UPPER TRIANGULAR TECHNOLOGY AND THE ARF-KERVAIRE INVARIANT VICTOR P. SNAITH Abstract. This paper introduces the upper triangular technology (UTT) into classical homotopy theory. This is a new and easy to use method to calculate the effect of the left unit map in 2-adic connective K-theory; * *the map which is the basis for operations in bu-theory. By way of application, UTT is used to give a new, very simple proof of a conjecture of Barratt- Jones-Mahowald, which rephrases K-theoretically the existence of framed manifolds of Arf-Kervaire invariant one. 1.Introduction 1.1. The upper triangular technology (UTT) referred to in the title con- sists of the following two results. Let bu and bo denote the stable homotopy spectra representing 2-adically completed unitary and orthogonal connective K-theory respectively. Thus the smash product, bu ^ bo is a left bu-module spectrum and so we may consider the ring of left bu-module endomorphisms of degree zero in the stable homotopy category of spectra [2], which we shall denote by Endleft-bu-mod(bu ^ bo). The group of units in this ring will be denoted by Autleft-bu-mod(bu ^ bo), the group of homotopy classes of left bu-module homotopy equvialences and let Aut0left-bu-mod(bu ^ bo) denote the subgroup of left bu-module homotopy equivalences which induce the identity map on H*(bu ^ bo; Z=2). Let U1 Z2 denote the group of infinite, invertible upper triangular matrices with entries in the 2-adic integers. That is, X = (Xi,j) 2 U1 Z2 if Xi,j2 Z2 for each pair of integers 0 i, j and Xi,j= 0 if j < i and Xi,iis a 2-adic uni* *t. Theorem 1.2. ([20] x2.1) There is an isomorphism of the form ~= : Aut0left-bu-mod(bu ^ bo) -! U1 Z2. Let _3 : bo -! bo denote the Adams operation. This isomorphism is defined up to inner automorphisms of U1 Z2. Given an important automorphism in Aut0left-bu-mod(bu ^ bo) one is led to ask what is its conjugacy class in U1 Z2. By far the most important such automorphism is 1 ^ _3. ___________ Date: 26 September 2006. 1 Theorem 1.3. ([10] x1.1) Under the isomorphism the automorphism 1 ^ _3 corresponds to an ele- ment in the conjugacy class of the matrix 0 1 1 1 0 0 0 . . . BB C BB CC BB 0 9 1 0 0 . . .CC BB CC BB 0 0 92 1 0 . . .CC. BB CC BB 0 0 0 93 1 . . .CC BB CCC @ . A .. ......... ...... To be precise, the proof in ([10] x1.1) shows that 1 ^ _3 can be conjugated to this form in its first N columns for arbitarily large N. In Example 3.2 I shall explain what this means in practice. The main purpose of this paper is to illustrate UTT at work in an appli- cation. By way of illustration I shall use UTT to give an elementary proof of the following result, whose terminology and proof will be given in x3. Theorem 1.4. Let m be a positive integer and let 8m-2 : S8m-2 - ! RP 8m-2 be a mor- phism in the 2-local stable homotopy category. Then the bo e-invariant of 8m-2 is (34m - 1)=4 (modulo 24m-1) if and only if m = 2q and 8m-2 is q+2 detected by the Steenrod operation Sq2 . Any 2-adic stable homotopy class " 2 ss8m-2(S0) Z2 lifts canonically (via the Kahn-Priddy Theorem [3] [12]) to an element of the stable homotopy of RP 1 and thence to 8m-2. Detection by a primary mod 2 cohomology operation can only occur if m is a power of 2 and is equivalent to " being represented by a framed manifold of Arf-Kervaire invariant one ([8], [11], [18], [21]). The existence or otherwise of framed manifolds of Arf-Kervaire invari- ant one is a classical unsolved problem in homotopy theory. As explained in [13], the alternative formulation of Theorem 1.1 is equivalent to a conjecture of [7] which was first proved in [14] by a very difficult study of Ad-theory and is proved in [21] by a straightforward but brutally long-winded use of BP-operations. The attempted proof of [13] contains a gap caused by lack of control of the filtration in an Adams spectral sequence. Intuitively, the UTT proof of Theorem 1.1 is conceptually simple