On the spectrum bo ^ tmf Scott M. Bailey March 2, 2009 Abstract M. Mahowald, in his work on bo-resolutions, constructed a bo- module splitting of the spectrum bo^ bointo a wedge of summands related to integral Brown-Gitler spectra. In this paper, a similar split- ting of bo^ tmf is constructed. This splitting is then used to under- stand the bo*-algebra structure of bo*tmf and allows for a description of bo*tmf. 1 Introduction All cohomology groups are assumed to have coefficients in F2 and all spectra completed at the prime 2 unless stated otherwise. Let A denote the Steenrod n algebra, and A(n) the subalgebra generated by {Sq1, Sq2, . .,.Sq2 }. Con- sider the Hopf algebra quotient A==A(n) = A A(n)F2. Here the right action of A(n) on A is induced by the inclusion and the left action on F2 by the augmentation. Algebraically, one can consider the subsequent surjections A ! A==A(0) ! A==A(1) ! A==A(2) ! A==A(3) ! . . . and ask whether each algebra can be realized as the cohomology of some spectrum. The case n 3 requires the existance of a non-trivial map n+1-1 0 S2 ! S which cannot occur due to Hopf invariant one. For n < 3, however, it is now well-known that each algebra can indeed be realized by the cohomology of some spectrum: H*HF2 ! H*HZ ! H*bo ! H*tmf 1 There are maps realizing the above homomorphisms of cohomology groups tmf ! bo ! HZ ! HF2 In particular, the spectrum tmf is at the top of a "tower" whose "lower floors" have been well studied in the literature, culminating with Mahowald's [6] understanding of the spectrum bo ^ bo and Carlsson's [1] description of the cohomology operations [bo, bo]. More difficult questions arise: What is the structure of tmf ^ tmf? What are the stable cohomology operations of tmf? We would like to understand the spectrum bo^tmf for a variety of reasons. First, it might serve as a nice intermediate step towards understanding the spectrum tmf ^ tmf. Furthermore, determining its structure comes with an added bonus of understanding operations [tmf , bo] which may provide some insight into understanding the cohomology operations of tmf . Second, the splitting of bo ^ tmf has been instrumental to the author in demonstrating the splitting of the Tate spectrum of tmf into a wedge of suspensions of bo. Let B1(j) denote the jthintegral Brown-Gitler spectrum, whose homology will be described as a submodule of H*HZ. Such spectra have been studied extensively in the literature (see [2], [5], [9], for example). In particular, MahowaldW[6] demonstrated the splitting of bo-module spectra bo ^ bo ' 4j W 8i+4j j 0 bo^ B1(j). Let = 0 j i B1(j). The main theorem of this paper is the following Theorem 1.1. There is a homotopy equivalence of bo-module spectra bo ^ ! bo^ tmf (1) The splitting is analogous to that of bo ^ bo of Mahowald and even MO<8> ^ bo of Davis [3]. Its proof, therefore, contains ideas and results from both. Section 2 deals with demonstrating an isomorphism on the level of homotopy groups, which first requires an understanding of the left A(1)- module structure of H*tmf . In Section 3, we construct a map of bo-module spectra realizing the isomorphism of homotopy groups. Section 4 uses this splitting along with pairings of integral Brown-Gitler spectra to explicitly determine the bo*-algebra structure of bo*tmf and also identifies the coho- mology bo*tmf. 2 2 The algebraic splitting The E2-term of the Adams spectral sequence converging to the homotopy groups of bo ^ tmf is given by Ext s,tA(H*(bo ^ tmf), F2)) sst-s(bo ^ tmf). (2) The