THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RP n NITU KITCHLOO AND W. STEPHEN WILSON Abstract.In [HK01], Hu and Kriz construct the real Johnson-Wilson spec- trum, ER(n), which is 2n+2(2n - 1) periodic, from the 2(2n - 1) periodic spectrum E(n). ER(1) is just KO(2)and E(1) is just KU(2). We com- pute ER(n)*(RP1 ) and set up a Bockstein spectral sequence to compute ER(n)*(-) from E(n)*(-). We combine these to compute ER(2)*(RP2n) and use this to get new non-immersions for real projective spaces. Our l* *owest dimensional new example is an improvement of 2 for RP48. August 17, 2007 1.Introduction We have three main goals in this paper. First, we want to introduce the second real Johnson-Wilson cohomology, ER(2)*(-), as a real problem solving tool. Sec- ond, we develop computational tools for ER(2)*(-) and third, we apply ER(2)*(-) and our computational tools to prove some new families of non-immersions for re* *al projective spaces. The lowest dimensional example is RP 48. Our concern is with the real Johnson-Wilson cohomology, ER(n)*(X), developed in [HK01 ], [Hu01 ], [KW ], and [KW07 ]. It is 2n+2(2n - 1) periodic and cons* *tructed from the 2(2n - 1) periodic spectrum E(n). ER(1) is just KO(2)and E(1) is just KU(2). Our long term goals are to develop and apply ER(2)*(-) but much of our preliminary work stands for ER(n)*(-) in general. The coefficient ring, E(n)*(S0), is given by Z(2)[v1, v2, . .,.vn1] where the* * degree of vk is -2(2k- 1). E(n) is a complex orientable theory and as such it has a fo* *rmal group law with coefficients made up from the vk's and a two series, X Xn k (1.1) [2](u) = akuk+1 = Fvku2 . k 0 k 0 We prefer to grade our cohomologies over Z=(2n+2(2n - 1)). For E(n)*(-) we just set v2n+1n= 1. We start off with a theorem that goes back to the equivariant roots of ER(n). Theorem 1.2. Let ~(n) = 22n+1-2n+2+1. Then there is a u 2 ER(n)1-~(n)(RP 1) and ER(n)*(RP 1) ' ER(n)*[[u]]=([2](u)) where the vk are replaced by vER(n)k2 ER(n)(~(n)-1)(2k-1)(S0). In our special case of interest, ER(2), vER(2)2= 1 and we rename vER(2)1as ff. Our relation becomes (1.3) [2](u) = 2u +F ffu2 +F u4 1 2 NITU KITCHLOO AND W. STEPHEN WILSON and it maps to the same relation in E(2)*(RP 1) for our 48 periodic E(2). To do this we have to replace the usual x2 2 E(2)2(RP 1) with the image of the u 2 ER(2)-16(RP 1), which is v32x2 and replace v1 with the image of ff 2 ER(2)1* *6, v52v1. Since v2 is a unit this is not a problem. For our applications we need ER(2)*(RP 2n) and the equivariant approach doesn* *'t work here. The stable cofibration of [KW07 ]: (1.4) ~(n)ER(n) _x__//_ER(n)____//E(n), gives us a long exact sequence: (1.5) ER(n)*(X)d_____x____//ER(n)*(X)dH HHHH vvv @ HHH zzvvaevvv E(n)*(X) where x lowers degree by ~(n) and @ raises degree by ~(n) + 1. This is a classic exact couple andnleads+us1directly to a Bockstein spectral sequence for x-torsi* *on. We know that x2 -1= 0 so there can be only 2n+1 - 1 differentials. We set up this spectral sequence and compute d1. In the case n = 1 it can be used to comp* *ute KO*(2)(X) from KU*(2)(X). In this case there are only 3 differentials. For the * *case of interest to us, n = 2, there are only 7 differentials and because for many o* *f our spaces our E1 term, E(2)*(X), is even degree, we have only 4 differentials beca* *use the d2r are odd degree. We use the Bockstein spectral sequence to compute ER(2)*(RP 2n) after setting up the spectral sequence. This breaks up into 8 cases depending on n modulo 8. * *The descriptions can get lengthy (times 8) but can be read off directly from the Bo* *ckstein spectral sequence which is quite compact. Here we will be content to describe t* *he part of ER(2)*(RP 2n) we are really interested in, namely ER(2)16*(RP 2n). We like to describe our groups with what we call a 2-adic basis, i.e. a set * *of elements such that any element in our group is written as a finite sum of these elements with coefficients 0 or 1. Usually we can compute 2 times an element by using 1.3. Keep in mind that, using this notation, the usual E(2)*(RP 2n) is given by ff* *kuj, with 0 k and 0 < j n and the 48-periodic version is given by vi2ffkuj, with 0 i < 8. We always use reduced cohomology. Theorem 1.6. ER(2)16*(RP 2n) consists of the elements ffkuj, with 0 k and 0 < j n, and, when n = 1 or 6 modulo 8, ffkun+1, n = 2 or 5 modulo 8, ffkun+1, and un+2, n = 3 or 4 modulo 8, un+1, un+2, and un+3, and no others. Our applications use the four cases, n = 1, 2, 5 and 6, modulo 8, where we ha* *ve ffkun+1. Here we have a purely algebraic, no topology implied or used, surjecti* *on ER(2)16*(RP 2n) -! E(2)16*(RP 2n+2) n = 1, 2, 5, 6 mod 8. This is the key to getting our non-immersion results. We also need the isomorph* *ism given above for ER(2)16*(RP 2n) -! E(2)16*(RP 2n) n = 0, 7 mod 8. THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 3 From [Jam63 ] we know that if there is an immersion of RP 2ninto R2k then there is an axial map: K -2k-2 2K -2n-2 RP 2nx RP 2 -! RP . Don Davis, in [Dav84 ], shows that there is no such map when n = m + ff(m) - 1 and k = 2m - ff(m) where ff(m) is the number of ones in the binary expansion of m. We use the commuting diagram: (1.7) E(2)*(RP 2n) E(2)*E(2)*(RP 2K -2k-2)oo___ E(2)*(RP 2K -2n-2) OO OO | | | | | | ER(2)*(RP 2n) ER(2)*ER(2)*(RP 2K -2k-4)oo_ ER(2)*(RP 2K -2n-2). Don Davis shows, in his case, that the u2K-1-n which is zero on the right for E(2)*(-) is non-zero on the left. We use the commutativity of the above diagram to get our non-immersions, which are an improvement of 2. The abelian group structure of the tensor product is extremely complicated but by relying on [Dav* *84 ] we avoid ever having to consider it. We restrict our attention to the cases n =* * 0 and 7 modulo 8 where the right hand vertical map is an isomorphism in degrees 16*. So is the left term of the tensor product. For