lHDn(A)=_lHQn(A) = HC~rl(A )s H n(A'‘+'/(\-x, 1 +y ),b ), where the last isomorphism holds if it is a field of characteristic 0 . We can define dihedral (and quatemionic) homology for a simplicial ring. Suppose R is a simplicial hermitian ring with | e R. Then ZR is a dihedral simplicial abelian group. Let P+ be the simplicial projective resolution of ZR. Then k P+ gives rise to a double chain complex. The homology of the total complex of this double chain complex is defined to be HDn(R). Applying Lemma 1.1.9, we see that HDn(R) = HC+n(R). We also have _xHDn(R) = HCT(R). 20 § 1.3 Algebraic AT-theory of simplicial rings. Given a (discrete) ring R, Quillen ([Q2]) defined the higher algebraic AT-theory by K((R) = jr. BGL(R)+ for i > 1, where + means Quillen's plus construction (cf. [Q2], [H-H], [Wag]) with respect to the perfect normal subgroup E(R) of GL(R). Recall the Volodin algebraic AT-theory ([Su], [G2]). For each partial ordering on {1, •••, n), define the triangular group T^(R) = [M e GLn(R) I Mij= 8ij unless /<;'}. Let T°(R) = linji T^(R). Define the Volodin space X(R) by X(R) = U BT^iR). Suslin showed ([S]) that X(R) —► BGL(R) —► BGL(R)+ is a homotopy fibration sequence. This implies that the Volodin algebraic AT-theory K^(R) := niAX{R) (/ > 3) is isomorphic to Ki BGL(R)+ = K^R). Waldhausen ([Wl]) defined the algebraic AT-theory of simplicial rings: Let /?* be a simplicial ring with 1. Let p : /?* —► be the simplicial map which assigns to each element of /?* its corresponding path component. The simplicial monoid A GLn(R+) is defined by the pullback GLn(R.) - > Mn{Rm) r r G L^tCqR,) —> M/J(^ 0R*) Let GL(R+) = lin^ GLn(A*). Notice that nQGL(R *) = GL(ntfl*). GL(R *) is some A collection of path components of Mn(R*). In particular Jti GL(R+) = n^M{R^) = A Minfin) for for i > 1 and the functor GL preserves equivalences. The i-th algebraic AT-theory of R* is defined by AT,(/?*) = Ki BGL(R*) for i > 1. 21 As in the discrete ring case we can construct the Volodin space for a simplicial ring R* (cf. [G3], p. 359). For a partial ordering cr on {1, •••, n), define the simplicial group 7^(/?*) whose p- th level is T^(R*)p = T^(Rp). Let T^iR*) = liry T^(R+). Define the Volodin space X(R*) by X(R>) = U BT°(R+) In chapter III we show (Goodwillie also mentioned in [G3]) that X(R*) —► BGL(R*) —► BGL(R.)+ is a homotopy fibration sequence. This is also a cofibration sequence ([H-H], Theorem 2.5). That is, BGL(R*)+ ~ BGL(JR+)/X(/?*). Following [G3], we may make a functorial definition: K(R+) .= BGL(Rt) /X(R*). Define K^R*) := K(R+). For any map f : /?* —► 5* of simplicial rings, define the relative theory Kfif) by Kt .j(fiber(/if(/?*) —> AT(5*))- CHAPTER II VOLODIN HERMITIAN AT-THEORY FOR DISCRETE RINGS In this chapter we define Karoubi's hermitian algebraic AT-theory and construct the Volodin hermitian AT-theory for discrete rings. We also show that they are equivalent. Throughout this chapter/? is a (discrete) ring with 1 having an involution t : R —► R. Denote t(a) by a for all a eR. Let e be an element in the center of R such that ee = 1. § 2.1 Karoubi's Hermitian Algebraic AT-theory Definition 2.1.1 Let M be a right /?-module. A sesquilinear form on M is a bilinear map B : M xM —► R such that B (ma, nb) = aB (m, n)b. Sesquilinear forms constitute an additive group Sesq (M) which has an involution sending B to B, where B is defined by B(mj, m2) = B(jm2,m 1) . The group of n*n matrices Mn(R) has an involution sending a matrix X to X*, where X* =lX. A matrix X is called e- hermitian if X = -£ X . Definition 2.1.2 Let P be a finitely generated projective right /^-module. Denote Hom^(P, R) by P*. Then P* is a right /?-module with (f • a)(p ) = a f(p), ♦ ♦ where a eR, peP, / eP. A hyperbolic form Bp on P ® P is a sesquilinear form defined by B p : ( P © P*) x (P © P*) > R B P ((p,f), 07 . *)) =f(q) + e g(p ) 22 23 The pair ( P © P*, Bp ) is called hyperbolic module and denoted by H(P). We now focus on a special case when P is a free (right) fl-module which we denote by Rn. Pick up a basis eV ',en for R n and the duAl basis e * ,—,e* for (/?”)*. Let p = Z ei ut e R n and / = Z e*bj e (Rn)*. Then we have / (p) = (Ze*bj )(Zeiai) = } bj(e*)(Zeiai) = Z bt a. It follows that the matrix associated with BRn is J n = ^ eJ ^ . Let* = (p,/) = (av - ,an , b v ••• ,bn ) and y =(cv - ,cn , d v - ,dn ) which belong to R ” © (R n)*. Then we have x*Jny = Z biCi + z Z a i di =Zi l b: lC: + e I S d:11 d: = / (q) + e g ip) = B Rn ((P>/)> (q> 8)) = B Rn y) Definition 2.1.3 Let Aut (H(R n)) denote the group of automorphisms of the hyperbolic module ( Rn ® (/?”)*, BRn ), i. e., Aut (H(Rn)) = { q>:/?"© (/?”)* —►/?"© (/?")* | We may identify A ut (H(R n)) with a subgroup of GL2n (/?) which we call the E-hermitian orthogonal group and denote by zOn n {R). We can easily show the following lemma: a{3 Lemma 2.1.4 A 2nx2n matrix g I in GL2n (R) belongs to zOnn{R) if and a p ' ■1 f S ep* only if — - * * y 8 U f a 'a P ' -I 8* ep Proof. It is clear, for ;;jc .I;;)- (??)*—JS »j y\ e y a Remark 2.1.5 Notice that the above group eOn n(R) is a special case of the general A -quadratic group GQ(R, A) in [Ba]. Here A is a form parameter which is defined to be an additive subgroup of R satisfying i) {a - to |ae/? } cv\ c{ae/f \ a = -e a } ii)a A a c A If we choose A to be A fnax = [a e/? I a = -e a }, then t On n(R ) equals GQiR, Amax). Moreover, if 2 is a unit in R , then the form parameter is unique, that is, = Amu: We explicitly find the commutator subgroup of t On n(R) and show that it is perfect and has a "Steinberg-like" perfect universal central extension. Definition 2.1.6 A matrix X in Mn (R) is called z-hermitian if X = - e X and e- * ^ot O' hermitian if X = - z X . Notice that O g j e e ° n if and only if 8 = (a*)'1. Let £En n(R) denote the subgroup of eOn^(R) generated by the matrices of the form a | n ( i o w e o ^ 1° U ’ lv Let Ejj(a) be an elementary n xn matrix having a at the (/ ,j)-cntry and zero at all other entries. Let & (a) = Ei}{a) - zE^a)* = E ^a ) + E -i-z a ) and E.J(a) = E^a) - zE ^af = Ei}(a) + Ejt(- e a ) Note that E^. (a) is e -hermitian and £ .e(a) is z-hermitian. Define E,,{a ) = I + £.(a) ij ij V y ' (/ and j are distinct). We now easily see that eEnjt (R) is generated by the following (e -hermitian) elementary matrices: /E. / I 0 I I £?. («) lij(a) = rij (a) = ij ^ .®(fl) i I 0 I J I y y Let A = (ae R\ a = - z a) and A = {a e R\ a = - z a). Notice that ru (a) is defined only for a in A and /i(- (a) is defined only for a e A. Now we have the Steinberg relations: Lemma 2.1.7 ( [Ba], lemma 3.16 ) 1. [rij(a),lkl(b)] = 1 ({i ,j ) n { k ,l) = 0 )) 2 . W a l l * (ft)] = Hjk (-e a b ) (i, j, and k distinct) 3. [r,y (a), Ijj (ft)] = H- (ab )ru (-zab a ) (i =t= j ) 4. [rjj (b), ljt (a)] = Hy (ba)lu ( z aba) (i * j ) E l. (a + ft) = (ajffy- (ft) (/ + j ) E2. [//jy (a), % (ft)] = 1 (/ * / andy * k ) E3. [Hjj (a), Hjk (ft)] = Hik (ab) (i,j, and k distinct) L I. lv (a + ft) = lij (a)/,y (ft) 26 L2. lij(a)lr s (b) = lrs(b)lij (a) L3. /0 (a) = ///(-e 5 ) L4- [/y(fl),/ / „ ( 6 )] = 1 (r*s,r*i,j) L5a. [/(y (a), // ^ ( 6 )] = (ab) (i, j, and k distinct) L5b. [lij (a), (b)) = la (ab - e a b ) (i * j ) L6 . [lu (a), (ft)] =1^ (a b )^ (bob) (i * k ) R l. rij(a+ b) = rij(a)rij(b) R2. rtj (a)rkl (b) = rkl (b)rtj (a) R3. rtj (a) = rjt (-e a) R4. [r|y (a), Hkl (b)] = 1 (k */,/*/ J ) R5a. [r- (a), H y (&)] = rik (-a b ) (i, j, and k distinct) R5b. [/-■• (a), Hjj(b)] = ru (-a b + ea b) (i * j ) R6 . [ru (a), Hh (b)\ = rik (-a b)rkk(b a b ) (i * k) Lemma 2.1.7 directly implies the following: Corollary 2.1.8 eEn n (R) is perfect, i. e., [ eEn n (R), t En „(/?)] = zEn n (R). Denote linji £En n (R) by eE (R). We now also figure out the commutator subgroup of eO(R). Lemma 2.1.9 [ eO(R), eO(R)] = ^E(R), that is, the commutator subgroup of e- hermitian orthogonal group is