Ext -group appearing in the above spectral sequence can be simplified via a change-of-rings isomorphism: Exts,tA(1)(H*tmf , F2)) sst-s(bo ^ tmf). (3) Therefore, it suffices to understand the left A(1)-module structure of H*tmf . Computations and definitions simplify upon dualizing. Indeed, the dual Steenrod algebra, A*, is the graded polynomial ring F2[,1, ,2, ,3, . .].with |,i| = 2i- 1. An equivalent problem after dualizing is determining the right A(1)-module structure of the subring H*tmf A*. The homology of tmf as a right A-module is given by Rezk [8] H*tmf ~= F2[i81, i42, i23, i4, . .].. (4) The generators ii = O,i, where O : A* ! A* is the canonical antiautomor- phism. Define a new weight on elements of A* by !(ii) = 2i-1 for i 1. For a, b 2 A* define the weight on their product by !(ab) = !(a) + !(b). Let Ntmfkdenote the F2-vector space inside H*tmf generated by all monomials of weight k with Ntmf0= F2 generated by the identity. Lemma 2.1. As right A(2)-modules, M H*tmf ~= Ntmf8i i 0 Proof. Certainly, the two modules are isomorphic as F2-vector spaces. To see there is an isomorphismPof right A(2)-modules, note that the right action of the total square Sq = i 0Sqi on the generators of H*tmf is given by: i81. Sq= i81+ 1; i42. Sq= i42+ i81+ 1; i23. Sq= i23+ i42+ i81+ 1; Xn i in . Sq= i2n-i i=0 3 for n > 3. Since !(1) = 0, modulo the identity the total square preserves n-1 2n-1 the weight of the generators of H*tmf . Note that i21 Sq = 1, hence the total square over A(2) cannot contain a 1 in the expansion for dimensional __ reasons. |__| Consider the homomorphism V : A* ! A* defined on generators by ( 1, i = 0, 1; V (ii) = ii-1, i 2. Restricting V to the subring H*tmf A* clearly provides a surjection Vtmf : H*tmf ! H*bo. Let Mbo(4i) denote the image of Ntmf8iunder the homomorphism Vtmf. It is generated by all monomials with !(iI) 4i. The following proposition is clear. Proposition 2.2. As right A(2)-modules Ntmf8i~= 8iMbo(4i). (5) Proof. Due to the weight requirements, Vtmf is injective when restricted to Ntmf8i. Indeed, the exponent of i1 in each monomial is uniquely determined __ by the other exponents. |__| Additionally, if we denote by Nbokthe F2-vector space inside H*bo gener- ated by all elements of weight k with Nbo0= F2 generated by the identity, we have a similar lemma: Lemma 2.3. As right A(1)-modules, Mn Mbo(4i) ~= Nbo4j. j=0 Further restricting V to the subring H*bo provides a surjection Vbo : H*bo ! H*HZ. Let MHZ(2j) denote the image of Nbo4junder V . This sub- module is generated by all monomials with !(iI) 2j. As in Proposition 2.2 we have the identification Proposition 2.4. As right A(1)-modules, Nbo4j~= 4jMHZ(2j). (6) 4 Goerss, Jones and Mahowald [5] identify the submodule MHZ(2j) H*HZ as the homology of the jth integral Brown-Gitler spectrum: Theorem 2.5 (Goerss, Jones, Mahowald [5]). For j 0, there is a spectrum B1(j) and a map g B1(j) -! HZ such that (i) g* sends H*(B1(j)) isomorphically onto the span of monomials of weight 2j; (ii)there are pairings B1(m) ^ B1(n) ! B1(m + n) whose homology homomorphism is compatible with the multiplication in H*HZ. Remark 2.1. The submodules Mbo(4i) are the so-called bo-Brown-Gitler modules. There is a family of spectra with similar properties, having these modules as their homology. Proposition 2.2 demonstrates that as an A(2)- module, H*tmf is a direct sum of these modules. On the level of spectra, however, tmf ^ tmf does not split as a wedge of bo-Brown-Gitler spectra. Combining the results of Lemmas 2.1 and 2.3 with Theorem 2.5, H*tmf as a right A(1)-module can be written in terms of homology of integral Brown- Gitler spectra: Theorem 2.6. As right A(1)-modules, M H*tmf ~= 8i+4jH*B1(j). 