the right term of the ten* *sor product we need -k - 2 to be 1, 2, 5 or 6 modulo 8 in order to get a surjection* *. Our non-immersion theorem is: Theorem 1.8. When the pair (m, ff(m)) is, modulo 8, (2, 7), (7, 2), (6, 3), (3,* * 6), (7, 1), (4, 4), (3, 5), or (0, 0), then RP 2(m+ff(m)-1)does not immerse(*) inR2(2m-ff(m)+1). When the pair (m, ff(m)) is, modulo 8, (4, 3), (1, 6), (0, 7), or (5, 2), then RP 2(m+ff(m))* in R2(2m-ff(m)+1). This theorem is not for free from Don Davis's work. The injection of the tens* *or product, E(2)*(RP 2m) E(2)*E(2)*(RP 2n) -! E(2)*(RP 2m^ RP 2n), is ancient knowledge. For us to make use of Don Davis's computation, which tells us the obstruction is non-zero in the tensor product, we need the corresponding* * part of the tensor product in ER(2)*(-) to inject into the cohomology of the product. That is where our work comes in. We do not compute the entirety of the ER(2) cohomology of the product but just enough to give us what we need. Looking closely at our theorem to decide if we really have anything new or not, let's take the pair (m, ff(m)) = (6, 3). Let m = 2 + 4 + 2i, i > 3, then 2n = 2(m + ff(m) - 1) = 2(2 + 4 + 2i+ 3 - 1) = 16 + 2i+1and 2(2m - ff(m) + 1) = 2(4 + 8 + 2i+1- 3 + 1) = 4 + 16 + 2i+2= 20 + 2i+2. Our result, in this case, sh* *ows that RP 16+2i+1* R20+2i+2. Looking at the best known results, compiled by Don Davis in [Dav ], we know that RP 16+2i+1* R18+2i+2but RP 16+2i+1 R23+2i+2. Furthermore, when i = 4, 5 and 6 we get results for very low spaces: RP 48* R84 RP 80* R148 RP 144* R276. 4 NITU KITCHLOO AND W. STEPHEN WILSON Our claim is that this alone makes a good case for ER(2)*(-) as a powerful tool. There are only 8 projective spaces, RP n, with n 50, where the best possible results are not yet known. For these 8 spaces there were a total of 26 gaps, no* *w 24. Our result is the first improvement in over 20 years for any RP nwith n 50. The pair (4, 4) gives i 106+2i+1 RP 62+2* R where i > 5. The lowest cases here are RP 126* R234 RP 190* R362. These are nice because they get onto Don Davis's tables, [Dav ], just barely, b* *ut at least this way we know we have something new, which would be difficult to te* *ll otherwise. From the second part of the theorem we only get an improvement of one dimen- sion. The pair (4, 3) gives m = 4 + 2i+ 2j with 2 < i < j. We get i+1+2j+1 12+2i+2+2j+2 RP 14+2 * R . With i = 3 we get j+1 44+2j+2 RP 30+2 * R with lowest examples: RP 62* R108 RP 94* R172 RP 158* R300. With i = 4 we get j+1 76+2j+2 RP 46+2 * R with lowest examples: RP 110* R204 RP 174* R332. For the i = 3 and 4 cases above there is now just a gap of 1 between known non-immersions and immersions. i Don Davis, [Dav ], keeps track of the best results for RP d+2 for 0 d < 64. 24 of these 64 are best possible at this time. We have improved the results for* * 4 of the 40 remaining: d = 16, 30, 46 and 62. We conjecture that this machine can improve results for d = 32, 48, 49, 54, 56 and 57, but these could be computati* *onally intensive and so don't fit into this paper. There is a bit of a saga associated with the RP 48case. The theory tmf is cle* *arly stronger than ER(2) so the question arose as to why [BDM02 ] didn't see this r* *esult. When they looked again at their results they realized that they had actually st* *ated a family that included this case but had overlooked it when converting to the tables [Dav ]. A closer look at [BDM02 ] revealed a simplification that had no* *t been justified. That allows us to technically slip in with this result before they m* *anaged to patch up some of their theorems, which now include this. The theory tmf is very complicated and because of this complexity it cannot presently approach our other results. The paper begins by computing ER(n)*(RP 1) using the equivariant approach. We then set up the Bockstein spectral sequence for computing ER(n)*(-) from E(n)*(-). We use the Bockstein spectral sequence to compute all of the (8 cases) of ER(2)*(RP 2n). When this is done we extract, from the Bockstein spectral sequence, just what we need about ER(2)*(-) of products. Then we wrap things up by producing our non-immersion results. THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 5 The authors had worked with ER(n) with an emphasis on ER(2) before with an eye to eventually attacking non-immersion problems. This project really got und* *er- way at the Bendersky-Davis 60th birthday conference at Newark, Delaware, April 2005, where, over lunch, the second author was inspired to work on the problem by Jesus Gonz'alez and Martin Bendersky who have continued to correspond with the authors throughout the project. Don Davis then joined the group, and without his help and tables we would never even know if we had new results. Thanks to a* *ll three. At the end of the paper we have a short section explaining how to compute ER(2)*(RP 2n) using the Atiyah-Hirzebruch spectral sequence. This was how it was first done and it required some interesting twists. Then it was thought that the Bockstein spectral sequence approach was unworkable. The Atiyah-Hirzebruch spectral sequence approach broke down when it came to studying the products; things just got too complicated. We then resurrected the Bockstein spectral se- quence approached which proved successful. One of the unexpected and unnec- essarily complicating factors was our choice of product spaces to study first to learn about products. RP 16was essential for the study of ER(2)*(RP 2n) for the Atiyah-Hirzebruch spectral sequence approach and it is "nicer" than other RP 2n. RP 16xRP 16was thus chosen on the grounds that it should be both elementary and educational. It turns out that great simplification occurs when one space is bi* *gger than the other, or, phrased differently, complications occurred for RP 16xRP 16* *that only occur when the spaces are the same. Much time was lost on these irrelevant complications. 