because it amounts to inflicting the relevant mod 2 Adams spectral sequence with a "mixed Hodge structure"; that is, a direct sum decomposition (corresponding to the entries in U1 Z2) compatible with the usual Adams filtration. Incidentally Theorem 1.4 poses the existence of framed manifolds of Arf- Kervaire invariant one in a form which is similar to the formulation of the exi* *s- tence of framed manifolds of Hopf invariant one in terms of 2 2m-1 : S2m-1 -! RP 2m-1 (see [19]). These only exist for dimensions 1, 3, 7 and currently framed manifolds of Arf-Kervaire invariant one have been con- structed in dimensions 2, 6, 14, 30, 62 (see [18] and [15] - I believe that [16] has a gap in its construction). Accordingly the following conjecture seems reasonable: Conjecture 1.5. Framed manifolds of Arf-Kervaire invariant one exist at most in dimen- sions 2, 6, 14, 30 and 62. The paper is organised in the following manner. The basis of Theorems 1.2 and 1.3 is the left bu-module splitting of bu ^ bo into a sum of spectra of the form bu ^ (F4k=F4k-1) for k 0. In x2 the mod 2 Adams spectral sequence for the homotopy of bu ^ (F4k=F4k-1) ^ RP 2mis described. The crucial result is the multiplicative structure stated in Theorem 2.6 and proved in x2.8. Combined with results from [10], Theorem 2.6 yields Proposition 2.10, which evaluates the effect on homotopy of the maps corresponding to the super- diagonal entries of an upper-triangular matrix. These maps correspond to the 1's in the matrix for 1 ^ _3 in Theorem 1.3. In x3 Theorem 1.3 is combined with Proposition 2.10 to transform expressions for the bo e-invariant of 8m-2 into a series of 2-adic equations in Example 3.4 (and Propositions 3.5 and 3.6) from which Theorem 1.4 is easily deduced in x3.8. I am very grateful for help and advice to Huajian Yang, my postdoc at McMaster University in 1996-98, to Jonathan Barker, my PhD student at the University of Southampton in 2003-2006 and to Francis Clarke. 2. K-theory examples 2.1. Let bu*(X) (resp. KU*(X)) denote the reduced, connective (resp. pe- riodic) complex K-theory of a (based) CW complex X. When X equals the zero-dimensional sphere we have bu*(S0) ~=Z[u] and KU*(S0) ~=Z[u 1] where deg(u) = 2. Let RP ndenote n-dimensional real projective space. Let Z=t denote a cyclic group of order t with generator w The following result is well known. Proposition 2.2. For 1 m 1 8 >>>0 ifj is even, >>< buj(RP 2m) = > Z=2i if1 j = 2i - 1 < 2m, >>> >: Z=2m if2m < j = 2i - 1. In addition, the generators may be chosen to satisfy uv2i-1= 2v2i+1for 1 i m - 1. 3 Proof The Atiyah-Hirzebruch spectral sequences for computing bu*(RP 2m) and KU*(RP 2m) collapse for dimensional reasons. This implies that buj(RP 2m) has the correct order. It also implies the injectivity of the canonical maps hj(RP 2m) -! hj(RP 2m+2) (h = bu, KU), ~* : buj(RP 2m) -! KUj(RP 2m). However, by the universal coefficient theorem for KU and the results of ([5] p.107) we have KU2i-1(RP 1 ) ~= Z=21 so each bu2i-1(RP 2m) is cyclic. The relation uv2i-1= 2v2i+1follows from Bott periodicity and the fact that the injection ~* commutes with multiplication by u. 2 2.3. Ext*,*B(H"*(RP 2m; Z=2), Z=2) Let B = E(Sq1, Sq0,1) denote the exterior subalgebra of the mod 2 Steen- rod algebra A [22] generated by Sq1 and Sq0,1= [Sq1, Sq0,1]. There is an isomorphism of bigraded algebras Ext *,*B(Z=2, Z=2) ~= Z=2[a, b], the poly- nomial algebra on a and b with bideg(a) = (1, 1), bideg(b) = (1, 3). Also H"*(RP 2m; Z=2) = .=(x2m+1) with Sq1(xn) = nxn+1, Sq0,1(xn) = nxn+3. Consider the bigraded Z=2[a, b]-module Ext*,*B(H"*(RP 2m; Z=2), Z=2). De- note the non-zero element of Ext0,2i-1B(H"*(RP 2m; Z=2), Z=2) by "v2i-1for 1 i m. Proposition 2.4. For 1 m 1 the bigraded Ext *,*B(Z=2, Z=2)-module Ext*,*B(H"*(RP 2m; Z=2), Z=2) is equal to Z=2[a,_b]<"v1,_"v3,_._.,."v2m-1>_ . {ai"v2i-1, b"v2i-1- a"v2i+1} Proof We prove this by induction on m. When m = 1 we have Ext*,*B(H"*(RP 2; Z=2), Z=2) ~=Ext*,*B(E(Sq1)[1], Z=2) ~=Z=2[b]<"v1> where X[n] denotes X with a dimension shift by n so that X[1] = X in the notation of [2] and [20]. We have a short exact sequence of B-modules (x2.