0 j i The E2-term of the Adams spectral sequence (3) then becomes isomorphic to M 8i+4jExts,tA(1)(H*B1(j), F2)) sst-s(bo ^ tmf). (7) 0 j i This is precisely the Adams E2-term converging to the homotopy of bo ^ . The chart can be obtained by applying the following theorem of Davis [4] which links bo^ B1(n) to Adams covers of bo or bsp, depending on the parity of n. 5 P s __ TheoremP2.7 (Davis [4]). If __n= (n1, . .,.ns), let |__n| = i=1ni and ff(n ) = s __ V s i=1ff(ni), and B1(n ) = i=1B1(ni). Then there are homotopy equivalences ( _ _ bo2|n|-ff(n), if |__n| is even; bo ^ B1(__n) ' K _ _ _ bsp2|n|-1-ff(n),if |__n| is odd; where K is a wedge of suspensions of HF2. Figure 1: Ext s,tA(H*(bo ^ tmf), F2)) sst-s(bo ^ tmf) The charts for bo and bsp are well known. Using the above theorem along with the algebraic splitting of H*tmf , we see that Adams covers of bo begin in stems congruent to 0 mod 8 while Adams covers of bsp begin in stems congruent to 4 mod 8. The first 32 stems of the chart for bo ^ tmf is displayed in Figure 1 modulo possible elements of order 2 in AdamsJfiltration s = 0 corresponding to free A(1)'s inside H*tmf . The symbol appears in Figure 1 to reduce clutter. It is used to mark the beginning of another Z-tower. In general, all Z-towers are found in stems congruent to 0 mod 4 while those supporting multiplication by j occur in stems congruent to 4 mod 8. Theorem 2.8. There is an isomorphism of homotopy groups ss*(bo ^ tmf) ~=ss* (bo^ ) 6 Proof. The E2-terms of their respective Adams spectral sequences have been shown to be isomorphic. Both spectral sequences collapse. Indeed, the classes charted in Figure 1 cannot support differentials for dimensional and natural- ity reasons. Each element of order two in Adams filtration s = 0 correspond to copies of A(1) inside H*tmf . These summands split off, obviating the __ existance of differentials. |__| 3 The topological splitting Theorem 1.1 concerns a bo-module splitting of the spectrum bo ^ tmf. The following observation will aid us in studying bo-module maps. Lemma 3.1. Let X and Y be spectra. Then [bo ^ X, bo^ Y ]bo= [X, bo^ Y ] Proof. Let ubo : S0 ! bo and mbo : bo ^ bo ! bo denote the unit and the product map of bo, respectively. Given f : bo ^ X ! bo ^ Y and g : X ! bo^ Y , the equivalence is given by the composites f 7! f O (u ^ 1) g 7! (mbo^ 1) O (1 ^ g) __ |__| The spectra (bo, mbo, ubo) and (tmf , mtmf, utmf) are both unital E1 -ring spectra [7]. This induces a unital E1 -ring structure (bo ^ tmf, m, u). This structure will play an important role in the proof of the main theorem. We begin by defining an increasing filtration of via: `n 1` n = 8i+4jB1(j) (8) j=0i=j Notationally, it will be convenient to let B(j) = 12jB1(j), so that the fil- tration (8) can be rewritten as `n ` n = 8iB(j). (9) j=0i 0 7 The proof of Theorem 1.1 will proceed inductively on n. We will assume the i-1 existance of a bo-module map %2i-1: bo^ 2 ! bo^ tmf which is a stable A-isomorphism through a certain dimension. The inductive step will be then i+1-1 to construct a bo-module map %2i+1-1: bo ^ 2 ! bo ^ tmf which is a stable A-isomorphism through higher dimensions. To do this, we will employ the pairings given in Theorem 2.5(ii). Define the map gm,n : 8nB(m) ! bo^ tmf (10) to be the restriction of %2i-1to the summand 8nB(m). Denote by gm = gm,0. Lemma 