2. Equivariant results Recall from [HK01 ], that there is a real spectrum E(n) corresponding to the Johnson-Wilson spectra. In particular, E(n) consists of a bigraded family of Z=* *(2)- spaces E(n)(a,b). We denote by ER(n)(a,b), the homotopy fixed point space of the Z=(2) action on E(n)(a,b). The collection of spaces E(n)(k,0)form a (naive) Z=(2)-equivariant omega spectrum, and we define the spectrum ER(n) as the cor- responding homotopy fixed point spectrum ER(n)(k,0). Furthermore, it is shown in [HK01 ] that the real spectrum E(n) satisfies a strong completion theorem, in t* *he sense that the canonical map: ' : E(n)(a,b)-! Map (EZ=(2)+ , E(n)(a,b)) is a Z=(2)-equivalence, where EZ=(2) represents the free, contractible Z=(2)-co* *mplex. For a space X with a Z=(2)-action, we may define bigraded cohomology groups ER(n)a,b(X) [HK01 ] as the groups ER(n)(a,b)(X) = ss0Map Z=(2)(X, E(n)(a,b)) The strong completion theorem has a few useful consequences: Proposition 2.1. Let X be a pointed space with the trivial Z=(2) action. Then t* *he map ' above induces an isomorphism: ER(n)(k,0)(X) -! ER(n)k(X) The proof of the above proposition follows directly from the strong completion theorem, and is left to the reader. Another useful consequence of the strong co* *m- pletion theorem is the following: 6 NITU KITCHLOO AND W. STEPHEN WILSON Proposition 2.2. Let X and Y be pointed Z=(2)-spaces. Assume that f : X ! Y is a Z=(2)-equivariant map that is a homotopy equivalence (non equivariantly). Then f induces an isomorphism: f* : ER(n)(a,b)(Y ) -! ER(n)(a,b)(X) Proof.By the strong completion theorem, we may write the groups ER(n)(a,b)(Z) as ss0Map Z=(2)(EZ=(2)+ ^ Z, E(n)(a,b)), for an arbitrary Z=(2)-space Z. Now consi* *der the map Id ^ f : EZ=(2)+ ^ X -! EZ=(2)+ ^ Y Since the spaces EZ=(2)+ ^X and EZ=(2)+ ^Y are free Z=(2)-spaces, the map Id^f is a Z=(2)-homotopy equivalence. It follows from the above previous observation, that f* is an isomorphism. 3. Cohomology of Projective spaces We shall use the above propositions to describe the ER(n) cohomology of the i* *nfi- nite projective space. To this end, we need to consider the complex projective * *space CP 1, with the action of Z=(2) given by complex conjugation. The space CP 1 supports the Z=(2)-equivariant tautological complex line bundle fl. Moreover, f* *l is real-oriented, in the sense that it admits a real Thom class t 2 ER(n)(1,1)(T h* *(fl)). Let u 2 ER(n)(1,1)(CP 1) denote the Euler class of fl. The standard argument using the Atiyah-Hirzebruch spectral sequence may be invoked in the real setting to show that ER(n)(*,*)(CP 1) ' ER(n)(*,*)[[u]], as a bigraded ring [HK01 ]. Now consider the real bundle fl 2. The Euler class of fl 2 is simply [2](u). * *Let RP"1 denote the unit sphere bundle of fl 2. Notice that RP"1 may be identified with the space of (real) lines in C1 and as such, it supports a nontrivial Z=(2* *)-action given by complex conjugation. Let f : RP 1 ! RP"1 denote the inclusion induced by the R1 C1 . Notice that f is a Z=(2)-equivariant map with RP 1 having a trivial Z=(2)-action. Moreover, f is a (non equivariant) homotopy equivalence. * *It follows from the previous proposition that: Lemma 3.1. The map f : RP 1 ! RP"1 induces an isomorphism f* : ER(n)(a,b)(RP"1 ) -! ER(n)(a,b)(RP 1) We may calculate the cohomology of RP"1 using the Gysin sequence for the bundle fl 2: . .-.! ER(n)(a-1,b-1)(CP 1) [2](u)-!ER(n)(a,b)(CP 1) -! ER(n)(a,b)(RP"1 ) -! ER(n)(a,b-1)(CP 1) -! . . . Since [2](u) is clearly not a zero divisor in ER(n)(*,*)(CP 1) we conclude that ER(n)(*,*)(RP 1) ' ER(n)(*,*)[[u]]=[2](u). At this point, let us recall the invertible class y(n) 2 ER(n)(-~(n),-1)[KW07* * , Claim 4.1], where ~(n) = 22n+1-2n+2+1. We have the vER(n)k2 ER(n)(-2k+1,-2k+1) and get elements y(n)-2k+1vER(n)k= vER(n)k2 ER(2k-1)(~(n)-1)(S0). We may nor- malize u to be in degree (1 - ~(n), 0) by redefining u as u y(n). Using the fi* *rst proposition, we get: THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 7 Theorem 3.2. Let ~(n) = 22n+1- 2n+2 + 1. Then ER(n)*(RP 1) is isomorphic to ER(n)*[[u]]=([2](u)) where u 2 ER(n)1-~(n)(RP 1) and the vk are replaced by vER(n)k2 ER(n)(~(n)-1)(2k-1)(S0). We may also calculate the ER(n) cohomology of spaces of the form X ^ RP 1 using similar ideas. Let X be a space with a trivial Z=2-action. As before, we * *may show that ER(n)(*,*)(X ^ CP 1) ' ER(n)(*,*)(X)[[u]]. Again, we consider the real bundle 0 x fl 2 over X ^ CP 1 with Euler class [2](u). We have the Gysin sequen* *ce for this bundle: - ! ER(n)(a-1,b-1)(X)[[u]] [2](u)-!ER(n)(a,b)(X)[[u]] -! ER(n)(a,b)(X ^ RP"1) -! ER(n)(a,b-1)(X)[[u]] -! . . . Since we know that ER(n)(a,b)(X ^ RP"1) is isomorphic to ER(n)(a,b)(X ^ RP 1) from an earlier proposition, we would be done provided we knew that [2](u) was not a zero divisor in ER(n)(*,*)(X)[[u]]. For this we require an algebraic lemm* *a (let v0 = 2): Lemma 3.3. Let M be a ER(n)(*,*)-module such that M has no infinitely I- divisible elements, where I is the ideal (v0, vER(n)1, . .,.vER(n)n-1) i.e " IkM = 0, k Then [2](u) is not a zero divisor in M[[u]]. Proof.Filter M by submodules 0 = \Mk . . .M2 M1 M0 = M, where Mk = IkM. Notice that vER(n)iMk Mk+1 for i < n. Now let f(u) 2 M[[u]] be a power series with the property f(u)[2](u) = 0, then working in M=M1[[u]], this equality reduces to vER(n)nf(u) u2n = 0, which implies that f(u) belongs to M1[* *[u]]. Continuing with M1=M2[[u]] and so forth, we conclude that f(u) 2 \kMk[[u]] = 0. It follows from the above lemma and the Gysin sequence that that Theorem 3.4. Let X be a space with the property that ER(n)*(X) has no infinitely vER(n)idivisible elements for i < n, (e.g. X is finite or X = RP 1). Let u be t* *he class defined earlier, then we have an isomorphism: ER(n)*(X ^ RP 1) ' ER(n)*(X)[[u]]=([2](u)) 4.The