the e-hermitian elementary subgroup. Proof. See [Ba], p31. Note that A - Amax = [ae R | a - - E a }. Definition 2.1.10 Let eSt (R) denote the free group generated by the elements 27 f%j (a), 7tj (a), ly (a) modulo the relations given in Lemma 2.1.7. ^St (R) is called the e-hermitian Steinberg group. Lemma 2.1.11 ^St (R) is perfect and the canonical map t St (R) ---- ► t E(R) Xij I---- ► x-j (x y = H y (a), rtj (a) or I- (a)) is the perfect universal central extension. Proof. See [Ba], p. 50. Definition 2.1.12 We now define (Quillen-like higher) hermitian algebraic K- theory: we define the i-th hemitian algebraic K-theory ^LfR) by = k , b c O ( R ) + oai) Here + means Quillen's plus construction with respect to the perfect normal subgroup eE(R) (which is actually the commutator subgroup) of eO (R ). We denote B £0(R )+ by fL(R). Recall that this is an infinite loop space. Note that ^ ( R ) = zO(R) j zE (R) and ^ 2(R) = ker (£St(R) —► £E (/?)) § 2.2 Volodin Hermitian algebraic AT-theory In this section we construct Volodin hermitian algebraic AT-theory which is analogous to (ordinary) Volodin algebraic AT-theory, and show that Volodin hermitian AT-theory is equivalent to (Karoubi's) hermitian AT-theory defined in section 2.2. We construct hermitian Volodin space ^i(R) such that itt _1(£X_(/?)) = (R), the i-th Volodin hermitian AT-theory, for i > 3, and show that 28 J((R) —► B fiiR ) —* B £0 { R f is a homotopy fibration sequence so that =*;B/>(«)+ = *, .,(,*(«» = Ll- J m for i a 3. The mainstream of the proof resembles that of Suslin for ordinary Volodin space ([Su]). Definition 2.2.1 Let G be a group and {G^} be a family of subgroups. Denote by V(G,[Gk}) a (abstract) simplicial complex (and also its geometric realization) whose vertices are the elements of G and g0, — ,gp ( gi =*= gj) form a p-simplex if and only if all g'^gj lie in the same Gk for some k (i.e., each Gk acts on G by right multiplication). Denote by W(G,{Gk}) the geometric realization of the simplicial set whose p-simplices are the sequences (g0, ••• £ p ) of elements of G such that all g ^g j lie in the same Gk and the face and degeneracy maps are omitting and repeating respectively. Note that WXG.fG^}) = y B(G, Gk, *), where B(G, Gk, *) is a two-sided bar construction. The homeomorphism is given by (g0, - ,gp ) !—► (g0, g'Jgv g]lg2, - , g'pAgp )■ Define the Volodin space X((Gk)) to be W(G,{Gk))/G = y BG k . Lemma 2.2.2 Let W be a CW-complex and V be a simplicial complex. Let f: W —» Vbe a cellular map. Then if f X(A) is contractible for every closed simplex A in V, then f is a homotopy equivalence. Proof. Let Vi be the /-skeleton of V and let Wi = f l(V^. We show by induction that for each n, for every finite subcomplex of Vn, f : W%=fl(V^) —► V® is an equivalence. For n = 0, f : Wq —► V ^ is clearly an equivalence for each finite subcomplex of Vq . We assume, by induction, that / : W?= f l (V?) —► V? is 29 an equivalence for every finite subcomplex of Vi for 1 Xi+l= XiUdtf1 An c Vn. Then we can easily see that f l(Xi+]) —► Xi+1is also an equivalence from the following pushout squares (actually cofibration squares): / f@A) - > / * ( A") dAn A” * / *«+i f 1(dA) —► dAn is an equivalence by induction hypothesis. /* ( An) —► A” and and f l(Xj) —► Xj are also equivalences by our assumptions, s o /J(XJ+1) —► X i+1 is an equivalence. Hence every finite subcomplex of Vn is homotopy equivalent to f l {V^). Now we see that for each n Wn = = ^n *s 811 equivalence. Moreover, W = lirji Wn ► Uqi Vn = V is also an equivalence. Combined with the fact that for a nonempty set X the simplicial set whose p- simplices are all sequences (x0, — , xp) of elements of X (with the same degeneracy and face maps as above) is contractible, Lemma 2.2.2 implies that the obvious map 30 WXG.fG*}) —► VXGJG^}) is a homotopy equivalence. Deflnition 2.2.3 Let a be a partial ordering on {1,2,— ,2 n). A triangular subgroup T^n(R) of GL2n(R) is defined by 7?b(R) = (M e GL2n(R) I M y = Sy unless i < j } Then for each partial ordering a, the subgroup T^n(R) C \fin n(R) of £On n(R) is called e-hermitian triangular subgroup and denoted by ^ n(R). Remark 2.2.4 Let i,j, k, I e {1,2,— ,n}. Then we have a) if Hij(a)e £T ^ j(/?), then / b) if r-ia) e ^ n(R), then j < n + i an d i c) if /y(a) e gT ^ m(R), then n + i < j an d n+j d) if both li}{a) and r ^ b ) e ^ n(R), then either i) [i,j] PI [k,l] =0 or ii) k =i and i * j,k * I and n+j All generators and relations of Lemma 2.1.7 are contained in some e-hermitian triangular group. Thus the hermitian Steinberg group is the amalgamated free product of the e-hermitian triangular groups. We now define Volodin Hermitian A!--theory Definition 2.2.5 The i-th Volodin hermitian K-theory is defined by 31 The connected component of V {pnA{R), { ^ n(R)}) containing the identity matrix is n(R), { ^ n(R)}), since every e-hermitian elementary matrix can be expressed by a finite product of e-hermitian triangular matrices and conversely every finite product of e-hermitian triangular matrices belongs to £EnA(R). Moreover, * V(eEn,n(R>)' is ^ universal covering. We may generalize these as the following lemma: Lemma 2.2.6 Let G be a group and {Gk) be a collection of subgroups of G. Let H be the subgroup of G generated by Gjfs. Then i) V(G, {G^}) is the component of the vertex 1 is V(H, {Gk}). ii) Suppose the collection {G^} is closed under finite intersection. Then 7t{X( {Gk}) is isomorphic to the amalgamated free product *G^. . iii) The universal covering of X({Gk}) and hence also of V(H, [Gk}) is V( *Gk , {G*}). We now compare (Karoubi) hermitian AT-theory pfR ) and Volodin hermitian AT-theory ^ ( R ) . First note that / > ) = lhy = PW/PW = / / ( « ) A (*) = (.tor (*»„,„(«) — = ker ( / « « ) - * p ( R ) ) = PJR) A m = = ^.fi^W ^St^R), (e7'(* )))) for i a 3 The group p ^ ^ R ) acts on W(£Stn n(R), { ^ n(R)}) (which equals 32 y B (^tn n(R), [ J ^ n(R))t *)) by Geft) multiplication. This action is free and the corresponding quotient (coinvariant) space is ^ /?(*,{ },*) = I ^ B ^ T ^ n(R) which we denote by ^Cn^(R). Denote lir^ ^Cn Volodin space o f hermitian K-theory. Now we see that TtjQXXft)) = £ t(R ) and 7tM (J((R)) = ^ ( R ) for i £ 3. In order to show that ^(R ) = p(R) (= niB£0{R )+), it suffices to show that J((R) — > B p { R ) — ► B fi{ R )+ is a homotopy fibration sequece. First we need the following lemma: Lemma 2.2.8 gX( R) is simple in dimension £ 2, i. e., n x{^({R)) = p t(R ) acts trivially on ttf^CiR)) for i ^ 2 . Proof We show that tt^{^C(R)) = p ( R ) acts trivially on W = lii^W(£5tnin(R)> {£T ^ n(R))) up to homotopy, that is, for each x e p t(R ), the map x : W —► W is homotopic to the identity map. It suffices to show that the canonical inclusion is homotopic to yun for all y e £5'r/I+1/I+1(i?). Since eStn+l n+1(R) is generated by Hn+ij(a)’ Hitn+i(a)* rn + U ^’ we may assume that y is one of these. Then we have the following simplicial homotopy between un and y un : ((0 ,•••,(), 1 1 ), C^o> ■" I ^ y "" ’ ^ where (0 ,—,0 ,1,—,1) has s +1 0 's and t l ’s. 33 To verify that this homotopy is well-defined, we show that when x'^Xj e £T® n(R), x]](yxj) belongs to where a is an extension of or to { 1, —, 2 (n+l)}. The following calculations using Lemma 2.1.7 show this, i) if y = Hn+lj(a), then we have Hkl(b)Hn + u (a) if**/ H n+ uW H uib) = .Hkl(b)Hn + iti(a)Hn + h i(ab) if * = / ^ n +i , i W b ) = k ^ ) H n+ u(a) \ rU ^ H n + l,i(a) if I** H n + 1 t f r k P ) = [ rkl(b)Hn+i M H n+iM ab) if /=/ These equalities show that x'^yxj e / ” +](/l), where a is an extension of a to { 1, •••, 2 (n+l)} such that n+l / rlwrfrt-J "klW n + lM ) \f k*i *n+\AaWkl(b) - n+u Hkl(b)ln+u(a)ln+i,i(ab) if * = / '„ +U = w w « > rk l ^ ln + \ i(Q)H l n + l(£ if i = k > rk l ^ ln + \ i ^ Hk « + l ( 'e if i = l in+^ m = < rk l ^ ln + \ i(fl) if ** k ’ 1 rkl(W n+ 1 fC«)/n+i „+i( e aba)Hin+1(- eb