3.2. Let m = 2i and 0 n < m. Suppose there are bo-module maps fm : bo^ B1(m) ! bo^ tmf and fn : bo^ B1(n) ! bo^ tmf inducing injections on homology. Then there is a bo-module map fm+n : bo^ B1(m + n) ! bo^ tmf inducing an injection on homology. Proof. For all 0 n < m, Theorem 2.7 supplies equivalences of bo-module spectra bo^ B1(m) ^ B1(n) ' (bo^ B1(m + n)) _ K (11) where K is a wedge of suspensions of HF2. There are no maps [HF2, bo^ tmf] so that the composite m O (fm ^ fn) lifts as a bo-module map to the first summand fm+n : bo^ B1(m + n) ! bo^ tmf __ hence is also an injection in homology. |__| i-1 Corollary 3.3. Suppose there are bo-module maps %2i-1 : bo ^ 2 ! bo^ tmf and g2i : bo ^ B(2i) ! bo ^ tmf inducing injections on homology. Then there is a bo-module map i+1-1 %2i+1-1: bo^ 2 ! bo^ tmf inducing an injection on homology groups. Proof. For 0 m 2i - 1 and n 0, there are bo-module maps g2i+m,n inducing an injection in homology. These maps are obtained by applying Lemma 3.2 to g2iand the restriction of %2i-1to the summand gm,n : bo^ 8nB(m) ! bo^ tmf __ The map %2i+1-1is the wedge of these maps. |__| 8 The following observation will simplify our calculations inside the Adams spectral sequence. Lemma 3.4. Let X and Y be spectra. Suppose F : bo^ X ! bo^ Y is given by the composite (mbo^ Y ) O (bo ^ f) for some map f : X ! bo ^ Y . Then F*(rx) = rF*(x) if r 2 bo* and x 2 bo*X. __ Proof. By construction, the composite F is a bo-module map. |__| In particular, the bo-module map %2i+1-1constructed in Lemma 3.2 in- duces a map in homotopy groups in Adams filtration s = 0. The above lemma allows us to apply the bo*-module structure to extend the morphism into positive Adams filtrations. To complete the inductive step it suffices to construct a map g2i: B(2i) ! bo ^ tmf inducing an injection on homology. Indeed, we can then apply Corollary 3.3 to extend %2i-1to a bo-module map i+1-1 %2i+1-1: bo^ 2 ! bo^ tmf. To construct g2i, we will use g2i-1supplied by the inductive hypothe- sis. Once again we will attempt to use the pairing of integral Brown-Gitler spectra: B1(2i-1) ^ B1(2i-1) ! B1(2i) (12) to construct a map bo ^ B1(2i) ! bo ^ tmf. Unfortunately, Lemma 3.2 will not apply. Indeed, the above pairings (12) are not surjective in homology since the element corresponding to ii+3 inside H*B1(2i) is indecomposible. To handle this case, we turn to a lemma of Mahowald [6] made precise by Davis [3]: Lemma 3.5 (Davis [3]). If n is a power of 2, let Fn = 8n-5M2'^ B1(1). j There is a map Fn -! bo^B1(n)^B1(n) such that the cofibre of the composite 1^j mbo^1^1 ffi : bo^ Fn --! bo ^ bo^ B1(n) ^ B1(n) -----! bo ^ B1(n) ^ B1(n) is equivalent modulo suspensions of HF2 to bo ^ B1(2n). Define mi-1 : bo ^ B(2i-1) ^ B(2i-1) ! bo ^ tmf to be the bo-module map induced by the composite m O (g2i-1^ g2i-1). With Lemma 3.5 in mind, consider the diagram: i+4-5 _ffi//_ i-1 i-1 ____//_ i bo ^ 2 M2'^ B1(1) bo^ B(2 ) ^ B(2 ) bo^ B(2 ) k k mi-1|| k k gkk fflffl|uukk 2i bo ^ tmf 9 It suffices to show the composite mi-1ffi is nulhomotopic, since then mi-1 would then extend to the desired map g2i. The following theorem is essen- tially due to Davis [3, Prop. 2.8], however modified to our context. Theorem 3.6 (Davis, [3]). Suppose g2i-1: bo^ B(2i-1) ! bo^ tmf induces an injection on homology. Then i-1 i-1 ss2i+4-4 bo^ B(2 ) ^ B(2 ) ~=Z(2) (13) with generator ff2i+4-4whose image under (mi-1)] is divisible by 2. Proof. Since bo ^ tmf has the structure of an E1 -ring spectrum, the map mi-1 factors through the