Bockstein spectral sequence We begin with the stable cofibration 1.4 of [KW07 ]. (4.1) ~(n)ER(n) _x__//_ER(n)____//E(n), where x 2 ER(n)-~(n)and ~(n) = 22n+1- 2n+2 + 1. The fibration gives us the long exact sequence 1.5. Our long exact sequence i* *s an exact couple and so gives risento+a1spectral sequence whose differentials give * *us the x-torsion. We have that x2 -1= 0 so there are a finite number of differential* *s. Most of the details of the spectral sequence are fairly straightforward but s* *ince we will make extensive use of it we want to be careful about its basics, so we collect them in a theorem. We will need complex conjugation. E(n) is a complex 8 NITU KITCHLOO AND W. STEPHEN WILSON orientable theory and as such has a complex conjugation map on it that we denote by c. We always use reduced cohomology. We know that 2x = 0, a simple fact that isn't necessary in the spectral sequence but should be kept in mind. Theorem 4.2 (The Bockstein Spectral Sequence for ER(n)*(X)). (i)The exact couple of 1.5 gives a spectral sequence, Er, of ER(n)* modules, starting with E1 ' E(n)*(X). (ii) n+1 E2 = 0. (iii)The targets of the differentials, dr, represent the xr-torsion generato* *rs of ER(n)*(X) as described below. (iv)The degree of dr is r~(n) + 1. (v)Filter ER(n)*(X) by Ki, the kernel of xi. Then {0} = K0 K1 K2 . . .K2n+1-1= ER(n)*(X). (vi)Filter M = ER(n)*(X)=xER(n)*(X) by Mi the image of Ki so {0} = M0 M1 M2 . . .M2n+1-1= M. M=Mr-1 -! Er, r 1 injects andMr=Mr-1 ' imagedr. (vii) dr(ab) = dr(a)b + c(a)dr(b). (viii) n-1) d1(z) = v-(2n (1 - c)(z) wherec(vi) = -vi. (ix) Ifc(z) = z 2 E1 thend1(z) = 0. Ifc(z) = z 2 Er thendr(z2) = 0. (x)The following are all vector spaces over Z=(2): Kj=Ki, Mj=Mi, j i > 0, and Er, r > 1. Most of the theorem followsnimmediately+from1the basic properties of an exact couple and the fact that x2 -1= 0. We defer those proofs we need until after * *we have worked some simple examples. Remark 4.3. Note that the image of ER(n)*(X) ! E(n)*(X) gives the set of ele- ments that are targets of differentials and therefore always have all the diffe* *rentials trivial on them. Note also that anything in the image is invariant under the ac* *tion of c. Remark 4.4. Since ER(n)*(-) is 2n+2(2n-1) periodic we will consider it as graded over Z=(2n+2(2n - 1)). We have to do the same then with E(n)*(-) and we can accomplish this by setting the unit v2n+1n= 1 in the homotopy of E(n). THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 9 Remark 4.5. Recall that KO(2)= ER(1). For a very simple warmup exercise we can compute the coefficient ring, ER(1)*(S0), using the spectral sequence. Our E1 term is E(1)*(S0) made 8 periodic where E(1) = KU(0)so E1 is just Z(2)on generators vi1, 0 i < 4. We have that c(v1) = -v1 so d1(v1) = v-11(1 - c)v1 = v-11(v1 + v1) = 2. Similarly, d1(v31) = 2v21. Since c(v21) = v21we have d1(v21) = 0. We have the Z* *(2) free submodule generated by 2 and 2v21giving us our x1-torsion. Give the element of ER(1)*(S0) that maps to 2v21the name fi. All that is left for our E2 term is* * the Z=(2) vector space generated by 1 and v21. They have degree 0 and 4 respectivel* *y. The only differentials we have left are d2, which is odd degree so we don't hav* *e it, and d3 which has degree 4. Since we know 1 is in the image from ER(1)*(S0) the differential must be d3(v21) = 1. We have recovered the well known homotopy of KO(0). Our x is really j and we have j3 = 0 on 1. We have 2j = 0 = jfi and we have fi2 = 4. All of this read off from our spectral sequence for j torsion. 5. The Spectral Sequence for ER(2)*(-). In [KW ] we describe the homotopy of ER(2) in more detail than we need here. ER(2)*(S0), graded over Z=(48), is generated by elements, x, w, ff, ff1, ff2, a* *nd ff3 of degrees -17, -8, -32, -12, -24, and -36 respectively. Some relations are giv* *en by 0 = 2x = x7 = x3w = x3ff = xffi. As a module over Z(2)[ff], the homotopy can be described as having generators: 1, w, ff1, ff3, and ff2 with one relation: ffff2 = 2w, copies of Z=(2)[ff] on generators x, x2, xw, x2w, and copies of Z=(2) on x3, x4, x5, x6. The focus of our important computations which will rely on the spectral seque* *nce will be for the theory ER(2)*(-). The homotopy of ER(2) is non-trivial and taki* *ng a look at it from the perspective of the spectral sequence is well worth the ef* *fort, plus it helps us compute differentials in the future. Our spectral sequence begins with E1 = E(2)*(S0), which is just a free Z(2)[v* *1] module on a basis given by vi2for 0 i < 8. We know our d1. We get: d1(v2s+12) = v-32(1 - c)(v2s+12) = v-322v2s+12= 2v2s-22 Similarly, d1(v2s2) = 0. This seems like a good start but if we multiply v2s+12* *by v1 the differential suddenly becomes zero because c(v1v2s+12) = v1v2s+12. Complica* *ting matters even more, if we were to multiply by v21we would be back to our multipl* *i- cation by 2v-32. This is a problem that would persist in all of our computation* *s for ER(2)*(X). We need a better way to deal with this. Our solution to this problem is to introduce the element ff discussed for 1.3. The image of ff is v1v522 E(2)-32(S0) from [KW07 ] and [KW ]. (We'll discuss * *ff more in the next section.) Because v2 is a unit, this is a good substitute for * *our plain v1. Furthermore, it is invariant under c because it is in the image of th* *e map from ER(2)*(S0), or, we could just compute that v1v52is invariant because it has 10 NITU KITCHLOO AND W. STEPHEN WILSON an even number of v's. We now rewrite the homotopy of E(2) as Z(2)[ff, v21] but again set v82= 1. We can go back to our computation of d1 on v2s+12where E1 is now a free Z(2)[* *ff] module on vi2, 0 i < 8. Now we could compute d1(ffkv2s+12) = d1(ffk)v2s+12+ c(ffk)d1(v2s+12) = 0 + ffk2v2s-22, but this really follows automatically from the fact that the spectral sequence * *is a spectral sequence of