-) * 2m * 2 0 -! "H*(RP 2m-2; Z=2)[-2] - ! "H (RP ; Z=2) -! "H (RP ; Z=2) -! 0. By induction, for each non-negative integer r the resulting long exact sequence yields an upper bound for the sums of F2- dimensions X1 X dim F2(Ext s,tB(H"*(RP 2m; Z=2), Z=2)) dr s=0 t-s=r where dr = 0 if r is even, d2i-1= i for 1 i m and d2i-1= m for m i. On the other hand, if Z2 denotes the 2-adic integers, the Adams spectral sequence ([2]) for ss*(bu ^ RP 2m) Z2 = bu*(RP 2m) Z2 has the form ([10]; [20]) Es,t2= Exts,tB(H"*(RP 2m; Z=2), Z=2) =) but-s(RP 2m) Z2 4 and collapses for dimensional reasons, being concentrated where t - s is odd. Therefore Proposition 2.2 shows that dr is also a lower bound. The relations folllow from the fact that a and b represent 2 and u respectively in the Adams spectral sequence for bu*(S0) Z2. 2 2.5. Ext*,*B(H"*(X ^ (F4k=F4k-1); Z=2), Z=2) Consider the second loopspace of the 3-sphere, 2S3. There exists a model for 2S3 which is filtered by finite complexes ([9],[17]) [ S1 = F1 F2 F3 . . . 2S3 = Fk k 1 and there is a stable homotopy equivalence, an example of the so-called Snaith splitting, of the form 2S3 ' _k 1Fk=Fk-1. Consider the finite complexes F4k=F4k-1with the convention that F0=F-1 = S0, the 0-sphere. Let ff(n) denote the number of 1's in the dyadic expansion of the positive integer n. The results of Adams-Margolis ([2], [4]; see also [1* *0] and [20]) yield Ext*,*B(Z=2, Z=2)-module isomorphisms of the form Exts,tB(H"*(X ^ (F4k=F4k-1); Z=2), Z=2) ~=Ext s+2k-ff(k),t-2k-ff(k)B(H"*(X; Z=2), Z=2) for all s > 0. We shall need this isomorphism in the case where X is either a real projective space or a sphere. The case when X is a sphere is described extensively in [20] in connection with the left bu-module equivalence of 2-local spectra (see also ([10] x2) ^L: _k 0 bu ^ (F4k=F4k-1) -'! bu ^ bo. The groups Ext s,tB(H"*(RP 2m ^ (F4k=F4k-1); Z=2), Z=2), when depicted in the traditional Adams spectral sequence manner with s along the vertical and t - s along the horizontal axis, looks as in the figure below. The figure is interpreted as follows: the groups are F2-vector spaces which are possibly non-zero only when s = 0 and t - s 4k + 1 or a copy of Z=2 at each point with (s, t - s) = (v, 8k - 2ff(k) - 1 + 2w + 2v) with v = 1, 2, 3, . .a.nd 1 w m. 5 |6|6 || || s |||| || || || || | | | | | | | | | | | | | | | | | . . .. . . | | 2 | | | . . .. . . 1 | | _|______________o_____o_______________________o_____o_____o_________o 4k + 1 4k + 5 8k - 2ff(k) + 1 4k + 3 _______________- 8k - 2ff(k) - 1 + 2m t - s We have the following result describes the important aspects of the mod- ule structure over the bigraded algebra Ext *,*B(Z=2, Z=2) ~= Z=2[a, b]. Let "v2i-12 Ext0,2i-1B(H"*(RP 2m; Z=2), Z=2) be as in Proposition 2.4 and let ^z4k2 Ext0,4kB(H"*(F4k=F4k-1; Z=2), Z=2) be element represented as a homomorphism on mod 2 cohomology by the inclusion of the bottom cell of F4k=F4k-1 ([10] Theorem 2.12). Hence we have a (non-zero) external product "v2i-1^z4k2 Ext0,4k+2i-1B(H"*(RP 2m ^ (F4k=F4k-1); Z=2), Z=2) for 1 i m. 6 Theorem 2.6. In Exts,tB(H"*(RP 2m ^ (F4k=F4k-1); Z=2), Z=2) we have: (i) "vi^z4k6= 0 for each 1 i m, (ii) b"vi^z4k= 0 = a"vi^z4kfor i = 1, . .,.2k - ff(k), (iii) be"vi^z4k6= 0 for e 1 and i = 2k - ff(k) + 1, . .,.m. The proof of Theorem 2.6 will be given in x2.8 after some preliminaries concerning B-resolutions. 2.7. Resolutions For 1 m 1, the B-action on H"*(RP 2m; Z=2) = is given by Sq1(xi) = ixi+1 and Sq0,1(xi) = ixi+3. The beginning of a free B-resolution . .-.d!P1 -d! P0 -ffl!"H*(RP 2m; Z=2) -! 0 may be given by P0 = B and P1 = B< 5, 4, 6, . .,. 