quadratic construction on B(2i-1), i.e., there is a map j making the the following diagram commute: bo ^4D2(B(2i-1))4 iiii ijiiii | iiiii | ii fflffl| bo ^ B(2i-1) ^ B(2i-1)____mi-1_//bo^ tmf where D2(B(2i-1)) = S1 n 2 (B(2i-1) ^ B(2i-1)). Here the 2-action on S1 is the antipode and the action on the smash product interchanges factors. Using this factorization, it suffices to show that the induced map j] in homotopy sends the generator in dimension 2i+4 - 4 to twice an element of the homotopy of the quadratic construction. This is __ proved by Davis [3]. |__| Proof that Theorem 3.6 implies Theorem 1.1. Let [xi] 2 ssi(bo ^ tmf) for i = 0, 8, 12 denote the classes in bidegree (i, 0) in the E2-term displayed in Fig- ure 1. The class [x12] does not support action by j so that x12 extends to a map B(1) ! bo^ tmf. Upon smashing with bo, we get maps g0 : bo^ B(0)! bo^ tmf g0,1: 8bo ^ B(0)! bo^ tmf g1 : bo^ B(1)! bo^ tmf inducing injections in homology. In particular, Lemma 3.2 extends these to a bo-module map %1 : bo^ 1 ! bo^tmf which is also an injection on homology. 10 Lemma 3.4 extends this morphism to positive Adams filtrations. Figure 1 demonstrates that modulo possible order 2 elements on the zero line, this map accounts for all homotopy classes through the 23-stem. Hence, it is a stable A-equivalence in this range. For the purpose of induction, assume the existance of a bo-module map i-1 %2i-1 : bo ^ 2 ! bo ^ tmf inducing a stable A-equivalence through the (12(2i)-1)-stem. In particular, there is a map g2i-1: bo^B(2i-1) ! bo^tmf of bo-module spectra inducing an injection on homology groups. Define mi-1 and ffi as above. We will show mi-1ffi ' *. oooO ____Oo oooo oooo ooo ooooo O ____O____O____O_____________________________________* *__________oo __________________________________________________* *_____________________________________________________________________________* *_________________________ 0 1 2 3 4 Figure 2: H*(M2'^ B1(1)) Figure 2 shows the cell diagram for H*(M2'^ B1(1)). Since there are no elements of positive Adams filtration in stems congruent to {5, 6, 7} mod 8 in the Adams spectral sequence converging to ss*(bo ^ tmf), the composite i+4-5 mi-1ffi restricts to a map 2 M2'! bo^ tmf. Consider the composite i+4-5 a0 2i+4-5 mi-1ffi S2 -! M2'^ B1(1) ---! bo ^ tmf i+4-5 restricting mi-1ffi to the bottom cell of 2 M2'. There are no elements of positive Adams filtration in stems congruent to 3 mod 8 so this restriction extends to the top cell i+4-4 a1 2i+4-5 mi-1ffi S2 -! M2'^ B1(1) ---! bo ^ tmf. Theorem 3.6 indicates that the class (mi-1)](ffia1) is divisible by 2. Hence, __ this map is nulhomotopic. Applying Corollary 3.3 gives the result. |__| 4 The bo -homology of tmf Both bo and tmf have the structure of E1 -ring spectra, so that the smash product bo^ tmf also inherits such a structure. The splitting of bo^ tmf into 11 pieces involving integral Brown-Gitler spectra gives a nice description of its structure as a ring spectrum. Indeed, the pairing of the B1(j) is compatible with multiplication inside H*HZ of which H*tmf is a subring. In particular, the pairings of the integral Brown-Gitler spectra induce the ring structure of bo^tmf . The induced structure on homotopy groups is given by the following theorem: Theorem 4.1. There is an isomorphism of graded bo*-algebras bo*[oe, bi, ~i| i 0] ss*(bo ^ tmf) ~=__________________________2 F (14) (~bi - 8bi+1, ~bi- 4~i, jbi) where |oe| = 8, |bi| = 2i+4 - 4, |~i| = 2i+4 and F is a direct sum of F2 in varying