ER(2)* modules. After the d1 in our spectral sequence for ER(2)*(S0), all we have left for E2 is the free Z=(2)[ff] module with basis given by {1, v22, v42, v62}. We give n* *ames to the elements of ER(2)*(S0) that must be x1-torsion and map to 2v2s2. Let ffi 2 ER(2)-12i(S0) map to 2v2i2where ff0 = 2. In ER(2)*(S0), these elements generate a free x1-torsion submodule over Z(2)[ff]. Since all our remaining elements in E2 are in even degrees we can only have o* *dd differentials since the even ones have odd degrees. Our choices are d3, d5, and* * d7. The degree of d5 is 38. If we look at our E2 in degrees module 16 we find that * *we only have elements in degrees 0, 4, 8, and 12. The mod 16 degree of d5 is 6 and so must be zero. Note also that we must have two non-trivial differentials beca* *use v42= (v22)2 and we can apply our Theorem 4.2 to show that our first new differe* *ntial must be trivial on this. We need the differentials on the coefficients because we will use them regula* *rly in our other computations. We also want to demonstrate how much information can be extracted from the spectral sequence without much input. In our present case all we have done is replace v1 with the invariant ff = v1v52. Proceeding,* * we must have a d3 and it must be non-trivial on v22and v62. The degree of d3 is 4, (remember, we are graded over Z=(48)), so d3(v22) = ff3k+1v42for some k for deg* *ree reasons. Multiply by v42to get d3(v62) = ff3k+11. Unfortunately, we don't know * *k. However, if we keep going, we can compute our E4 = E7. It is just ffi1 and ffiv* *42 for 0 i 3k. Since we know that ffi1 is in the image, it must be the target * *of differentials and all that is left is d7(ffiv42) = ffi1. Since E8 must be zero,* * this is forced. At this stage we have to introduce a fact, namely that x3ff = 0. That forces our k above to be zero and our d7 to just be d7(v42) = 1. Name the element that maps to ffv42, w 2 ER(2)-8(S0). Our x3-torsion elements are given by ffk and wf* *fk. Finally, our only x7-torsion element is 1. In order to do this computation the only thing we had to use that didn't come directly from the spectral sequence was the fact that x3ff = 0. We can recover most, if not all, of the ring structure by looking at the image of ER(2)*(S0) in E(2)*(S0) (for example, ff22= 4 and ff2ff = 2w). We want another relation, note that in our spectral sequence we have w2 = (ffv42)2 = ff2v82= ff2. This is only modulo x but this is in degree -16. The degree of x is -17 and the* *re are no elements at all in degree 1, so there is no z such that we could possibl* *y have xz + w2 = ff2 so this relation, w2 = ff2 must be true on the nose. From our theorem and our computation: THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 11 Proposition 5.1. In the Bockstein spectral sequence for ER(2)*(X), the map d1 is an ER(2)*=(x) module map. d2 and d3 are Z=(2)[ff, w]=(w2 = ff2) module maps. d4, d5, d6, and d7 are only Z=(2) module maps. Proof.Of course all of these differentials are really still ER(2)* module maps * *but some of the elements of ER(2)* are zero in Er. For example, the ffi and 2 are a* *ll zero in E2. All that is left then is Z=(2)[ff, w]=(w2 = ff2) but w and ff go to* * zero in E4 leaving only Z=(2). Our Bockstein spectral sequences will be modules over ER(2)*. We collect some of the facts we will use repeatedly: Proposition 5.2. d1(v2s+12) = 2v2s-22 d3(v4s-22) = ffv4s2 d7(v42) = 1. d1(v2s2) = 0 d2(v2s2) = 0 dr(v4s2) = 0 3 r < 7 6. Proof of Theorem 4.2 The spectral sequence obtained from 1.5 is a classic example of an exact coup* *le. Everything but the facts about the differentials is automatic. Even the product rule for dr follows if we know it for d1. It is as if Bill Massey consulted us * *about what we needed before he wrote [Mas54 ]. We have complex conjugation for our involution on E(n)*(X) and the trivial involution, i.e. the identity, on ER(n)** *(X). Our situation then fits [Mas54 ] exactly. Assuming our formula for d1 we confirm the product formula for it: n-1) -(2n-1) -(2n-1) d1(ab) = v-(2n (1 - c)(ab) = vn (ab - c(ab)) = vn (ab - c(a)c(b)) n-1) 1 1 = v-(2n ((a - c(a))b + c(a)(b - c(b))) = d (a)b + c(a)d (b). We prove the part that assumes c(z) = z: n-1) -(2n-1) d1(z) = v-(2n (1 - c)(z) = vn (z - z) = 0. For the second case, d1(z2) = 0 because z2 is invariant under c. For r > 1, dr(z2) = dr(z)z + c(z)dr(z) = dr(z)z + zdr(z) = 2zdr(z) which is zero since we are working modulo 2 for r > 1. All that remains is to get our formula for d1 and prove our statements about mod 2 vector spaces. Let's continue to assume our formula for d1 and show our Z=(2) vector spaces. First, we note that 2x = 0 so 2ER(n)*(X) K1. This is all we need to show that Kj=Ki, and Mj=Mi, j i > 0 are Z=(2) vector spaces. To show that Er, r > 1 is a Z=(2) vector space, it is enough to show it for E2. We start with an arbitrary element y 2 E1 with 2y 6= * *0. Obviously, if d1(y) 6= 0 this situation does not persist to E2 so we can assume* * that d1(y) = 0. First we need n-1 -(2n-1) 2n-1 -(2n-1) 2n-1 2n-1 d1(v2n ) = vn (1 - c)vn = vn (vn + vn ) = 2 n-1 (which, by the way, shows 2x = 0). Consider the element v22 y, we have: n-1 1 2n-1 2n-1 1 d1(v22 y) = d (v2 )y + c(v2 )d (y) = 2y + 0. 