2m> where deg(oei) = i, ffl(oei) = xi and d( 2t) = Sq1oe2t-1+ Sq0,1oe2t-3and d( 5) = Sq1Sq0,1oe1. By Proposition 2.4 b"v2i-1is the only non-zero element in the group Ext1,2i+2B(H"*(RP 2m; Z=2), Z=2) for 1 i m. Therefore it must be represented by homomorphism hi 2 Hom B (P1, Z=2) given by hi( 2i+2) 1 (modulo 2) and hi( j) 0 otherwise. Let H(k) be the graded F2-vector space with basis yk,2k, yk,2k+2, yk,2k+4, . .,.yk,2k+1-2, yk,2k+3, yk,2k+5, . .,.yk,2k+1-1 where deg(yi) = i, with the "lightning flash" B-module structure given by Sq0,1yk,2k= yk,2k+3= Sq1yk,2k+2, . .,.Sq0,1yk,2k+1-4= yk,2k+1-1= Sq1yk,2k+1-2. We define the start of a free B-resolution d(k) ffl(k) . .-.! Rk,1-! Rk,0-! H(k) -! 0 by Rk,0 = B< k,0,2k, k,0,2k+2, . .,. k,0,2k+1-2> where deg ( k,0,i) = i and Rk,1= B< k,1,2k+1, k,1,2k+3, . .,. k,1,2k+1+1> where deg( k,1,i) = i. Also ffl* *(k) and d(k) are given by ffl(k)( k,0,2i) = yk,2iand d(k)( k,1,2k+1) = Sq1 k,0,2k, d(k)( k,1,2k+3) = Sq1 k,0,2k+2+ Sq0,1 k,0,2k, d(k)( k,1,2k+5) = Sq1 k,0,2k+4+ Sq0,1 k,0,2k+2, .. . . . .. .. d(k)( k,1,2k+1+1) = Sq0,1 k,0,2k+1-2. We are now ready to embark on the proof of Theorem 2.6. 7 2.8. Proof of Theorem 2.6 Part (i) follows since the exterior product of two non-zero B-homomorphisms to Z=2 is also non-zero and part (ii) follows because the elements in question lie in groups which are zero, by Proposition 2.4 and the discussion of x2.5. Part (iii) is more substantial. By naturality it suffices to work with m = 1. Let k = 2ffl1+ 2ffl2+ . .+.2ffltwith 0 ffl1 < ffl2 < . .<.fflt so we are in* *terested in i 2ffl1+1+ 2ffl2+1+ . .+.2fflt+1- t + 1 and 4k = 2ffl1+2+ 2ffl2+2+ . .+.2f* *flt+2. From ([2] pp.341-2) or ([20] p.1267) H"*(F4k=F4k-1; Z=2) ~= tj=1H(fflj + 2). We have a free B-resolution given by the tensor product . .!. tj=1, P aj=1Rfflj+2,ajd! tj=1Rfflj+2,0! "H*(F4k=F4k-1; Z=2) ! 0. We introduce the convention that fflj+2,1,2s+1= 0 = fflj+2,0,2sifs 2fflj+1or2fflj+2 s. With this convention the differential has the form d(fflj + 2)( fflj+2,1,2s+1) = Sq1 fflj+2,1,2s+ Sq0,1 fflj+2,1,2s-2. The element ^z4kis represented by the B-homomorphism gk 2 Hom B ( tj=1Rfflj+2,0, Z=2) given by gk( tj=1 fflj+2,0,2fflj+2) 1 (modulo 2) and gk( tj=1 fflj+2,0,wj) 0 otherwise. We must show that there does not exist a B-homomorphism f 2 Hom B (P0 ( tj=1Rfflj+2,0), Z=2) such that f . d = (0, hi gk) in the group of B-homomorphisms Hom B(P0 ( tj=1, P aj=1Rfflj+2,aj) P1 ( tj=1Rfflj+2,0), Z=2) when m = 1 and i lies in the range i 2k - ff(k). This will show that b"vi^z4k6= 0 from which be"vi^z4k6= 0 for e 1 follows because the isomorphism of x2.5 commutes with multiplication by b when s > 0. I shall first give the argument to prove that f does not exist and finally I shall explain where the i 2k - ff(k) is necessary. In degree 4k + 2i + 2, suppose that we have the relation (0, hi gk) = f . d. Then we shall apply f . d to all the B-basis elements in P* ( tj=1R(fflj+2,*)) in resolution degree 1 and homological degree 4k + 2i + 2 and add the results in two ways to get a contradiction. The basis elements in question are X { 2q+2 ( j fflj+2,0,2sj) | 2q + 2 + 2sj = 4k + 2i + 2 } j P and (where 0 aj 1 and jaj = 1) X {oe2q+1 ( j fflj+2,aj,2sj+aj) | 2q + 2 + 2sj + aj = 4k + 2i + 2 } j disregarding, of course, the ones of this list which are zero by the convention introduced above. 8 We have f(d( 2q+2 ( j fflj+2,0,2sj))) = f(Sq1oe2q+1 ( j fflj+2,0,2sj)) + f(Sq0,1oe2q-1 ( j fflj+2,0,2sj)) P and (where 0 aj 1 and jaj = 1) f(d(oe2q+1 ( j fflj+2,aj,2sj+aj))) = f(oe2q+1 . . .Sq1 fflj+2,0,2sj . .). +f(oe2q+1 . . .Sq0,1 fflj+2,0,2sj-2 . .). where in the last expression the Sq's appear precisely in the unique factor for which aj was equal to one. P Now fix a t + 1-tuple (q, s1, . .,.st) such that 4k + 2i + 2 = 2q + 2 + j 2sj and consider the sum P t 1 f(Sq1oe2q+1 ( j fflj+2,0,2sj)) + j=1f(oe2q+1 . . .Sq fflj+2,0,2sj . .). = Sq1(f(oe2q+1 ( j fflj+2,0,2sj))) = 0 because Sq1 acts trivially on Z=2 for dimensional reasons. Similarly f(Sq0,1oe2q-1 ( j fflj+2,0,2sj)) Pt 0,1 + j=1 f(oe2q+1 . . .Sq fflj+2,0,2sj . .). = 0. Therefore applying f . d to each of the basis elements listed above and adding the results yields zero modulo 2. Now consider what happens if we apply (0, hi gk) to each of the basis elements listed above and add the results. The sum equals 1 because the map is zero on P0* ( tj=1, P aj=1Rfflj+2,aj) and is also zero on 2q+2 ( j fflj+2* *,0,2sj) unless q = i and 2sj = 2fflj+2for j = 1, . .,.t. This contradiction completes the proof of part (iii) except that it remains to explain why we need the condition that i 2ffl1+1+ 2ffl2+1+ . .+.2fflt+1- t + 1. We require that i be large enough so that all the elements 2q+2 ( j fflj+2,0,* *2sj) and oe2q+1 ( j fflj+2,aj,2sj+aj) over which we want to sum are permissible within homological degree 2i + 2 + 4k. However if i 2ffl1+1+ 2ffl2+1+ . .+. 2fflt+1- t + 1 then Xt 2i + 2 + 4k 2ffl1+3- 2 + 2ffl2+3- 2 + . .+.2fflt+3- 2 + 2 2sj j=1 9 for all possible choices of the sj's involved in the sum. 2 2.9. The maps 'k,l As in x2.5 let bu and bo denote the 2-localised, connective unitary and or- thogonal K-theory spectra, respectively. Consider a left-bu-module spectrum map ' : bu ^ (F4k=F4k-1) -! bu ^ (F4l=F4l-1). This map is determined up to homotopy by its restriction, via the unit of bu, to (F4k=F4k-1). By S-duality this restriction is equivalent to a map of the form S0 -! D(F4k=F4k-1) ^ bu ^ (F4l=F4l-1), which DX denotes the S-dual of X. Maps of this form are studied by means of the (collapsed) Adams spectral sequence (see [20] x3.1) Es,t2= Exts,tB(H"*(D(F4k=F4k-1); Z=2) "H*(F4l=F4l-1; Z=2), Z=2) =) sst-s(D(F4k=F4k-1) ^ (F4l=F4l-1) ^ bu) Z2 where Z2 denotes the 2-adic integers. It is shown in [20] that such maps ' are trivial when l < k and form a copy of the Z2 when l k. Following [10] and [20] we choose left-bu-module spectrum maps 'k,l: bu ^ (F4k=F4k-1) -! bu ^ (F4l=F4l-1) to satisfy 'k,k= 1, 'k,l= 'l+1,l'l+2,l+1. .'.k,k-1for all k - l 2 and each 't* *+1,t is a Z2-module generator of the group of such left-bu-module maps. Let "z4k2 ss4k(bu ^ (F4k=F4k-1)) Z2 denote the element represented by the smash product of the unit j of the bu-spectrum with the inclusion of the bottom cell jk into F4k=F4k-1 (see [10] x2.12) j^jk S0 ^ S4k -! bu ^ F4k=F4k-1 and let v2i-12 ss2i-1(bu ^ RP1 ) Z2 = bu2i-1(RP 1 ) be as in Proposition 2.2. Then we have the exterior product v2i-1"z4k2 ss4k+2i-1(bu ^ RP 1 ^ (F4k=F4k-1) Z2 which is non-zero and is represented by "v2i-1^z4kin the collapsed Adams spec- tral sequence whose E2-term is described in x2.5. The following formula is central to the proof in x3.8 of our main theorem. Proposition 2.10. For l < k, for some 2-adic unit ~4k,4l, ('k,l)*(v2i-1"z4k) = ~4k,4l24k-4l-ff(k)+ff(l)v2i+4k-4l-1"z4l 10 Proof Since 'k,lis a left-b-module map we have ('k,l)*(v2i-1"z4k) = v2i-1('k,l)*("z* *4k) and, by ([10] Proposition 3.2) ('k,l)*("z4k) = ~4k,4l22k-2l-ff(k)+ff(l)u2k-2l"z* *4lfor some 2-adic unit ~4k,4l. The result follows since, by Proposition 2.2, v2i-1~4k,4l22k-2l-ff(k)+ff(l)u2k-2l"z4l= v2i+4k-4l-1~4k,4l24k-4l-ff(k)+ff(l* *)"z4l. 2 3. Applications 3.1. The main diagram In this section we are going to apply the results of the previous section together with the upper triangular yoga of [20] and [10] to the following partially commutative diagram to prove Theorem 1.4. In the diagram j is the unit of bu, c is complexification, ~ is the bu-multiplication and _3 is the Adams operation. The homomorphism ~* is equal to (~ ^ 1)* . (1 ^ c ^ 1)*. The diagram does not commute because the right-hand oblique vertical rectangle does not commute. However the upper and lower triangles, the back rectangle and the front left-hand oblique vertical rectangle do commute. 