a) if / = * =/ -l Hence x'; yx ; e S , ,(/?), where 2n+2 2 n + 2 <«+l+/ for all /= 1, —,n+l, / = 1, —,n. iv) if y = rn+i ,(a), then we have 34 Hkl(b)rn + lti(a) if / * / n+l,i' . Hkl{b)rn+xj{a)rn+i'k{-a if b) / = » rn+U ^ rkl^ = rd bK +\M lk l ^ rn + 1 i ^ Hn + 1 /(£ if * = *' * * / =< l k l( b>>r n +1 +1 *('e if ,V*’ * = 7 rn+l m m ® lk l ^ rn + \ /(fl) if ,V*' ,V 1 lkl(bK + 1 ifcOtf/,+1 i^ B + l n+l("£^ A) if7 = ■* = 7 Hence J/V*/ e e^n+1>n+1(^)* where rt+1 < i, n+i < 2n+2, I< 2n+2, n+1 < n+l+/ for all / = 1, — , n+1, / = 1, •••, n. Let a be a partial ordering on 2fl = {1, ”, 2n}. Then define oxm to be a lexicographical ordering on the set 2/1 x m. = {1, 2n}x[ 1, —, m) such that the order on m. = { 1, wi) is given by 1> — >m. Define an injection (Pj■: {1, 2/i} —► {1, 2n}x{l, m ) by (Pj(i) = i x j . Suppose <7 is an extension of a to in. = {1, —, 4n) such that cr= o on 2/t and if 0 O i A I ► A © 04*) \ where © means the diagonal sum. There is also an injection 7^1 : (R) * In,In 7 ♦ induced >>y «v Lemma 2.2.9 Ler <7^, •••, <7^ be partial orderings on {1, 2n}, pbe a natural number. If m is large enough, then the homomorphisms 35 H , ( U B eT \ R ) ) ► H: ( U (/?)) •/ i=l e n,n 7 ■/ i=l ~ 2nm,2nm induced by the injection 7^1 °/ arc zero fo r all j = 1, — , p. Proof. The proof is anlogous to [Su] p. 1564. The following claim implies this lemma through Mayer-Vietoris sequences and induction on m (See [Su] p. 1564). Claim 2.2.10 The homomorphism HXJ° (/?); F ) * (/?); F) J njt J fc 2nm,2nm induced by 7^1° / is zero for 1 <>j <> m and for any field F. We prove this claim by induction on m and Card(2n ). This claim is trivial for m = 1. Define yr 2 : 4lI —► 4n. x m by V2 = (P\ on min 4zt and y/2 = (p^ on 4r - min 4r . Let gj = 7^1°/ and g 2 = T ^ f . Then the images of gj and g2: ^T ^n( R ) ----- ► ^2nm2nnft^ commute- The homomorphism F ^ * H m^eT ^ 2 n m ^ ' F^ induced by M 2 is decomposed as Hm(gl-g2) = Hm(gi) © ^ m(g2)» since by the induction hypothesis the map ^,(eT^rt(F); F) —► ^ is zer° for 1 ^ ^ m"1, Now let u = n [e (-1) ® e (1)]. Then we have « , * 2 = (* 2)“((j>,')“° *). ie2n-mm 2n 2 1 1 2 where gj'= gj and n *s ^ obvi°us projection. Hence we have Hm(Sl) © H J g 2) = Hm{gvg2) = H j t g j * ) © X) Thus H m{g^) = Hfn(gi'')° Hm(n) = 0, since Card(2& - min2n) < Card(2n ). Corollary 2.2.11 ^(F) /j acyclic. Proof. Lemma 2.2.9 implies that the canonical inclusion n(R ) —► ^ ( F ) induces the zero map on homology. 36 Theorem 2.2.12 ^C(R) —► BeO(R) —► BeO(R)+ is a homotopy fibration sequence. Proof. Step 1: R) —► B^>t(R) —► BgSt(R)+ is homotopy fibration sequence. Let F be the fiber of the acyclic map B ^t(R ) —► B £ t(R )+ ( F is acyclic). Let JC(R) and F be the universal coverings of (X{R) and F, respectively. Then the following two sequences JC(R) — > ^iR ) — > B ^ t(R ) and F —► F —> B^t{R) are fibration sequences, since ^ (^ (F )) = gStiR) and ttxF = ^t{R ). There exists a map ^(R) —► F which makes the following diagram commute (up to homotopy) JC(R) —►B ^ t(R ) —► B ^ t i R f f i II II F -----> B ^ t( R ) ----- >fi£Sr(/?)+ since the composition map ^C(R) —► BgStiR) —► B£St(R)+ is null-homotopic. This follows from the fact that any map from an acyclic space to a nilpotent space is null- homotopic as can be seen by considering Postnikov decompositions and noting that ^(R) is acyclic and BgSt(R)+ is simply connected. Now consider the following diagram of fibration sequences JC(R) fXiR) B ^ tiR ) g i f i II F —► F —► B ^ tiR ) n x{ ^ ^ t { R ) ) = gStiR) acts trivially on kx J((R) for ik 1 and also on HfJ((R) ). Moreover n ^ ^ ^ t i R ) ) = fStiR) acts trivially on H-t F , since it acts through jti£B£St(R)+ which is trivial. Thus we can apply the comparison theorem of Serre spectral sequences (cf. [Me] Theorem 11.1 p. 355) which implies that the map 37 g : fX(R) —► F induces isomorphism on homology (Recall that (X(R) and F are acyclic). Hence g is a homotopy equivalence by the Whitehead theorem, so/ is also a homotopy equivalence by five lemma. Step 2 : R)—► BJ£(R)+ is a homotopy fibration sequence. Consider the diagram of fibration sequences: G — > B ^ 2{R) F \r j r j r ^(R) —> B^t(R) —> B^t(R)+ 1 I u H — ► B^(JR) —► fle£(/?)+ By the comparison theorem of spectral sequences, h -.B ^ iR ) —► F induces an isomorphism on homology, which means h is a homotopy equivalence since B ^ 2(R) ^ simple since , and so is F . Thus G is contractible, so the map R ) —► H is a homotopy equivalence. Step 3 : ^(R) — ► BeO(R) —► Bf){R)+ is a homotopy fibration sequence. Consider the diagram of fibration sequences: G — > R) —► F J iiR ) —► B ^ ( R ) —► B ^ i R f H — ► B eO(R) —► B eO(R)+ By the same argument as Step 2, the map ^ ( R ) H is a homotopy equivalence. CHAPTER in HERMITIAN VOLODIN CONSTRUCTION FOR SIMPLICIAL RINGS We first recall the definitions of hermitian algebraic AT-theory (for simplicial rings) of Burghelea and Fiedorowicz (cf. [B-F2]). Our goal in this chapter is to construct the hermitian Volodin space /?«,) and show that it is the homotopy fiber of the acyclic map B£0 (/?*) — > B £0(R+)*. § 3.1 Definitions of Hermitian AT-theory for Simplicial Rings Let /?* be a simplicial ring with an involution t : /?* —► R+. Let p : R * —► tTq/?* be the quotient map which assigns to each element of /?* its corresponding path component, p also denotes, by abuse of notation, the map which assigns to each matrix over /?* a matrix over TCq/?*. A Definition 3.1.1 We denote by £On „(/?*) the simplicial category whose objects are 2nx2n matrices A over R * such that A = E A* and p(A) = ® C for some C e GL2/l(tCqR*) and a morphism M : A —► B is a matrix M e GL2n(/?*) with M AM = ~ e ^ B. Let Syn \^(R *) denote the set of objects of the category „(/?*)• We denote by A A jO(R*) the simplicial category i l JX n(R+)- The operation of direct sum of matrices n> l A endows fi(R+ ) with the structure of a permutative category (cf. [M3], [F-P]). Thus the classifying space B fl(R * ) = IL 5 ^^(/?,) may be group-completed to be an 38 39 infinite loop space which is the plus construction. We denote R*) = B£0(R ^)+ = f2Ba(JLL B £On n(R*)). We define the i-th hermitian K-theory by A Remark 3.1.2 B £Onn(R*) is identified with the two-sided bar construction GL 2n ^ * ) * * ) • GL 2n ( R J a c t s o n S y tr£ ln ^ v i a 04, A/) I----- ► M*AM, where M e GL2n(/?*)» >4 e Sym£n(/?*). Notice also that fi(Syrr^ n(R *), GL^iR*), *) is homeomorphic to the Borel construction ~ £ A Symw n(R*) x^£ EGL2n(R+). We have a fibration sequence 2/i * ' (3.1.3) s7n£„(«.) —>■BcOn„(K,) — ► BGL2n(R.) Remark 3.1.4 The key idea of Definition 3.1.1 is that for a discrete ring R there is a canonical bijection between Sym^n(/?) and GL2niR)j£On n(R), where £On n(R) is the hermitian orthogonal group. We define a map S : GL2n(R ) >. Sym enn(R) A ■ - * * V o ) A Then S : GL2n(R)/£Onn(R) ----- ► Sym *n(R) is a bijection (here GL2n(R)/£On n(R) is the set of cosets). We have a fibration sequence SK n < * > BGL2n(R) which is anlogous to (3.1.3). We check that for discrete rings the definition 3.1.1 is equivalent to that of hermitian AT-theory defined in chapter 2. Let R be a discrete ring. Then we have W = Sy = GL2n(R)/£On n(R) 'XGL2n(R)EGL 2n(R) 40 = EGL2n(R)/fO nn(R) « BfinJR) A Here, of course, P n(R) is the e-hyperbolic category defined in Definition 3.1.1, and {O n n(R) is the e-orthogonal group. There is another direct proof of this. Let &n(R) be a subcategory of „(Z?) with a single object ^® EJ j and morphisms of Gn(R) are e-orthogonal matrices £Onn(R). Note that &n(R) is a skeleton subcategory of P n(R). Hence we have B £Onn(R) = B Zn(R) = B£Onn(R). Definition 3.1.5 There is alternative definition of hermitian Zf-theory (cf. [B-F2]). Define /^(Z ?*)to ^ a simplicial category whose objects are 2nx2n matrices A over Z?* ♦ ^ such that A = e A and p(A) e G L ^iK qZ?