12 NITU KITCHLOO AND W. STEPHEN WILSON Thus no multiplication by 2 survives to E2 which concludes our proof. Wenhave only one thing left to do, and that is to prove the formula d1 = v-(2n-1)(1 - c). We've put this off till last because it requires a review of * *the source of our fibration. This also gives us a chance to describe some of the ge* *neral properties of ER(n)*(X). In [KW07 ] we have bigraded spaces, E(n)a,bwith b = 0 giving our standard spectrum for E(n). Likewise we have ER(n)a,bwith b = 0 giving our spectrum for ER(n). There is ample opportunity for confusion here. Before we proceed, let's do a little review of all the elements named vk. Our unadorned element is k-1) 0 -2(2k-1),00 vk 2 E(n)-2(2 (S ) = E(n) (S ) where E(n) is the bigraded equivariant spectrum with complex conjugation, c, ac* *t- ing on it. The element k-1),-(2k-1)0 vE(n)k2 E(n)-(2 (S ) is invariant under the action of c and gives rise to k-1),-(2k-1)0 vER(n)k2 ER(n)-(2 (S ). We have an element oe 2 ss0(E(n)1,-1) with a non-trivial Z=(2) action on it. However, the element oe2n+1lifts to a un* *it in ss0(ER(n)2n+1,-2n+1) = ER(n)2n+1,-2n+1(S0). The first thing we want to show is how the invariant vE(n)k2 E(n)-2k+1,-2k+1(* *S0) is connected to our vk 2 E(n)-2(2k-1),0(S0) E(n)-2(2k-1)(S0). We have k+1 vk = vE(n)koe-2 . Since there are an odd number of oe's we get our c(vk) = -vk. In [KW07 , Claim 4.1] we produced an invertible homotopy element n-1 -2n+1(2n-1-1) -~(n),-1 0 y(n) = (vER(n)n)2 oe 2 ER(n) (S ) and multiplication by it gives an isomorphism ER(n)a+1+~(n),b(X) ' ER(n)a+1,b-1(X) Note that n-1) 2n+1(2n-1-1) ER(n) -(2n-1) -(2n-1) -1 y(n)-1 = (vER(n)n)-(2 oe = (vn oe ) oe n-1) which reduces to v-(2n oe-1 in E(n)~(n),+1(S0) . We get a new element k+1 (~(n)-1)(2k-1)0 vER(n)k= vER(n)ky(n)-2 2 ER(n) (S ) n-1)(2k-1) which reduces, in E(n)(~(n)-1)(2k-1)(S0), to vkv-(2n . In order for the map of ER(n) to E(n) to work nicely we would replace the n-1)(2k-1) element vk 2 E(n)*(S0) with vkv-(2n . In particular, when k = 1 and n = 2 we did this in the last sectionnwhen+we1renamed this element ff. vER(n)nis a unit and so is oe2 . Consequently, so is n+1 -2n+1(2n-1) ER(n) -(2n-1) 2n+1 (vER(n)n)2 oe = (vn oe ) , THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 13 and this is the periodicity element for ER(n)*(-) ER(n)*,0(-) and it maps to v2n+1nin E(n)*(S0). In [KW07 ] the fibration actually proven is ER(n)a,b-1-! ER(n)a,b-! E(n)a,b. The map, @, (6.1) E(n)a,b-! ER(n)a+1,b-1-! E(n)a+1,b-1 is evaluated in [KW07 , Proposition 1.6] as 1 - c with the understanding that * *the two ends are homeomorphic because they are both just loops on E(n)a+1,bwith different Z=(2) actions. Multiplication by our oe gives this homeomorphism. S* *o, implicit in [KW07 , Proposition 1.6] is oe-1@ = 1 - c. This map corresponds somewhat to our first differential. However, we work with the spectra ER(n) and E(n). Our boundary map, i.e. d1, is E(n)a -! ER(n)a+~(n)+1-! E(n)a+~(n)+1. To finish off our d1 we need the diagram: E(n)a,0(X)________'________//__________E(n)a(X) ____________________________________________________________* *_______________________________________ ________|______________________________________________________* *__________________________________________________|__________________________* *______________________ __________|______________________________________________________* *______________________________________|______________________________________* *_________ ____________fflffl|________________________________________________* *______fflffl|______________________-1 ____________________________________________________1_______________* *______y(n)/a+1,-1/_a+1+~(n) _____________________________@______________________d________________* *______ER(n)'ER(n)(X) (X) _____________________________________________________________________* *__________________________________| 1-c_____________________________________________________________________* *_____________________________________________________________________________* *__________||| _____________________________&&______________________________________* *______xx__________________________________________fflffl|fflffl|n _____________________________v-(2n-1)oe-1 _____________________________E(n)a+1,-1(X)//_E(n)a+1+~(n)(X) ____________________________________' _________________________________________________________|| _________________________________________oe-1'|'| _''__________________________________________fflffl|fflffl|v* *-(2n-1)n E(n)a,0(X)________'_____//E(n)a+1+~(n)(X) oe-1@ = (1 - c) so n-1) -1 -(2n-1) -1 -(2n-1) d1 = (v-(2n oe )@ = vn (oe @) = vn (1 - c) This concludes our proof. 7. Notational conventions Our descriptions of groups are usually by giving a "2-adic basis", i.e. a set* * of elements such that any element in our group is written as a finite sum of these elements with coefficients 0 or 1. For example, if we have Z=(2n) generated by u with the relation 2u = u2, our 2-adic basis would be uj, 0 < j n. In the case* * of infinite dimensional spaces we can have infinite sums but care must be taken ab* *out the topology. We frequently write our list of elements as efficiently as possible by using * *notation such as x{1,2}and x{0-2}to indicate the obvious list of elements, x and x2 in t* *he first case and 1, x, and x2 in the second case. This notation will be used in b* *oth superscripts and subscripts. Whenever we use ffl we mean it can be either 0 or 1. 14 NITU KITCHLOO AND W. STEPHEN WILSON Whenever we give names to new elements, the subscript given as part of the name is also the degree of the element. 8. The Bockstein spectral sequence for ER(2)*(RP 1) We begin by computing ER(2)*(RP 1) using the Bockstein spectral sequence. In principle, we already know this from 3.2. Note that until we start our work with products, many of our Bockstein spectral sequences are even degree. Our even differentials, d2r, are odd degree so they are all zero. This leaves us wi* *th only d{1,3,5,7}. Theorem 8.1. The Bockstein spectral sequence for ER(2)*(RP 1). E1 = E(2)*(RP 1) is represented by vi2ffkuj 0 i < 8 0 k 1 j. d1(v2s-52ffkuj) = 2v2s2ffkuj = v2s2ffk+1uj+1 modulo higher powers of u. E2 = E3 is given by: v2s2ffku v2s2uj 2 j 0 s < 4 0 k. d3(v4s-22ffku) = v4s2ffk+1u and for 2 j, d3(v4s-22uj) = v4s2ffuj = v4s2uj+2 modulo higher powers of u. E4 = E5 = E6 = E7 is given by: v42u{1-3}and u{1-3}. d7(v42u{1-3}) = u{1-3}. The x1-torsion generators are given by: ffiffkuj 0 i 0 k 1 j where ff0 = 2. The x3-torsion generators are given by: wfflffku, ffl + k > 0, wuj 1 < j, and uj 3 < j. The only x7-torsion generators are u{1-3}. Remark 8.2. This is consistent with the description in 3.2. Because 2x = 0, x t* *imes the relation 0 = 2u +F ffu2 +F u4 gives us 0 = x(ffu2 +F u4). This explains why no ffu2 shows up in our description. From the point of view of x-torsion it can be replaced with u4 plus other terms. Likewise, if we multiply by x3 and use the relation x3ff = 0 we end up with x3u4 = 0. Proof.The proof is straightforward. Since u 2 E(2)*(RP 1) is in the image from ER(2)*(RP 1) our differentials commute with multiplication by u (from the prod- uct formula). They also commute with multiplication by ff. We also have, from o* *ur computation of the spectral sequence for ER(2)*(S0) the differentials 5.2. The * *d1 differential creates a relation coming from our relation 0 = 2u +F ffu2 +F u4 w* *hen 2u is set to zero. So, in E2, we have ffu2 = u4 modulo higher powers of u. This explains some of our d3. All of our differentials follow. THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 15 We use the map ER(2)*(S0) ! E(2)*(S0) which takes ffi! 2v2i2and w ! v42ff to identify the xr-torsion generators. Corollary 8.3. The map ER(2)*(-) ! E(2)*(-) induces an isomorphism ER(2)16*(RP 1) -! E(2)16*(RP 1). Both have 2-adic bases given by ffkuj. Proof.E(2)16*(RP 1) has for a 2-adic basis ffiuj. Since ff and u both come from ER(2)*(RP 1) we have a surjection. From the Bockstein spectral sequence for ER(2)*(RP 1) we can just read off all of the elements in degree 16*. From the x* *1- torsion we have ff0ffkuj where ff0 = 2. These elements are, modulo higher filtr* *ation, ffk+1uj+1. From the x3-torsion we have ffku and uj for k > 0 and j > 3. There are elements in degree 8 mod 16 but with the degree of x equal to -17 they do n* *ot give rise to any more degree 16* elements. Likewise for the x7-torsion where we pick up only u{1-3}. Altogether we have ffiuj, the same as for E(2)16*(RP 1). Remark 8.4. In the next paper, if there is a next paper, we will need the sligh* *tly more delicate fact that ER(2)16*+8(RP 1) injects into E(2)16*+8(RP 1). 9. ER(2)*(RP 2) To start our computation of ER(2)*(RP 2) we revert to the Atiyah-Hirzebruch spectral sequence. Recall the homotopy of ER(2) from the beginning of Section 5. The Atiyah-Hirzebruch spectral sequence has elements only in filtrations 1 a* *nd 2. In filtration 1 we have wfflffkx{1,2}x1 and x{3-6}x1. In filtration 2 we h* *ave wfflffkx{0,1,2}x2, x{3-6}x2, ff{1,3}ffkx2 and ff2x2. Since all differentials in* *crease fil- tration by at least 2 the spectral sequence collapses. As ER(2)* modules this * *is generated by elements we call z-16 represented by xx1 (recall that the degree of x is -17) and z2 represented by x2. Remember, of course, that we are working in degrees indexed by Z=(48) for ER(2)*(-) and E(2)*(-). There is a surprising amount of detail to be had in ER(2)*(RP 2). We distill what we need down to: Theorem 9.1. We have elements z2, z-16 = u 2 ER(2)*(RP 2). A 2-adic basis for ER(2)*(RP 2) is given by x{0-2}wfflffkz-16, x{3-6}z-16, x{0-2}wfflffkz2, and x{* *3-6}z2 where 2wfflffkz-16 = x2wfflffk+1z2, u2 = x2z2, x6z-16 = ff2z2, x2ffk+1z-16 = ff3ffkz2, and x2wffkz-16 = ff1ffkz2. Proof.We consider the commuting diagram: ER(2)*(RP 1) _____//E(2)*(RP 1) | | | | fflffl| fflffl| ER(2)*(RP 2) _____//_E(2)*(RP 2). We know that the u 2 ER(2)-16(RP 1) factors through E(2)-16(RP 1) to u 2 E(2)-16(RP 2) and so we must have 0 6= u 2 ER(2)-16(RP 2) as well. The only element that could represent this u is xx1 = z-16. That means u2 is represented by (xx1)2 = x2x2. Recalling our relation, we have 0 = 2u +F ffu2 +F u4. This simplifies because u4 would have to be in at least the 4th filtration but every* *thing above the 2nd filtration is zero. Thus our relation is 0 = 2u +F ffu2, but sinc* *e 2u2 must be in filtration 3 or higher this is only 0 = 2u + ffu2 and since filtrati* *on 2 16 NITU KITCHLOO AND W. STEPHEN WILSON is all modulo (2) we can just as well use 2u = ffu2. From this, of course, we g* *et 2wfflffku = wfflffk+1u2, or, really, 2wfflffkz-16 = x2wfflffk+1z2. We still don't know all we want to yet. The ER(2) cohomology of this simple Moore space is unnecessarily complex. We can solve the next level of problem by looking at the long exact sequence: (9.2) ER(2)*(RP 2)______x______//ER(2)*(RP 2) eeLLL r LLL rrr @ LLL yyrraerrr E(2)*(RP 2). E(2)*(RP 2) is given by vi2ffku and is a Z=(2) vector space. We know that ffi! * *2v2i2 so ffiz2 must map to zero. That means these element are divisible by x. The only candidates, mainly for degree reasons, are x6z-16 = ff2z2, x2ffz-16 = ff3z2 and x2wz-16 = ff1z2. The long exact sequence 9.2 gives the Bockstein spectral sequence, so as long* * as we are using it, we may as well do it using the Bockstein spectral sequence dir* *ectly. We will be working with the Bockstein spectral sequence in general and we need * *to set this up for our future calculations. E1 ' E(2)*(RP 2) for the Bockstein spectral sequence for ER(2)*(RP 2) where we are working with our usual 48 periodic E(2). This E1 is: vi2ffku 0 i < 8 which is a vector space over Z=(2). We know, from the Atiyah-Hirzebruch spectral sequence, that we have two ER(2)* generators that map to here, z-16 and z2. We also know that z-16 ! u. Thus all differentials must be trivial on u. We use the product formula and the fact that d1 times even powers of v2 is zero and d1 on odd powers of v2 gives 2 times an even power which is also zero since we * *are working modulo 2. So d1(vs2u) = d1(vs2)u + c(vs2)d1(u) = 0 + 0. d1 is trivial in our spectral sequence. d2 is trivial because it is odd degree. Again we can use the product rule and d3(v{2,6}2) = ffv{4,0}2to get d3(v{2,6}2ffku) = ffk+1v{4,0}2u. We need to worry about the elements v2s+12u. For purely degree