11 ssj(bo ^ X) ________________- ssj(bu ^ bo ^ X) (j ^ 1 ^ 1)* | | | B | @ || B || @ | B | @ | B | @ || B || 3 @ 3 | B |(_ ^ 1)* @ (1 ^ _ ^ 1|)* B | @ || B || @ (c ^ 1)* | B | @ | B | @ || B || @ | ~*B |? @ | B @ |? B @ B @ B (j ^ 1 ^ 1)* ssj(bo ^ X) ________________- @ssj(bu@^ bo ^ X) BB @ B Z @ B Z Z @ B Z @ B Z @A B Z Z A@ B Z (c ^ 1) A @ B Z * A @ B Z Z A @ B Z A @ BN Z @R Z A Z ~*A Z A ssj(bu ^ X) Z Z A Z A | Z A | Z Z A 3 Z A (_ ^ 1)* Z A Z | Z A | Z Z" A || AAU|? ssj(bu ^ X) Example 3.2. RP 8m-1 Let K"U denote reduced 2-local periodic unitary K-theory and let bu, bo denote the associated connective K-theories. In this example, by way of illustration, we shall show how to use the results of [20] and [10] to calculate 12 the map (j ^ 1 ^ 1)* : bo8m-1(RP 8m-1) -! ss8m-1(bu ^ bo ^ RP 8m-1). 0 2t-1 0 2t-2 t-1 From [5] we have K"U (RP ) ~= KU" (RP ) ~= Z=2 and K"U 1(RP 2t-1) ~= Z2. The KU-theory universal coefficient theorem (proved by the method of [6]) shows that K"U 1(RP 2t-1) ~=Z2 Z=2t-1, KU" 0(RP 2t-1) = 0. The Adams operation _3 gives a stable operation on 2-local KU-homology and in the book review [19] a (then) new, one-line proof of the non-existence of maps of Hopf invariant one based on the formula _3(F2) = ((3t- 1)=2)F1 + 3tF2, _3(F1) = F1. The canonical map from bu2t-1(RP 2t-1) to K"U2t-1(RP 2t-1) is an isomorphism commuting with _3 so that bu2t-1(RP 2t-1) ~=Z2 Z=2t-1 with the _3 acting on the generators by the formulae _3(F2) = F2 + ut((3t- 1)=2)F1, _3(F1) = F1, where ut is an odd integer. When t = 4m the complexification map is an isomorphism giving, in the notation of Proposition 2.2, bo8m-1(RP 8m-1) ~=bu8m-1(RP 8m-1) ~=Z2<'8m-1> Z=24m-1 where the second summand is bo8m-1(RP 8m-2) ~=bu8m-1(RP 8m-2) and _3('8m-1) = '8m-1 + u4m((34m - 1)=2)v8m-3u, _3(v8m-3u) = v8m-3u. Proposition 2.4, the discussion of x2.5 and Theorem 2.6 easily imply (c.f. [2] Lemma 17.12) that the Adams spectral sequence Exts,tB(H"*(RP 8m-2^(F4k=F4k-1); Z=2), Z=2) =) but-s(RP 8m-2^(F4k=F4k-1)) collapses and that for 1 k 2m - 1 and 4m 4k - ff(k) + 1 bu8m-1(RP 8m-1 ^ (F4k=F4k-1)) ~=bu8m-1(RP 8m-2 ^ (F4k=F4k-1)) ~=Vk Z=24m-4k+ff(k) where Vk is a finite-dimensional F2-vector space consisting of elements which are detected in mod 2 cohomology (i.e. in Adams filtration zero) in the spectral sequence. If 8m - 1 8k - 2ff(k) + 1 then the group is a Z=2-vector space of the form bu8m-1(RP 8m-1 ^ (F4k=F4k-1)) ~=Vk Z=2 entirely in Adams filtration zero and if k 2m the group is zero. By means of the 2-local equivalence ^Lof x2.5 we have a direct sum decom- position ^L*: k 0 bu*(RP 8m-1 ^ (F4k=F4k-1)) -~=!ss*(bu ^ bo ^ RP 8m-1) 13 and by means of this identification we may write the element (j ^ 1 ^ 1)*('8m-1) 2 ss8m-1(bu ^ bo ^ RP8m-1 ) as a vector (w0, w1, . .,.w2m-1) with wk 2 bu8m-1(RP 8m-1 ^ (F4k=F4k-1)). In the diagram of x3.1 with X = RP 8m-1 and j = 8m - 1 the map (c ^ 1)* is an isomorphism which sends '8m-1 to itself. Hence w0 = '8m-1. According to the main theorem of [20] a left-bu-module self-equivalence of bu ^ bo inducing the identity on mod 2 homology determines a unique conju- gacy class in the upper triangular group with entries in the 2-adic integers. According to the main theorem of [10] the conjugacy class associated to the map 1 ^ _3 is equal to 0 1 1 1 0 0 0 . . . BB C BB CC BB 0 9 1 0 0 . . .CC BB CC BB 0 0 92 1 0 . . .CC. BB CC BB 0 0 0 93 1 . . .CC BB CCC @ . A .. ...... ......... In practical terms this means that, for any positive integer N, we may choose ^Lin x2.5 so that, for all k N, 1 ^ _3 maps the wedge summand bu ^ (F4k=F4k-1) to itself by 9k times the identity map, to bu ^ (F4k-4=F4k-5) by 'k,k-1and to all other wedge summands bu ^ (F4t=F4t-1) trivially if t N. If we choose ^Lin this manner, taking N very much larger than 8m - 1, we have (1 ^ _3 ^ 1)*((j ^ 1 ^ 1)*('8m-1)) = (1 ^ _3 ^ 1)*('8m-1, w1, w2, . .,.w2m-1) = ('8m-1 + ('1,0)*(w1), 9w1 + ('2,1)*(w2), 92w2 + ('3,2)*(w3), . .,.92m-1w2m-1* *). On the other hand this element is equal to (j ^ 1 ^ 1)*((_3 ^ 1)*('8m-1)) = (j ^ 1 ^ 1)*('8m-1 + u4m((34m - 1)=2)v8m-3u) = (', w1, w2, . .,.w2m-1) + u4m((34m - 1)=2)(v8m-3u, "w1, "w2, . .,."w2m-1) where (j ^ 1 ^ 1)*(v8m-3u) = (v8m-3u, "w1, "w2, . .,."w2m-1). 