*) such that ^ n(R*) is a full subcategory of denote IL £0 2n(R*) by p (R * ). Then JA.R*) is defined to be n^l B £0{R .)+ = DBe( II B p 2n(R*)), and ^ ( R . ) := *. f i R . ) for /> 1. n£l Burghelea and Fiedorowicz showed ([B-F2], Lemma 3.3) that the two definitions A — are equivalent away from 2, that is, the inclusion p(R*) —► £0(R+) induces an equivalence on the basepoint component of spaces ^(Z?*) ® Z[^] —► ^(Z?*) ® Z[^]. § 3.2 Hermitian Volodin Construction for Simplicial Rings Notation 3.2.1 L et = >AlE M2 „<*•> I 4 = C‘( ? o ) C fOT S°me C * 7?»(R,)I C S y r f / ™ A -A We denote by ^T „n(R*) the subcategory of P ^ iR * ) whose objects are 2nx2n matrices in ^^(Z?*) and morphisms are M : A —► B such that M e ^(^* ) an<3 41 M*AM = B. Here T^(R*) is the simplicial group of triangular matrices whose p- simplices are T 2 = O V = 1 Ce G L 2n& p> I C iJ = SV unless '■ ? j 1 c «%»(«•) As in Remark 3.1.2, we can identify fi£T ° n(R*) with B(e/6^n(R*), T2^(R*), *). Thus we have a fibration sequence (3.2.2) T°(R.) ► BeT where S : 7;c (/?.) —► p/6 a (/?.) is defined by S(M) = M*(°TeJ) M . 2n t- n,n q j Remark 3.2.3 Note that both p/6 a (/?«,) and T CT(/?*) are defined "dimensionwise." t ntn 2jx can be identified with simplicial set T^(R *) / 7^(/?*)fi£On n(R *) = r 2> * > /e Tn°(R ^ whose p-simplices are T ^ (R p) / £T°n(Rp), where £Tan(R*) = (R^)C\£On „(/?*) is the simplicial e-hermitian triangular group. A Thus we may identify B^T ° n(R*) with the classifying space of the simplicial group £Tw^(/?*) by the same argument as Remark 3.1.4. (Both arguments in Remark 3.1.4 apply.) That is, Definition 3.2.4 (Hermitian Volodin Space) Define := UB/ °„(«.) (c B eO „ J R .)) = U f l j a ( s .) O c n,nK ' We denote lir^pX^ n(R+) by J l (/?*) which we call hermitian Volodin space. 42 The space ^(T?*) is defined dimensionwise, directly from simplicial groups s0 ^ *s very to Volodin space defined in chapter 2. is, in A most cases, better to deal with rather than JC (/?*). There is an obvious map ^(T?*) — *■ which is identified with the inclusion (/?,) — >B£0 { R*) . Notation 3.2.5 We define the siplicial ring /?* ® Z[^] by the pullback diagram 7?,® Z[|] —► /?*®Z[i] 'r y —► TTqT?* ® Z[“] Note that %(/?* ® Z[^]) = 7Tq7?* and ( R * ® Z[^])° = /?? ® Z[^], where ( )° means the basepoint component. As we did in chapter 2 to show that the two hermitian 7f-theories arc equivalent, A A A we are going to show that (/?*) —► Bp(R„) —>B£0 (R*) is a homotopy fibration sequence. However we need an assumption that /?*=/?* ® Z[^] to make A ] everything reasonable. In fact, ^(T?*) —* e^(^* ® ^ 2 ^ *s an ecluivtilence away from 2 (cf. [B-F2], Proposition 3.9). Suppose that /?*=/?* ® Z[^]. Let A*= {a e /?* I a = - e a ) and A* = {ae /?J a = - e a ). Notice that we may identify gT^T?*) with a collection of path components of (7?*)rx (A *)5 x (A*/ for some integers r, s and t with r + s + t < n(n- 1)/2. Hence we have 7r0eT ^(7?*) = er ^(tTqT?,), moreover we have n xB£0 (R*) = £0 (7^ * ) (cf. [B-F2], Lemma 3.8). Now it is easy to see that 43 by using the Van Kampen Theorem . Thus we have = * , 0 J Ber y . ) ) = *,(U = TTj^ nn(no^*) = hermitian Steinberg group. Hence we have *rleY(fl*) = lir^ <*„,„(/?♦)) = I115 A „(^*) = A V**)- Note that the image of nlfX{R+) = A ^ o ^ * ) in nxB£0(Rf) = £O(k0R„) is the commutator subgroup (hermitian elementary group). Theorem 3.2.6 Suppose that R+= /?* ® Z[|], 77ie/i ^(R*) —► B £0(R+) —>B£0 (R*) fr a homotopy fibration sequence. The proof of this theorem is a little more difficult than that in chapter 2. We cannot use Suslin's argument directly, since the hermitian Volodin space ^(Rf) is not simple for i ;> 2 in general. First we prove an analogous theorem in general linear case. Theorem 3.2.7 X{Rf)—► BGL(R + )—► BGL(R*)+ is a homotopy fibration sequence, where X(R+) = U BT°(R+), the Volodin space for a simplicial ring R^. Proof. The Volodin space X(Rf) is acyclic, since X(JRf)p =X(Rp) is acyclic for each p. First we replace Z?* by a free simplicial ring B(T, T, Z?*) which is homotopy equivalent to Z?*, where T(R+) is the tensor algebra construction on free Z-module on the set Z?*. Note that T is a monad. Let = B(T, T, Z?*) and k = tTqZ7^ = n^fR^. Let f be the kernel of F^ — ► it, i. e., f is the connected component of 0. Then we take Z^-adic completion = ljm F Jl J 1 of F^ and show that is homotopy equivalent to F *,(/.") = «,(/." , 3) = , d), for = /B*n by freeness s « ,((/., 3)*n) s ((w.(/,. 3))*"),. = o for all i £ n, since « 0(/.) = %(/,) =0 . Thus — ► Fjl+ is ^-connected. So F0 is (weak) homotopy equivalent to = F j l . n . Claim 3.2.8. GL(Pj = GL(Pj. Let A be a matrix in G L^Pj, then we can find a matrix B such that A B e ker (GL(Pj —>• GL(n)), i. e., A B = 1 - M, where M e = ker(P ^—> j i ) = l|m /*//*” . We have A 5(1 + M + M2 + ••• ) = 1. Since 1 + M + M 2 + — converges and (1 - Af)(l + M + M2 + — ) = 1, A has a right inverse. Similarly it has a left inverse, so A e GL(Pj. Thus GL(Pt) = GL(Pj. Now we are going to show that B G L (P j+ = | p I—► BGL(Pp)+ \ . Note that | p I—► BGL(Pp)+ 1 is an infinite loop space, since for a levelwise connected simplicial space X*, |l2X,J =* Q |x*|. Hence 7^(1 p I— > BGL(Pp)+ 1) is abelian, so by the universal property of the plus construction ([Wag], p. 350) there exists a map g : | p I— > BGL(Pp) | + ----- > \ p I—>■ BGL(Pp)+ 1 making the following diagram commute up to homotopy. 45 1 p |- > BGL(PD) | | p I—► B G L { P y I 8 I P I BGL(Pp) | + The canonical map | p I—► BGL(Pp) \ ■■■-■-> | p \—► BGL(Fp)+ | is a homology equivalence, since it is so on each dimension. Since both / and h are homology equivalences, so is g. Thus g is a homotopy equivalence, for it is a map between infinite loop spaces. X(Pp) —► BGL(Pp) —► BGL(Pp)+ is a fibration sequence for each p (cf. [Su]). Thus by fibration lemma (Lemma 1.1.10) -* I p I *■ BGL(Pp) -* I p I *• BGLiPp)* is also a homotopy fibration sequence, so X(Pf) —► BGL(Pf) —► BGL(Pj+ is a homotopy fibration sequence. This implies that A A X(/?*) —► BGL(R*) —► BGLiRfy is a homotopy fibration sequence. Lemma 3.2.9 fXiR*) is acyclic. Proof. This is immediate, for ^C{Rf)p = ^(R p ) is acyclic for each p (Corollary 2.2. 11). The fact that GL(Pj equals GL(Pf) also plays a key role in the proof of Theorem 3.2.6. 46 Proof of Theorem 3.2.6. The proof is analogous to that of Theorem 3.2.7. It suffices to show that B^O(Pm) = B f l ( P j , where B£0 (P t) = \ p I—► Bfi(Pp) \. . A/ Recall that n(Pj is a simplicial category whose set of objects is Sym£„(£*) and a morphism M : A —► B is a matrix M e GL2n(Pj = GLln( P j such that A A M AM = B. Let £„ be a (simplicial) full subcategory of „(r*) with a single object ^® ^. Note that the set of morphisms of Gn equals £On „(£*), the simplicial e-orthogonal group. Hence we have B G n = B£On n(Pj. Since every morphism of is an isomorphism, BG n is a connected component of the classifying space of a skeleton subcategory of e< ^ ( P,)- In fact, B£On n(Pj is connected ([B-F 2 ], Lemma 3.8). Hence B ^ ^ P j ~ BUn = B£On n(Pt). So we have B fiiP j ~ B£0(Pj. Remark 3.2.10 Since B £0 ( R*) — > B^O(R*)+ is an acyclic map, A , ^(T?*) —► B £0 (R*) — > B£0 (R*) is also a homotopy cofibration sequence, that is ^(T?,) = B £0(R*)+ = B £0(R*) /£X(/?*). In chapter 4 we will use the hermitian Volodin construction ^(Z?*). There we identify ^(7?*) =B£0(R*) with B£0(R*) /^(R*). CHAPTER IV THE MAIN THEOREM The main theorem is, roughly speaking, that rational relative hermitian AT-theory is isomorphic to rational relative dihedral homology under a certain condition. In this chapter we construct a map from hermitian AT-theory to dihedral homology and state the main theorem. We show that the main theorem in the special case implies the theorem in general. § 4.1 A Map from Hermitian AT-theory to Dihedral Homology Let R, S be simplicial hermitian rings and / : R —► 5 be a map of hermitian rings (i.e., commuting with involutions ). Assume that e = ± 1. In this section we construct two maps Oq(/) : ^(f) ® Q —► H C*(f® Q) and P q (f): H C jx(j) ® Q —► H C *(f ® Q). The Main Theorem says that they are both isomorphisms under a certain hypothesis. Thus ^(f) ® Q and H CJx(j) ® Q are isomorphic via «q(/)> Recall that ® Q s AHDiA(f) ® Q, the rationalized relative dihedral homology. We first want to define a natural map «(/?)