dimensions. Proof. Theorem 2.7 gives homotopy equivalences bo^ B(n) ^ B(2i) ! K _ (bo ^ B(n + 2i)) for all n < 2i. In particular, the induced pairings ss*(bo ^ B(n)) ss*(bo ^ B(2i)) ! ss*(bo ^ B(n + 2i)) provide an isomorphism for all n < 2i, modulo possible order 2 elements in Adams filtration zero corresponding to free A(1) inside H*tmf . Therefore, the homotopy classes inside bo, 8bo and bo ^ B(2i) for i 0 generate the homotopy of ss*(bo ^ tmf). Hence, it suffices to examine the pairings bo ^ B(2i) ^ B(2i) ! bo^ B(2i+1). Figure 3 depicts the E2-term of the Adams spectral sequence converging to bo*B(1) along with its generators as a bo*-module. With these generators, we can determine the decomposibles inside bo*B(2i). Indeed, Lemma 3.5 provides us with a fiber sequence i+5-4 bo^ B(2i) ^ B(2i) ! bo^ B(2i+1) ! bo^ 2 M2'^ B1(1) (15) inducing a long exact sequence of Ext -groups. Figure 4 shows how to use (15) to form the E2-page of bo ^ B(2i+1). The arrows represent subsequent differentials and the dotted lines non-trivial extensions. The classes in black are those contributed by bo ^ B(2i) ^ B(2i), i.e., the decomposible classes 12 Figure 3: Ext s,tA(1)(H*B(1), F2) hit by multiplication by elements in the summand bo ^ B(2i). Those in red i+5-4 (or grey) are contributed by bo ^ 2 M2'^ B1(1). Denote by bi+1 the class found in bidegree (2i+5- 4, 0) and ~i+1 the class in (2i+4, 1). These two elements are thus indecomposible in the ring ss*(bo ^ tmf). Note that the class in bidegree (2i+5- 8, 0) corresponds to the element b2i. In particular, ~b2i= 8bi+1. Also note that ~bi+1 = 4~i+1 and jbi+1 = 0. Figure 4: Ext s,tA(1)(H*B(2i+1), F2) __ |__| Remark 4.1. The splitting of bo^ tmf can also be used to give a description of the bo-cohomology of tmf . Indeed, Lemma 3.1 gives that [tmf , bo] = [bo ^ tmf, bo]bo. Since Theorem 1.1 provides a splitting as bo-module spectra, 13 one has the following chain of equivalences of bo*-comodules: bo*tmf = [tmf , bo] = [bo ^ tmf, bo]bo " # ` = 8nbo ^ B(m), bo m,n 0 bo " # ` = 8nB(m), bo m,n 0 M = -8nbo*B(m) m,n 0 A complete description of the summands bo*B(m) is given by Carlsson [1]. The comultiplication on bo*tmf is once again induced by the pairings of inte- gral Brown-Gitler spectra. It would be interesting to determine the explicit generators and relations as a bo*-coalgebra. References [1]Gunnar Carlsson, Operations in connective K-theory and associated co- homology theories, Ph.D. thesis, Stanford, 1976. [2]Fred R. Cohen, Donald M. Davis, Paul G. Goerss, and Mark E. Ma- howald, Integral Brown-Gitler spectra, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1299-1304. [3]Donald M. Davis, The splitting of BO<8> ^ bo and MO<8> ^ bo, Trans. Amer. Math. Soc. 276 (1983), no. 2, 671-683. [4]Donald M. Davis, Sam Gitler, and Mark Mahowald, The stable geomet- ric dimension of vector bundles over real projective spaces, Trans. Amer. Math. Soc. 268 (1981), no. 1, 39-61. [5]Paul G. Goerss, John D. S. Jones, and Mark E. Mahowald, Some gener- alized Brown-Gitler spectra, Trans. Amer. Math. Soc. 294 (1986), no. 1, 113-132. [6]Mark Mahowald, bo-resolutions, Pacific Journal of Mathematics 92 (1981), no. 2, 365-383. 14 [7]J.P. May, Infinite loop space theory, Bull. Amer. Math. Soc. 83 (1977), no. 4, 456-494. [8]Charles Rezk, Supplementary notes for Math 512 (ver. 0.18), http://www.math.uiuc.edu/~rezk/512-spr2001-notes.pdf, July 2007. [9]Don H. Shimamoto, An integral version of the Brown-Gitler spectrum, Trans. Amer. Math. Soc. 283 (1984), no. 2, 383-421. 15