14 Equating coordinates we obtain a string of equations ('1,0)*(w1) = u4m((34m - 1)=2)v8m-3u 2 bu8m-1(RP 8m-1), (9 - 1)w1 + ('2,1)*(w2) = u4m((34m - 1)=2)w"12 bu8m-1(bu ^ RP 8m-2^ (F4=F3)), (92 - 1)w2 + ('3,2)*(w3) = u4m((34m - 1)=2)w"22 bu8m-1(bu ^ RP 8m-2^ (F8=F7)), .. . . . . .. .. .. (9k - 1)wk + ('k+1,k)*(wk+1) = u4m((34m - 1)=2)w"k2 bu8m-1(RP 8m-2 ^ (F4k=F4k-1)), .. . . . . .. .. ... There is a relation between wi and "wiof the form 2wi = r8m-1w"ifor all i where r8m-1 is an odd integer. For we have a cofibration OE 8m S8m-1 -ss!RP 8m-1 -! RP in which ss is the canonical projection. Also OE*('8m-1) generates bu8m-1(RP 8m) so, by Proposition 2.2, OE*(2'8m-1 - r8m-1v8m-3u) = 0 for some odd integer r8m-1. Therefore 2'8m-1 - r8m-1v8m-3u originates in bu8m-1(S8m-1 ) and (1 ^ j ^ 1)*(2'8m-1) = (2'8m-1 - r8m-1v8m-3u, 0, 0, 0, . .).. If we write u04mfor the 2-adic unit u4mr-18m-1our string of equations simpli- fies to ('1,0)*(w1) = u4m((34m - 1)=2)v8m-3u 2 bu8m-1(RP 8m-1), (9 - 1)w1 + ('2,1)*(w2) = 2u04m((34m - 1)=2)w1, (92 - 1)w2 + ('3,2)*(w3) = 2u04m((34m - 1)=2)w2, .. . . . . .. .. .. (9k - 1)wk + ('k+1,k)*(wk+1) = 2u04m((34m - 1)=2)wk, .. . . . . .. .. ... 15 We conclude the discussion of this example with the following result con- cerning the homomorphism ('k,k-1)* bu8m-1(RP 8m-2^(F4k=F4k-1)) Z2 -! bu8m-1(RP 8m-2^(F4k-4=F4k-5)) Z2. Proposition 3.3. In the notation of Proposition 2.10 and Example 3.2 (i) ('1,0)*(v8m-5z"4) = ~4,022v8m-3u, (ii) If 2 k 2m - 1 and 4m 4k - ff(k) + 1 then ('k,k-1)* Vk Z=24m-4k+ff(k) -! Vk-1 Z=24m-4k+4+ff(k-1) satisfies ('k,k-1)*(v8m-4k-1"z4k) = ~4k,4k-424-ff(k)+ff(k-1)v8m-4k+3"z4k-4whe* *re ~4k,4k-4is a 2-adic unit. In particular, ('k,k-1)* is injective on Z=24m-4k+ff(k) in cases (i) and (ii). Proof These formulae follow from those of Proposition 2.10, concerning RP 1 together with the injectivity of the map from bu8m-1(RP 8m-2^(F4k=F4k-1)) to bu8m-1(RP 1 ^ (F4k=F4k-1)), which follows from the Adams spectral sequence via Proposition 2.4 and x2.5. The formulae make sense because ff(k - 1) = ff(k) - 1 + 2(k). 2 Example 3.4. The maps 2j In this example we shall study stable homotopy classes of maps of the form 2j : S2j -! RP 2j with mapping cone C( 2j) such that, on 2-local connective K-theory bu2j+1(C( 2j)) ~=Z2<'2j+1> Z=2j for some 2-adic unit uj (_3 ^ 1)*('2j+1) = '2j+1+ uj((3j - 1)=4)v2j-1u. In other words, the _3 e-invariant (see [1]) of C( 2j) is half that of RP 2j+1in Example 3.2, which is the mapping cone of ` in the canonical cofibre sequence RP 2j-! RP 2j+1-! S2j+1- `! RP 2j. For simplicity we shall restrict ourselves to the case when j = 4m - 1. In this case there are isomorphisms of 2-local K-groups bo8m-1(C( 8m-2)) ~=bu8m-1(C( 8m-2)) ~=bu8m-1(RP 8m-1) and for each k 1 bu8m-1(C( 8m-2) ^ (F4k=F4k-1)) ~=bu8m-1(RP 8m-2 ^ (F4k=F4k-1)). 16 Furthermore if (j ^1^1)*('8m-1) = ('8m-1, w01, w02, . .,.w02m-1) the e-invariant condition yields, as in Example 3.2, a string of equations ('1,0)*(w01) = u004m((34m - 1)=4)v8m-3u 2 bu8m-1(RP 8m-1), (9 - 1)w01+ ('2,1)*(w02) = u004m((34m - 1)=4)w"12 bu8m-1(bu ^ RP 8m-2^ (F4=F3)), (92 - 1)w02+ ('3,2)*(w03) = u004m((34m - 1)=4)w"22 bu8m-1(bu ^ RP 8m-2^ (F8=F7)), .. . . . . .. .. .. (9k - 1)w0k+ ('k+1,k)*(w0k+1) = u004m((34m - 1)=4)w"k2 bu8m-1(RP 8m-2 ^ (F4k=F4k-1)), .. . . . . .. .. ... where u004mis a 2-adic unit and the "wk's are the same as in Example 3.2. Suppose that we have an element w 2 Vi Z=24m-4i+ff(i) we shall write w ' 2N if w = (x, 2N (2t + 1)v8m-4i-1"z4i) for some integers t and N < 4m - 4i + ff(i). Also we observe that in Proposition 3.3(i) or (ii) we may choose Vk so that ('k,k-1)*(Vk) = 0. Choosing Vk in this manner will simplify our subsequent calculations. Proposition 3.5. Let m = (2p + 1)2q with p 1. In the notation of Examples 3.2 and 3.4 (i) For 2 k 2q+1, 