®Q : eL| which makes the following diagram commute: 47 48 H C+(R* Q) A (« )* Q Q ) where T is called the Dennis trace map for hermitian AT-theory. We recall the definition of the Dennis trace map r : Ki ► Hi and its lifting a : K i ► H Ci ([G3], p 369 - 371). Let R be a discrete ring. Then there is a natural injection from the simplicial abelian group ZBGLn(R) to the cyclic bar construction Cyc(Z[GLn(R ) ] ; Z[GLn(R )\ ) of the group ring Z[GLn(R)\. It is given by (4.1.1) (X v Xp) I ►(X 1 - X pr 1 ®Xp where (Xj, — , Xp) e Z BpGLn(R). It induces a map t : C+BGLn(R ) ► N. Cyc(Z[GLn(R ) ] ; Z [GLn(R)] ) = T °^Z {G L n(R)} Composition of this map t with the standard trace map Tr yields a map (4.1.2) C+BGL(R) — T*°Z[GL(R)] T*°R, where Tr : T*’°Mn(R) ----- ► T*’°R is defined by x0® -® x p I ► ? X0(/0,ij) ® ... ® xp(/p,/0) j which induces an isomorphism on Hochschild homology ( Morita Invariance). The map (4.1.2), composed with the Hurewicz map, induces the Dennis trace map x x : K t R = 7T, KiR) ----- ► //. K(R) = H fiG L iR ) ► H^R) 49 Now we define the Dennis trace map for simplicial rings. Since GL(R) is not a simplicial group (it is just a simplicial monoid, so we cannot define a map like (4.1.1) A directly), we replace it by the two sided bar construction B(G, J, GL(R)) which is a A simplicial free group and homotopy equivalent to GL(R) (cf. [B-Fl], [I7]). Here J is the James construction and G is the (based) free group construction. The Dennis trace map for a simplicial ring R is induced by the composition of the maps: C+BGL(R) * ----- C+BB(J, J, GL(R)) ► C+BB{G, J, GL{R)) t J l i ’°Z[GL(R)] <----- t 2 ,0Z[B(7, J, GL{R))] ------► T^°Z[B{G, J, (&(/?))] Tr v Here the horizontal maps are induced by the simplicial monoid maps GL(R) ----- B(J, J, G L(R)) ► B(G, J, GL(R)) which are equivalences. Theorem 4.1.3 ([G3], Theorem n.3.1) For any simplicial ring R there is a natural map a making the diagram commute H~CfR) K,(R) — ^ H,(R) This theorem is an immediate result of the following lemma: Lemma 4.1.4 ([G3], Lemma II.3.2) For any group G there is a chain map a making the diagram commute 50 : r ° ° ’0z [G] C+BG 1 > 7 ^ ° Z [G] and a is unique up to chain homotopy. Now we are ready to define our Dennis trace map x for hermitian A'-theory and its lifting a. First we rationalize everything, and we are going to construct the maps T and a making the diagram commute ( r _oo’0Z[GL(/?)]®Q)+ ( r " oo’0(/?®Q))+ a n K c +b £o (R)®q — l - > ( t 2 ’°z [(2 ,(/?)]®q)+ - ► (r 2 ’°(/?®Q))H Here Tr is the standard (Morita) trace map. Notice that Tr is Z/2 - equivariant, that is, Tr commutes with the involution y which is defined by aQ ® flj ® — ® an —► (—l)n^”+1>/^ a0® an ® ••• ® ~ax Now consider the following diagram: T~°°'0Z[GL(R)]®Q -A- (7^°°'0Z[GL(/?)]®Q)+- ^ (7^°°-0(/?®Q))+ a K n n C*5GL(/?)®Q t2, 0Z[GL(/?)]®Q (7^°Z[GL(/?)]®Q)+- ^ (rj,0(fl®Q))+ 4 C*BfO(R)» q 51 Here F is induced by the map forgetting the objects ( A/j—► - Mn) I ► (* \ *) where A ,e GL(R), W, e Sym(R). The map p is the projection onto Z/2 - invariant and the maps a and t are as above (actually a and t are a(fl)®Q and t(R)®Q, repectively). The desired maps T and a are defined by t := t • F and a := a • F A Remark 4.1.5 For a discrete ring R, B £0 (R) is identified with B fl(R ) (here fi(R ) is a subgroup of GL(R)) via B eO(R) ------► B£0(R ) Hence for a discrete ring R we may regard the forgetting map F as the canonical inclusion. This means J is defined as in (4.1.1). Remark 4.1.6 (Analogous to [G3], n.3.3) We can also construct naturally x and a in the relative case: Let / ®Q : /?®Q----- ► S®Q be the map induced by / : R —► S. We have a diagram Q)®Q = H~C*(f ®Q) a (f» Q)®Q y f n 52 We now define a map Oq(/) : ^Li(f)®Q ► H C t(f ®Q) by the composition a(/®Q)®Q - + - + — ► M* Q ) ® Q — ^ H C t( f»Q)®Q = H Cj R —> R * Q I I S —► S ® Q Meanwhile, the other map Pq (/) ’• HCT^ij) ® Q ► H Ct(f®Q) is defined by H C ^tf) ® Q = H C T tf* Q) Q) where p(f® Q) is relative version of (3 in the Connes-Gysin sequence. § 4.2 The Main Theorem Theorem 4.2.1 (Main Theorem) Let f : R ► S be a map of simplicial hermitian rings with e = ±1. Suppose that the induced homomorphism nJR ► n0S is a surjection with nilpotent kernel. Then the map aq (f): /-,(/) ® Q -^>fTCjV®Q) ® Q is an isomorphism. 53 It is easier to show that fiq(f) is an isomorphism. The following lemma implies it. Lem m a 4.2.2 (Analogous to [G3], Lemma 1.3.3) Suppose f : R —► S is a map of simplicial hermitian Q-algebras satisfying the same assumption as Theorem 4.2.1. Then HP^(f) (resp. HP+(f)) vanishes, and hence (3(f) : HC~ (J) —> H C+(f) (resp. —► H C~(fj) is an isomorphism. Proof. HPJj) vanishes ([G3], Lemma 1.3.3), and HP^if) c HP+(f). The Main Theorem is equivalent to the following lemma: Lemma 4.2.3 If f : R —► S is as in Theorem 42.1, then 0Cq(f) : gL;(/)®Q —► H C*(f®Q) is an isomorphism. § 4.3 Reduction to a special case In this section we show that to prove Lemma 4.2.3 it suffices to prove it in a special case. The key lemma is Lemma 4.3.3, in which we need the assumption that R = R ® Z[^] and 5 = 5® Z[^]. This assumption is actually not a restriction in Main Theorem, for we deal with rationalized hermitian A'-theory gL*(/?) ® Q and we have an equivalence ^L(R) ® Z[^] —► J^(R ® Z[^]) ® Z[^]. Note also that (fl® Z[±])® Q = /?® Q . The reduction to a special case will be done in two steps. Lemma 4.3.1 (Reduction I) If Lemma 4.2.3 holds for every surjection (of simplicial rings) with square-zero kernel then it holds in general. 54 Proof. See [G3], p. 374 - 375. For a suijective map of simplicial rings with square-zero kernel, we show that the relative hermitian AT-theory can be computed "dimensionwise." For a simplicial ring R we denote by K'{R) a simplicial set whose p-simplices are K{Rp), algebraic AT-theory of discrete ring Rp. Define £>\R) similarly. K\R) and ,\R) are defined "dimensionwise." / ? Lemma 4.3.2 Let I —> R —► S be an extension of simplicial rings with I = 0. Then the square K\R) — ► AT(/?) I I AT'(S) —► AT(S) is homotopy-cartesian, i.e.,fiber{ \ K\R) I — ► | AT'(S) I ) —► fiber{ \ K(R) \ —► I K(S) | ) is a homotopy equivalence. A Proof. Using the fibraion sequence X() —► BGL() —► AT() and the fibration lemma, it suffices to show that both the following squares are homotopy-cartesian for each p : X{Rp) X (/0„ BGL(Rp) —► B(GL(R)p) \ X(SP) — ► X(S)p BGL(Sp) — ► B(GL(S)p) The square on the left-hand side is trivially homotopy-cartesian since the Volodin space is defined dimensionwise i.e., X(Rp) - X(R)p. By using the assumption I2 = 0 we have the diagram The vertical sequences arc exact sequences. Note that M(I) — *■ GL(R) —*• GL(S) is an exact sequence, where the first map sends M to M + 1. Thus the above square on the right-hand side is homotopy-cartesian. For hermitian AT-theory we have a similar lemma to Lemma 4.3.2. There is a map (L'(R) —► (f(R) induced by the inclusions B£0(Rp) = B(Sym(Rp), GL(Rp), *) C - * B(Sym{R)p, GL{R)p , *) = B£0(R) p Lemma 4.3.3 Assume that R = R ® Z[^] and S = S ® Z[^]. Let I —► R -A - S be an extension of simplicial hermitian rings with I2 = 0. Then fiber^fL'iR )]—► l^'CS)!) — >■ fiber(\^(R )\—► 1^(5)!) is a homotopy equivalence. A Proof. Using the fibration sequence ) —► B£0 {) —► ), it suffices to show that the following two squares are homotopy-cartesian for each p: p((Rp) J((R)p B£0(Rp) - > Bfi{R )p JCiSp) - > JC(S)p B eO(Sp) — B p (S )p The square on the left is trivially homotopy-cartesian, since the hermitian Volodin construction ) is defined dimensionwise. For the square on the right, it suffices to Sym(R ) —► Sym(R) I J Sym{Sp) —> Sym(S)p is homotopy-cartesian, since we have a fibration sequence Sym( ) —► £ £<9() —► B G L (). Recall that Sym(R) = {/I e M(R) \ A = eA* and p(A) = C* jc for some C eGLC^/?)}, and Sym(Rp) = [A e GL(Rp) IA = C* ^ C for some C e GL(Rp) } which is identified with GL(Rp)/£0(Rp) via GL(Rp)/p (R p ) ► Sym(Rp) A Ut S(y = M 6 M(/p I + oI)'4 = °)- Note that M(Ip)/S(Ip) —► Sym(Rp) —► Sym(Sp) is a fibration sequence, where the first map is induced by A I ► (A+1)*^ ^04+1) = A * ^ + + ( j o*) • where A e M(Ip) and 1 is the identity matrix. Note also that Sym(Ip) —► Sym(R)p —► Sym(S)p is a fibration sequence, where the first map sends a matrix M lo M + (® eJ Y We now have the following commutative diagram M(Ip)/S(Ip) - > Symdp) \r | Sym(Rp) —► Sym(R)p r r Sym(Sp) —> Sym(S)p 57 Here the top horizontal map is defined by A I ► (i E()) + (i E() )^ ‘ ^ 0te l^at this map is a (additive) group isomorphism. The injectivity is trivial and the surjectivity comes from the fact that 21 = 1 (since R =R ® Z[^] ). For the second reduction we need the following lemma: Lemma 4.3.4 (Analogous to [G3], Lemma III.2.2) Assume R =R ® Z[^], S = , f ~ S ® Z[—]. Let I — > R —► S be an extension of simplicial rings with I = 0. If, for each fp : R p—► Sp , the map cc^(fp) is an isomorphism, then cc^if) is an isomorphism. Proof. Let T+ be a chain complex. We define a subcomplex T„ by Tn = Tn for n > 0, Tn = 0 forw<0 and T0 = her (d :T0 —► 7 j). Then HnT+ = forn>0 and HnTm= 0 for n < 0. The diagram f^°'°(R ® Q )+ CL-> 77°°,0(/? ® Q)+ \r | 7^°°,0(S ® Q)+ 77°°,0(5 ® Q)+ induces an isomorphism Hn (Tf°°,0(R ® Q)+—► T~°°'0(S ® Q )+) H C+(f® Q). Now review the construction of the maps a and a . a is induced by a "weak chain map" ChfLBGL(R) 77°°,0(/?) T~°°'0(R) (See [G3], p. 378). The map a is induced by Ch£LBeO(R) ® Q ) Ch.(ZBGL(R )® Q ) — ^ f ® Q) Consider the diagram of simplicial Q-algebras: Z p(R ) ® Q = ZB fiiR )® Q ZBGLXR) ® Q = N''NZBGL(R) ® Q = N lChZBGL(R) ® Q Af ® Q) -^-£ - N _1 f ; 00*0^ ® Q )+ = f F(K ® Q)+ The first map is induced by the quotient map Bp(JR) —► B£0(R) /p(R) = p(R)- The map F' is induced by the forgetting map F. The functor N 1 is an inverse of the equivalence of categories N. This diagram, combined with the Hurewicz map p (R ) —► ZgL( R), gives a map of homotopy groups < A « ) « Q =t c j m ® Q ----- ► ittF(R ® Q )+ - H,TT\ r ® Q)+ —»• ® Q )+ = H~C*(R ® Q) (Recall that for any simplicial abelian group At, = HjCh(At)). This map is a. Likewise the map p t f ) ® Q ► //,• (f^°°'0(R ® Q )+— ► f~°°'0(S ® Q)+) is Oq(/) which is identified with the map of rational relative homotopy groups induced by the map of pairs Qt(R) - » jLtf)) -»•(F(R ® Q)*—► F(5 ® Q)+) To prove this Lemma 4.3.4 it suffices to show that fib e rO p m —► l/,(S)l) —► fib e rm R ® Q)+l —► IF(S ® Q)+l) is a rational homotopy equivalence. Recall (Lemma 4.3.3) that fiber(\p\R)\ —> \p'(S)\) —► fib e iV p m —► ^ (S )!) 59 is a homotopy equivalence. The hypothesis of this Lemma 4.3.4 implies that fiber(\J^Rp)\ -► y,(Sp)l) -+f,beH\F(Rp« Q)*l - * IF(Sp® Q )+l) is a rational homotopy equivalence for each p, so fiberi\^(R)\ —► I / , '(5)1) —► fiber{\F{R ® Q)+l — ► IF(5 ® Q)+l) is a rational homotopy equivalence. Hence fiber(\fl(R)\ —► 1/(5)!) —► fiber(\F(R ® Q)+l —> IF(5® Q)+l) is a rational homotopy equivalence. Now we are ready for the second reduction which is analogous to [G3], p 376 - 377. Lemma 4.3.5 (Reduction II) Theorem 4.2.2 is true if it holds in the special case when R and 5 are discrete, the extension is split, and I is a free S-bimodule. Proof. We may replace 5 by a free simplicial ring B(T, T, 5) which is homotopy equivalent, where T(S) is tensor algebra construction on 5. Let S = B(T, T, S) and R is defined by the fiber product R -A* S I r / R S By the homotopy invariace of £L () ® Q and H Cf( ) ® Q, it suffices to show that aQ(f) is an isomorphism. Thus we may assume that S is free. By Lemma 4.3.4 it suffices to show that for each fp : Rp —► Sp, 0Cq(fp) is an isomorphism. Each f f extension R Sp splits, since Sp is free. Thus now we may assume that R —> S is a split extension of discrete rings with I = 0, where I = kerif). Note that if 72 - 0 then I is an (/?//)-bimodule. We now show that I also may be replaced by a simplicial free 5-bimodule. Let A be a set, and 5 be a ring. BS(A) is 60 defined to be a free S-bimodule on A {i.e., typical elements are finite sums. Z) siajSjc). Then Bs is a monad. Now we can form a simplicial free 5-bimodule /* = B(B$, Bs, I) with I/* I = I. Set Rp = Ip @ Sp, then we have a new extension /* —► R+ —► S„, where 5* is a constant simplicial ring with Sp = S for all p. I/* I - I implies that I /?* I -R. Applying Lemma 4.3.4 again to this extension, we are done. CHAPTER V CALCULATIONS IN THE SPECIAL CASE In this chapter we calculate ^>n+l(f)^ Q and HCT(f)® Q separately in the / special case in which I —► R —► 5 is a split extension of discrete hermitian rings with / = 0, and / is a free 5-bimodule. We show that they are isomorphic via 0Cq(J)'1 ° Pq(J). Recall that HCT{f) ® Q is isomorphic to the relative dihedral homology HDn(f) ® Q. The stream of the proof is similar to that in [G3]. Because of the nature of our situation the calculations in this chapter are a little more complicated. We use the classical invariant theory (cf. [L-P], [We], [P]). § 5.1 Notations. Let V be a vector space over Q. Let G be a finite group acting on V". Denote by VG and the coinvariants and the invariants of G-action on V, respectively. These two vector spaces are isomorphic. In particular, if G = Z/2, i.e., if there is an involution i : V — > V, then V splits into V+ ® V , since ^ e Q. — v+vv-v v+v + Denote /(v) by v. For all v e V, v = —^— + —2 — ’ w^e r e — 2— e ^ anc* V - v — + //2 —2 — e V . V equals V ' which is identified with Vzj2. Let 5 be a discrete ring with involution. Let B l t - , B r be 5-bimodules. We can endow B j & ••• ® Br with a 5®r-bimodule structure as follows: For any permutation p e we define 61 62 (Sj ® ••• ® 5r)(Xj ® — ® xr) := SjJCj ® ••• ® srxr si e S, xi e Bi (*! ® - ® xr)( 5 ! ® - ® sr) := x xsp(1) ® - ® x lSpir) s( e S, xt e Denote 7?j® "®7?r with this S^-bimodule structure by Dip; Denote it by Dip) for simplicity if there is no confusion. We have an involution t : D(p) —► D(p) given by t : jCj ® ••• ® xr I ► (-1 )r We can similarly define an involution on 5®r. Denote the Hochschild homology //^(5®r; Dip)) by WJip). Note that WQ(p) is the quotient of B x ® — ® Br by the relations x x ® — ® xp ® — ® xr ~ x x ® ••• ® sxp(l) ••• ® xr Denote the class of JCj ® ••• ® xr in WQip) by [jCj, •••, xr] . Each permutuation X e Er yields an isomorphism W+ip) ■■■- > W^iX'lpX) which in WQ(p) is given by [xv - , xr]p I ► sgn{X)[xm , - , xKr)]kApX Thus the centralizer C(p) of p acts on W+ip). Denote by W+ip) the coinvariant of C(p)-action, and denote the class of [jclt — , xr]p by O j , ••• , x r]p in W^ip). § 5.2 Dihedral Homology Calculation. / Suppose that / —► R — > S is a split extension of discrete hermitian rings with ___ 7 = 0 , and / is a free S-bimodule. Since 7/C7(- ® Q) = 77C* (-) ® Q, we may assume that /? = 7? ® Q, S = S ® Q and 7 = 7® Q. The splitting of / implies that HC+iR) = 77C*(S) © HC++l(f). Regarding R as a graded ring with grq/? = S, grxR = I, gr^R = 0 for i £ 2, the cyclic homology group is graded 63 //C*(/?) = (Sgr^C .iR ) with grJhfC+iR) = HC*(S). Goodwillie showed in [G3] that HC.(f) = © W^.r(pr; I), where pr is the cyclic permutation (1 ,2 ,-, r). Since / is a free S-bimodule, D(pr\ I) is a free S^-bimodule; hence we have HCn(f) = WQ(pn ; /). Notice that grnHCnA(R) = HCn(f). The isomorphism 0 : w 0(P n'r> 8rnHCn-\(R) = HCrtf) is explicitly given by U p •" , ----- * ( x \ ® ) xt e I c R where x x ® ••• ®jcn e C* is the the image of xx ® ••• ® e Zn_jR = /?