'k,k-1(wk) ' 24+q and 'k,k-1(w0k) ' 23+q. (ii) w2q+1' 1. (iii) Under these hypotheses 8m-2 does not exist. Proof First we observe that k 2q+1 implies that 4m = p2q+3+2q+2 4k-ff(k)+ 1 so that we may apply Proposition 3.3. Therefore the relations ('1,0)*(w01) = u004m((34m - 1)=4)v8m-3u and 2((92m - 1)=4) = 3 + 2(2m) - 2 = 2 + q implies w01' 2q and similarly w1 ' 21+q (once we observe that the condition 1 + q < 4m - 4 + ff(1) is fulfilled). The relation (9 - 1)w1 + ('2,1)*(w2) = u4m((34m - 1)=2)w"12 V1 Z=24m-3 17 implies that ('2,1)*(w2) ' 24+q and similarly ('2,1)*(w02) ' 23+q, which starts an induction on k. Suppose 2 k < 2q+1 that 'k,k-1(wk) ' 24+q 2 Z=24m-4k+4+ff(k-1). Therefore wk 2 Z=24m-4k+ff(k)satisfies wk ' 2ff(k)-ff(k-1)+q= 2q+1- 2(k). Then, since 2(9k - 1) = 3 + 2(k), (9k - 1)wk + 'k+1,k(wk+1) = (92m - 1)wk implies that 'k+1,k(wk+1) ' 24+q, as required. Similarly 'k+1,k(w0k+1) ' 23+q. Since ('2q+1,2q+1-1)*(v8m-2q+3-1"z2q+3) = ~2q+3,2q+3-42q+4v8m-2q+3+3"z2q+3-4we see that w2q+1' 1 and that w02q+1cannot exist. 2 Proposition 3.6. Let m = 2q. In the notation of Examples 3.2 and 3.4 (i) For 2 k 2q, ('k,k-1)*(wk) ' 24+q and ('k,k-1)*(w0k) ' 23+q. (ii) In Z=2 V2q, w02q' 1. Proof This time we observe that k 2q implies 4m = 2q+2 4k - ff(k) + 1 so that Proposition 3.3 applies and therefore part (i) follows as in Proposition 3.5. For part (ii) we have ('2q,2q-1)*(v8m-2q+2-1"z2q+2) = ~2q+2,2q+2-423+qv8m-2q+2+3"z2q+2-4 and the result follows. 2 Remark 3.7. (i) In Proposition 3.6 ('2q,2q-1)* has the form Z=2 V2q- ! Z=2q+4 V2q-1 so that the first component of ('2q,2q-1)*(w2q) is zero and so is that of w2q. (ii) When m = 2q we have shown that w0k' 2q- 2(k)for 1 k 2q. 3.8. Proof of Theorem 1.4 Let m = 2q then we have '8m-1 2 bo8m-1(C( 8m-2)) giving a stable homo- q+3-1 topy class '8m-1 : S2 - ! bo ^ C( 8m-2). Let ' : bu -! HZ=2 be the canonical cohomology class. Then, if h8m-1 2 H2q+3-1(bo ^ C( 8m-2); Z=2) is the mod 2 Hurewicz image of '8m-1, it is represented by either of the com- positions q+3-1'8m-1 (j^1^1) ('^1^1) S2 -! bo^C( 8m-2) -! bu^bo^C( 8m-2) -! HZ=2^bo^C( 8m-2) or q+3-1'8m-1 0 ("j^1^1) S2 -! S ^ bo ^ C( 8m-2) -! HZ=2 ^ bo ^ C( 8m-2) where "jis the unit for HZ=2. We have an isomorphism of Z=2-vector spaces H*(bo; Z=2) ~= k 0 H*(F4k=F4k-1; Z=2) 18 and 8 >>>Z=2<'8m-1> if j = 8m - 1, >>< Hj(C( 8m-2); Z=2) ~=> Z=2 if 1 j = 8m - 2, >>> >: 0 otherwise. If the e-invariant is correct then we have shown that h8m-1 2 H*(bo; Z=2) H*(C( 8m-2); Z=2) has the form 2q+2-2X h8m-1 = "z0 '2q+3-1+ "z2q+2 v2q+2-1+ x8m-1-j vj j=1 with x8m-1-j 2 H*(F2q+2+t=F2q+2-1+t; Z=2) with t 0. q+2 q+2 If X 2 A2 is an element of the mod 2 Steenrod algebra of degree 2 we write X* for the dual homomorphism on mod 2 homology, which decreases dimensions by 2q+2. Since H*(bo; Z=2) is a cyclic A-module generated by 1 in dimension zero there exists an X such that X*("z2q+2) = "z0. In fact, since the Hurewicz image of the bottom cell in H4k(F4k=F4k-1; Z=2) equals ,4k1, unravelling the homology isomorphism between H*(bu; Z=2) H*(bo; Z=2) and H*(bu; Z=2) H*(^l 0 F4l=F4l-1; Z=2) described in [20] one can show that Sq4k*("z4k) = "z0. q+2 Since h8m-1 is stably spherical we have 0 = Sq2* (h8m-1) so that q+2 "z0 Sq2* ('2q+3-1) q+2 = "z0 v2q+2-1+ "z2q+2 Sq2* (v2q+2-1) P2q+2-1 j 2q+2-j P 2q+2-2 2q+2 + j=1 Sq*("z2q+2) Sq* (v2q+2-1) + j=1 x8m-1-j Sq* (vj) P2q+2-1 P 2q+2-2 a 2q+2-a + a=1 j=1 Sq*(x8m-1-j) Sq* (vj) q+2 which implies, comparing coefficients of "z0, that Sq2* ('2q+3-1) = v2q+2-1so q+2 that '8m-1 is detected by Sq2 on its mapping cone, as required. That is:- ~= 2q+3-1 H2q+3-1(C( 2q+3-2); Z=2) -! H2q+3-1(S ; Z=2), q+2 ~= Sq2* : H2q+3-1(C( 2q+3-2); Z=2) -! H2q+2-1(C(`2q+3-2); Z=2), q+3-2 ~= H2q+2-1( RP 2 ; Z=2) -! H2q+2-1(C( 2q+3-2); Z=2). q+2 0 Conversely, if this detection by Sq2 is correct then w2q' 1 and therefore the e-invariant is right. 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