®n and { } denotes homology class in H ^ c f a ) = HCnA(R). Recall that the involution on Zn ]R is given by y : jtj ® ••• ® xn I ► (-l)n('n' 1)/2 x x ® xn ® — ® x2 and we have endowed D(pn ; I) with an involution defined by t : JCj ® — ® xn I ► ( - i ) ^ 1)/2 xn ® Jt^.j® - ® xx Now we have a commutative diagram The involution t maps [jr,, ••• , x ] n to Hn [(.1)»f»+l>2 ^ X|)^ = [(.1)^.+ 1^(.1)«-1 Xi> ^ Thus the above diagram commutes as follows: 64 -y [ - ( - l ) ^ " ‘,)/2 Jtj, *2lo --1------“ { (-1)'I<’"'U/2 ® ® " ® *2 } ^n Hence we have an isomorphism on Z/2 - invariants (5.2.1) W+Q(pn ; /) - | > grnHC~_x{R) = where Wj(p„ ; I) := D (p „ ; 0 ) ^ ) - The isomorphism and expressed explicitly by •Ft 'FI I 0. ► { Xx ® ••• ®Xn } - { ( . l )n('n-1^ 2 xJ ® ^ ®... ® ^ } The map B in the Connes-Gysin sequence maps the latter to (5.2.2) {£(-1)("+1)('‘-1)1®jci® - ® X: ,} j= l J - {( - i) n(rt' 1)/2£ (_i)(n+1)(n-;+2)i(gi x :,® - ® £ } j= l J J = { 21 (-l)(n+1)(>-1)l® JC.-® - S * : , } y=l 1 J' + { ( - l ) " ( n+1)/2 £ ( - l ) ( " +1)0 -l)l® X: J® - ® X: } y=i e g r ntt(R ) (cf. [L-Q], Prop. 1.11) (Note that — - + (n+l)(/-l) = 1 + —^ ^ + (/j+l)(n-y+2) mod 2.). This will play a key role in the proof of the commutativity of diagram (5.4.10). 65 We have shown that HC~(f) = W^(pn ; I). In section 5.4 we will calculate eLn+i(f) ® Q which turns out to be isomorphic to Wg(pn ; I). § 5.3 A Theorem for homology calculation We show in this section that the natural suijection A r£0(B) —► (A r£®(B)) induces an isomorphism on HA£0(S), - ), which will play a key role in the hermitian AT-theory calculation in section 5.4. First let us define a Lie algebra £©(jB) which corresponds to £0(B). ( f ep* a / T f Definition 5.3.1 Given a matrix e Mln(B), denote — * * by y 8 ey a y 8 Recall that the hermitian orthogonal group f i nA{B) is defined by £On n(B) = [M e Mln{B) \ MM* = MUM = 1} We define £©„„(#) by = I M e m 2 I -M*= M } = [ {y Pa* y MlniB) ^ P = ~£P*' 7= ~ £ Y' ] = [M e M2n(B) | M* ^ ) + ( j o = 0 } i.e., e®nn(B) is the invariant of Z/2-action M I—>-M** on M2n(B). £® (B ):= lij$£ We now define hyperoctahedral group and refer to the classical invariant theory (cf. [L-P]). 66 ♦ * Definition 5.3.2 Let 5 2m be the permutation group on {1, ••• , m, 1 ,••• , m }. The hyperoctahedral group H2m , which is a subgroup of S2m, is defined to be the semi-direct product (Z/2)m X Sm, where Sm acts by the same permutation on the i's and /*’s, and (Z/2)m is generated by r\i ’s each of which permutes i and i*. Hence Hm is the subgroup of S2m whose elements commute with *. Remark 5.3.3 The Classical Invariant Theory, (cf. [L-P], [P]) We denote by Q[S2njH m f S2/n-module whose underlying space is Q[52ot///ot] and on which the action of S2m is multiplication on the left if e = 1 and multiplication on the left times the sign of the permutation if e = -1. The main fact in the classical invariant theory which we arc going to refer to is that there is an H - equivariant isomorphism (gKQfn)^ (Q) ► Q lS2m/Hmf (cf. [L-P] Prop. 4.1) In particular for each ere 'Lm = Aut {1, ••• sn) c Aut {1, ••• , m, 1 ,••• , m ^ - S2m the corresponding linear map = W ) ^ = ((«/(Q ) o(q))(z/ 2/ = ( (s «Q >% (Q)® <^>W )' = (Q[S2JH,) ® where A e S2fjHr runs through a system of representatives for orbits of (Z/2)r-action on S2rjH r and 5(A) is the stabilizer of A , that is 5(A) = {77 e (Z/2)r | 77 A = A , i.e., A- 1 77 A e / / r } (Z/2)r acts on Q[52r///r] by multiplication on the left, as we regard S2rjHr as a set of all right cosets. (Z/2)r actson < 8 >ZL as follows: if 77. is the y-th generator of (Z/2)r y=i J J , then rjj : x l ® ® xr I—► x x ® ••• ® -Xj ® ••• ® ;tr . In (5.3.4) we may choose the representatives as A e £r, where I,r = Aut ({1, - ,r}) That is, for all ere S2r there exist re (Z/2)r and A e Zr such that xoHr = XHr . Moreover for two single cycles Aj = (iv •••, ik) ,7^ = (jv '- ,jfi e Zr , TAj//r = h1Hr for some x e (Z/2)r if and only if Aj = A 2 or Aj = A^1. Let Tr c £ r be a set of the representatives. Then (5.3.4) = $(.® Sy)s(J) , A A where L(A) := ( 77,-, T]kril e (Z/2)r | A(i) = t, t—► / 1—► k ) c (Z/2)r. Note that the stabilizer of A e I r is L(A). Thus we have (5.3.5) (,®£©(B>))£0(Q)£ ©(.®By)tW). Note that the latter does not depend on the choice of representatives. The projection of the isomorphism (5.3.5) onto the A - factor is given by (5.3.6) X t e - ® X m I— » S {^(i,, iw ) ® - ® Xr(ir ;„„) ) ^ Notation 5.3.7 Define VQ(p ) to be the quotient of WQ(p) = / / Q(5®r; D(p)) by the relations 68 i) ••• jcr]p ~ - , xr]p if p(i) = i. ii) [xl,-pci,‘-pcj , ••• rxr\p ~ [xlt— -,-X j,-,xr]p if (i,j) belongs to a disjoint cycle expression of p , i.e., i j i. Denote by [[jtj, •••, xr]]p the class of [jCj, — , xr]p in VQ(p). Note that W0 ^ L ( P) = W Notation 5.3.8 Let B be an S-bimodule. Let A e £G(B) and M e £0(S). Then note that M 'lAM e e©(#). Let i f = .®£©(By). £0(S) acts on Ur by conjugating in each factor. Define the trace map tr : i f > © WQ(p) whose projection onto p- factor is given by X l ®...« Xr I— Z lp(I)), . x r(ir, ipW)]p Define a map 0 : I f ► © VQ(p) to be the composition of two maps peYr u r W where the second map is the natural quotient map and Tr is a set of representatives defined in the construction of the isomorphism (5.3.5). Note that the map 0 is S)- equivariant, where ^0(S) acts trivially on ^ VQ(p ). Now we are ready for the following main theorem of this section. Theorem 5.3.9 For any ring S and any free (5 ® Q)-bimodule Bv -,B r the map 0: I f —► ^ VQ(p) induces an isomorphism ; U') ; ffi V0( p ) ) . P*lr Proof: It suffices to show that 0 induces an isomorphism on Lie algebra homology Hn(£0 (S ); - ) (See [G3], Theorem IV.3.2). Recall that for a right £©(S)-module V, the Lie algebra homology //„(e©(S); VO is defined to be Tor+£<^^(V , Q), where 69 U£0(S) is the universal enveloping algebra of £©(S). The standard chain complex for computing the Lie algebra homology V) is the Koszul complex C .(p {S ),V ) which is defined by C„(e©(S); V) = V 0 A"£©(S). (cf. [G2], p.388) By Proposition 6.4 of [L-Q], we have an isomorphism tf„(£® (S ); If) = H„(C.(e®(5); U ' ) ^ ) Now we are going to analyze the chain complex C*(£©(S); I f ) Above all we define an r-multisimplicial vector space V(p) = V^(p; Bv -, Br) by V n (P) = B i ® ® - ® Br ® (See [G3], p. 383) Then we have H,Ch(V(p)) = HmCh(Diag(V{p))) = WmQr, Blf- ,f lr)f since Diag( V(p)) may be identified with the cyclic bar construction C„(£®(S);If )^ (Qj = ® (CA, V(p))t(p) ® C.(£© (S); Q)£0(q) T/i/5 is not only a graded vector space isomorphism hut also a chain isomorphism. Proof of Claim. By the same reason as 5.3.5, we have (If 9 p a n 0(Q) ■ ® ((;® s , r+ii where L(X) = ( 77,., T]k77, e (Z/2)r+n I A(i) = 1, k I F * it >. This isomorphism is explicitly given by (5.3.11) {X, ® - ® X m} I ^ { E x ,! / ,, i1(1)) ® ••• ® Xr(/r, (1W) ) * We have C„(£® (S );(/)e0(Q)= ((A ® A"£®(X))£0(Q) = (( ««Q)® = ( Q I V « / Wr+n (5.3.12) = f «®«, ) ®S ° \ a , where A e ^(r+n/^r+n 11105 through a system of representatives for orbits of (Z/2)r +n>3 Sn action on S2(r+n/Hr+n a°d S(A) 1S the stabilizer of A, that is 5(A) = {ere (Z/2)r+nX Sn I a A = A , i.e., A ^ c A e / / r+/J). Now we may choose the representatives as A e £r+n , where A e I,r+n = Aut ({1, - ,r+n)) cA ut ({1, - , r+n, 1*,- , (r+n)*}) = 52(r+n). Moreover, such A is determined by a permutation p e Tr c l f and the integers [pj > 1 | 1 Ak(j) e {1, - , r) and p(J) = A?>+1(/), that is, (5.3.13) A : y I > r + \+ ^ p k \—► r + 2+^Cp^l—► ••• |— I—► p(/) n for all y'=l, — ,r. Hence Ae I, x l where £p.=p and p + q = n. p H . 1=1 7 The group L(p) = ( 7]^ , e (Z/2)r | p(0 = /, /: A ) acts on ( <8Bj ) 1—> ••• ® -xi ® ••• ® jcr ® — ® (-s,-^®—® -£,• t) ® — ® (^rj® —® ^ ) ii) TjiJil acts nontrivially only if pk =pl = 0 as follows: 77* 77, :*,® -®jfr® ju -®jr^ |—► Xj® ••• ® -xk ® ••• ® -X[ ® —® xr ® Sj j ® ••• ® S r p Now we have (analogous to [G3], IV.3.8) 71 (5.3.14) It remains to show that (5.3.14) is also a chain isomorphism. We arc going to write down the inverse map explicitly. Consider (at) ® m e Chp(V(p))L(pj ® C?(eO(S); Q)£0(Q) where x = jtj® (r, ••• ) ® ••• x r ® (sr j® ■■■ ®srp ) and Y = (II y,l - I Yq) e Cq(£(S)(S); Q ) , Yk e £©(S) with k = 1, We are going to find z e C„(£0(5); Ur) such that [z] corresponds to (jc) <2> (y) under the isomorphism (5.3.14). Choose r + p distinct natural numbers { Ijj | 1 zero lj ,-th row and column for every Ijj. Let E = E(a\ (ij)) be an elementary matrix in M(B). Then E®E* = E(a-, (ij)) © E(-a- (/,/)) is an elementary matrix of e©(£), where © means the diagonal sum. Now set Xj = E(Xj ; (lj0, fa)) © E(-Xj ; (ljv lj0)) , 1 and Sjj = E(sjj ; (/y/y l+1)) © £(-Sy ,•; (/y /+1, / y , ) ) , 1 z = (X1®-®A-r|5u l - ISr.p ll',1 - |r 4) € C„(£©(5);Ur). If A e Er+/|, then we apply the map (5.3.11), then the image will be zero unless A is of the form in (5.3.13). If A = /zU which is identified with {*} <8> {K} e Chp( v '(P))L(p) ® Cq(eO(S)\ Q ) ©(qj- 72 Moreover d{z] corresponds to d([x) ® {L}) by the same calculation as that in the proof of [G3] theorem IV.3.2, so the graded vector space isomorphism (5.3.14) is a chain isomorphism. Now we have H„( £©(S); (A) = (£).'Hp(ChV(p)) ®W f (t® (S );Q )0(Q) P+?-" £ p+q-n = H'o(P)i w ® W„(C®(S); Q ) . Since £>(p; B ,, -,S r) is a free 5®r-bimodule. = 0 r W ® H „(£0(S);Q) This completes the proof of Theorem 5.3.9. Corollary 5.3.15 The natural quotient map Ur —>► Urg(S) induces an isomorphism H.( £0(S); U f ► h,( p(S)- lf 0(S)) Proof. Consider the following diagram of £0(5)-modules V - » U f a 0 \ J 0 Only I f has nontrivial £0(S)-action. It suffices to show that (5.3.16) u f a = h 0( ur) H0(eo(sy. Corollary 5.3.17 Let B be a free (S ® Q)-bimodule. Then the natural surjection A r£0(B) —► (Ar£0(B))induces an isomorphism on - )• Proof. It is clear since the Enaction and the gO^-action commute. 73 The key facts which we will use in the next section are the isomorphism (5.3.16) 0 : Ur0(Sx —=-► © V0(p) and Corollary 5.3.17. c ' ' p e T r § 5.4 Calculation of p n+i(f) ® Q- f Let / —► R —► S be as in the introduction of Chapter 5. In this section our goal is to compute £L/J+1(/) ® Q. Consider the following diagram of fibration sequences: v r B£0(R) - > p (R ) y y BfiiS) p(S) Fj and F2 are homotopy fibers. Let //*() denote rational homology. Note that = nn(F2) ® Q which is isomorphic to the set of primitives in Hn(F2). Thus we are going to compute Hn(F2) and analyze its coproduct structure to find its primitives. It is easy to see that 0—► e©(^) —► eO{R) — *’ £0(S) —► 1 is an exact ry sequence, where the map e©(/) — > P (R ) is defined by M I—► 1 + M. Since I = 0, this is a homomorphism. Hence we have that Fj is homotopy equivalent to 5£0(7). Thus = a " £0 ( / ® q >. Consider the diagram of spaces 74 (A) (B) (C) (5.4.1) B£0(R) —► E — ► /,(/?) B£0(S) = fieO(S) ^(5) The columns (A), (B), (C) are homotopy fibration sequences, where E is the pullback. First consider (C). Kl£L(S) = fE^S) acts trivially on Hm(F2), since (C) is a fibration of //-spaces. In (B), itxB£0(S) = eO(S) acts trivially, since £ is a pullback. Compare the Serre spectral sequences of (B) and (C): £?.(B) =H,(B£0(S)) 8 H,(F2) 4 Y 4 Y ii E?*(C) = H ^ i S ) ) <8> Hm(F2) Since they are isomorphic on the £?*-level, so are they on the £^-level by the comparison theorem. Thus //*(£) = //*(gL(/?)) = //„(££<9(/?)). Now use the relative version of the Serre spectral sequence for the pair of fibrations (A) and (B): (5.4.2) £**((B), (A)) = H.(Bf0(S);H.(F2,Fl)) " => " //*(£, B£0(R)). //* (£ , BeO(R)) = 0, since B^O(R) —► £ induces a homology isomorphism, so O (B ), (A)) = 0. Note that H+iFJ = A*£(S>(I ® Q) and £0(S) acts on //*(£j) by conjugation. Corollary 5.3.17 says that H,(BcO(S); //.(F,» -=-» H,(BeO(S); //.(F ,)^ ). Hence we have H .(B p (S ) ; H,(F2, FJ) H.(B£0{S) ; 77, (F2, F , ) ^ (S) ). Lemma 5.4.3 Hi(F2,Fl) 0^ = 0 for all i. Proof. Let q be the smallest integer such that Hq(F2, Ff) *0. This means Eq * 0 in the Serre spectral sequence (5.4.2). Since q is the smallest integer, we have En„ =E~ *0, which is a contradiction. "•V u,<7 Remark 5.4.4 Lemma 5.4.3 implies H^(F}) = Hm(F2), since £0(S) acts trivially on F2. Hence we have (5.4.5) H n(F2) = Now we analyze (An£©(7® Q)) 0^ . Recall that (e© (7® Q )*“) o(S) *s isomorphic to ^ VQ(p; 7® Q) = ® W0(p\ I ® Q)L^ by (5.3.16) (here B x = • = Br =I® Q). This isomorphism, in fact, comes from = (Q[S2„///„]®(/®Q)®',)(z/2)„ Likewise we can compute (A £©(7® Q ))^ ^ by computing (A >(/® Q »^(Q) = (Q [V "J ® <'® where v runs through system of representatives for conjugacy classes in and N(v) (c Hn) is the stabilizer of v . We may choose the representatives v in a unique way as follows: 76 *1 *1 k v = (1,2, -,/jX/j, ij+1, -, z,+z2) - (X,y»X,y+1* i° adisjoint j= 1 7=1 7=1 k cycle expression, where ^i- < it and z'j < z'2 ^ ••• ^ z'^. Denote this unique set of 7=1 representatives by Yn. Let v e Yn. Then ( (/ ® is a quotient of (/ ® Q)*” by the following relations: i) Jtj® ® ••• ® Jtn ~ jCj® -••®-jci- ® **• ®JCn if v(z) = z — - v v li) Jtj® —®a:j- ® ;cJ+1® ••• ®Jtrt ~ ^j® —®-jc1-®-jc|-+1®— ®xn if z |—-> z'+l I—► z iii) *i® - ®*„ ~ sgnip) *p(1)® - ®*p(#l) if p e C(v) iv) jtj® -® jq® ••• ®jciy+1® ®J£tj ~ sgn(o) Xj® -®-jc/.+i® ••• ®-3c, ® ••• ®jcrt if Note that ii) is a special case of iii) and iv), and in fact, L(v) c N(v) , where L(v) = , VjVk I */) = (Z/2)n. Hence we have (5.4.6) (A\ S(f • Q))^ a €>_ ) ■ where W0(v)N^ is the quotient of Wq(v) by the same relations as the above. Denote the class of [[xv - ,xr]]v by [[xv ••• , xr]]y e W0(v) . Let A+ (resp. A„) be the free graded (resp. free commutative graded) Q- algebra generated by £©(/ ® Q) in degree 1. Then this is a Hopf algebra with the coproduct A* = ;t®l + l®;t for x e Ax (resp. ylj). 77 For all i = l , - , n, let Ei = E (|r{-; 0',p(0)) © E(.-\x iI (P(0. 0)- Then n { e (A*) o(S) corresponds to [[xv ••• ,x r]] e V0(p) under (A„) 0(5) = j=i £ £ ©r w n To see the coproduct on /4n, consider J’J/Sj- e /4n with e Aj for all /. i=\ Then we have n n (5.4.7) AfpfE,-) = II(£i® 1 + 1 ® £,) i= 1 >'=1 pel^Xlj 0^1 P=l where p runs through all (p, ^)-shuffles in £fl. The coproduct on © VT0(v) corresponding to (5.4.7) is given as follows: VfLji A([x,. - , *r] ) = % , sgn(o) - , x a(p)!it « [ x ^ y . j ^ ] , where [G3], p. 397 for more details.) Since the set Yn is the unique choice of representatives of conjugacy classes in , we have the coproduct on © VF0(v)^^ as follows: A( [[i,,-,xr]]v) = £ [[xi’ "■ ’ *P]],r 0 [[XP+1* • *«]]£ (*; {)eL where v = (tt, £ )e 'Lp x.'Lq . Hence we can easily see the following lemma: 78 Lemma 5.4.8 The primitive part of corresponds to W0(pn)N^ ^ under the isomorphism (5.4.6). Moreover, we have WQ(pn)N( = W*(pn) , Pn) where pn = (1, 2, - ,n). Remark 5.4.9 Loday and Procesi also computed Prim((.A”e© (/\))^Q^ ) ([L-P], Proof of Theorem 5.5), where A is a Q-algebra. Let us summarize their calculation. Recall that (A n £®(A)) ~ (Q[® A9n)/j . Let Un be the conjugacy class of pn = (1, •••, n) in Sn. Let Vn be the subset of S2n/Hn which consists of elements of the form ( ug), u e (Z/2)n, g e UnaSn. Then the primitive part of degree n of Moreover, there is an //n-equivariant bijection Hn/Dn = Vn ([L-P], Lemma 6.5) Thus we have Prim((A"£0(/l))£O(Q) ) = (Q[V„] ®A 9 " ) ,," - (Q lHJDn]®A*%' -(*«%. The action of Dn= (x,z\ xn =y2 = 1, yxy'1 = jc'1 ) on A9n is via Hn, which is explicitly x (av - ,an) = (-l)"'1^ , av - ,anA) y (flj, - ,an) = (-i)"(«+1)/2( ^ ... ^ ^ Notice that = (A®71)w. . , where N(p ) is as above. 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