reasons the image of z2 must go to a finite sum of ff3kv52u elements. All differentials must be trivial on this image element, in particula* *r, d3. Since there is no ff torsion and d3 commutes with ff, this implies that d3(v52u* *) = 0. As before, d3(v{2,6}2ffkv52u) = v{4,0}2ffk+1v52u and the image of z2 must be v52u (we may have to alter our choice of z2 a bit f* *or this) with ffkz2 ! v52ffku and wffkz2 ! v2ffk+1u (recall that w ! v42ff). Our E4 term is quite small, just v{0,1,4,5}2u. Because v{0,5}2u are both in the image, degree reasons force us to have no d4, * *d5, or d6, but we see that d7(v42u) = u and d7(v2u) = v52u. From the Bockstein spectral sequence perspective we have no x1-torsion gener- ators. Our x3-torsion generators are given by wfflffkz{2,-16}with ffl + k > 0 * *and, THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 17 finally, our x7-torsion generators are z2 and z-16, or, as we write for efficie* *ncy's sake, z{2,-16}. We can solve our Atiyah-Hirzebruch spectral sequence extension problems yet again using this approach. We now know we must have x6z-16 6= 0. The only possibility is for x6z-16 = ff2z2. Likewise, we know that x2 must be non-zero on all the wfflffkz-16 when ffl + k > 0. Since they only have x times them non-zer* *o in filtration one of the Atiyah-Hirzebruch spectral sequence these elements must a* *ll be in filtration 2 and we get the answer we have already obtained from the long exact sequence. The Moore space will be our basic building block. Corollary 9.3. Consider the cofibration: S1 ____//_RP_2___//S2. The long exact sequence (9.4) ER(2)*(S1)oo____'_____ ER(2)*(RP 2) JJ ss99 JJJ sss @ JJJ$$J ssssae ER(2)*(S2) is given by @('1) = 2'2, ae('2) = z2, and '*(u) = x'1. 10.The Bockstein spectral sequence for ER(2)*(RP 2n=RP 2n-2) For our computation we need the Bockstein spectral sequence in detail. Stating the complete Bockstein spectral sequence for even a simple space is highly tech- nical. We need to give the Er terms for r =1-7, compute the differentials and find corresponding xr-torsion generators in ER(2)*(X) that map to the image of dr. After this is done we have to solve extension problems and locate any speci* *al elements of interest to us. Normally we won't need to do all of this, but as th* *ese spaces are our basic building blocks we need to know them quite well. We have already done the case of ER(2)*(RP 2) and we know that: 2n-2ER(2)*(RP 2) ' ER(2)*(RP 2n=RP 2n-2) so we can just write down the answer. Note, in particular, that the right hand * *side inherits a multiplication by u from the left hand side. Theorem 10.1. We have elements z2n-18, z2n 2 ER(2)*(RP 2n=RP 2n-2). A 2-adic basis for ER(2)*(RP 2n=RP 2n-2) is given by elements x{0-2}wfflffkz2n-18, x{3-6}z2n-18, x{0-2}wfflffkz2n, and x{3-6}z2n where 2wfflffkz2n-18 = x2wfflffk+* *1z2n. Furthermore, uz2n-18= x2z2n, x2ffk+1z2n-18= ff3ffkz2n, x2wffkz2n-18= ff1ffkz2n, and x6z2n-18= ff2z2n. This follows automatically from the suspension isomorphism but we want to carefully write down the differentials and representations. E1 = E2 = E3 ' vi2ffkun 0 i < 8, 0 k. 18 NITU KITCHLOO AND W. STEPHEN WILSON d3(v62ffkv5n2un) = ffk+1v5n2unoo__ ffk+1z2n d3(v22ffkv5n2un) = v42ffk+1v5n2unoo_wffkz2n d3(v62ffkv5n+32un) = ffk+1v5n+32unoffk+1z2n-18o_ d3(v22ffkv5n+32un) = v42ffk+1v5n+32unwffkz2n-18oo_ E4 = E5 = E6 = E7 ' v{0,3,4,7}2v5n2un. d7(v42v5n2u) = v5n2unoo____z2n_ d7(v72v5n2u) = v5n+32v5n2unoz2n-18o_ 11. The Bockstein spectral sequence for ER(2)*(RP 1=RP 16K) We need ER(2)*(RP 1=RP 16K) for our applications in this paper. It is es- sentially the same computation as for ER(2)*(RP 1) but the proof requires more care. Theorem 11.1. The Bockstein spectral sequence for ER(2)*(RP 1=RP 16K). E1 = E(2)*(RP 1=RP 16K) E(2)*(RP 1) is represented by vi2ffkuj 0 i < 8 0 k 8K < j. d1(v2s-52ffkuj) = 2v2s2ffkuj = v2s2ffk+1uj+1 modulo higher powers of u. E2 = E3 is given by: v2s2ffku8K+1 v2s2uj 8K + 2 j 0 s < 4 0 k. d3(v4s-22ffku8K+1) = v4s2ffk+1u8K+1 and for 8K + 2 j, d3(v4s-22uj) = v4s2ffuj = v4s2uj+2 modulo higher powers of u. E4 = E5 = E6 = E7 is given by: v42u8K+{1-3}and u8K+{1-3}. d7(v42u8K+{1-3}) = u8K+{1-3}. There is an element z16K-16 2 ER(2)*(RP 1=RP 16K) that maps to u8K+1 2 ER(2)*(RP 1). The x1-torsion generators are given by: ffiffkz16K-16uj 0 i < 4 0 k 0 j where ff0 = 2. The x3-torsion generators are given by: wfflffkz16K-16, ffl + k > 0, wz16K-16uj, 0 < j, and z16K-16uj, 3 j. The only x7-torsion generators are z16K-16u{0-2}. THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 19 Proof.The E1 term of the spectral sequence injects to that for ER(2)*(RP 1) so d1 is induced. d3 is a trickier issue. We look at the element in E3 we have named u8K+1. If we have d3(u8K+1) = v22z 6= 0 then v22z must map to zero in E3 for ER(2)*(RP 1). Since we had an injection on E1, v22z must go to 2y for some y. The only such elements are v222ffku8K = v22ffk+1u8K+1 modulo higher powers of u. Now we use the map RP 16K+2=RP 16K! RP 1=RP 16Kwhere u8K+1 goes to z16K-16 in the spectral sequence. In the Bockstein spectral sequence f* *or ER(2)*(RP 16K+2=RP 16K), z16K-16 has d3 trivial but v22ffk+1z16K-16 is non-zero, so our d3 must be zero on u8K+1. d3 then follows from d3(v22) = ffv42. Likewise* *, our d7 follows by comparison with RP 16K+2=RP 16K. The same argument gives quite a different result when m 6= 8K. 12. The Bockstein spectral sequence for ER(2)*(RP 6) Before we proceed to ER(2)*(RP 2n) we need to do the equivalent of starting an induction. This will be a little different from what we have done before and wi* *ll show some of what is to come. We need just a simple fact about ER(2)*(RP 6). Proposition 12.1. In ER(2)*(RP 6) the elements u1-3are x7-torsion and d7 takes v42u1-3 to them in the Bockstein spectral sequence. Proof.We begin by computing d1 in our spectral sequence where E1 is represented by vi2ffkuj for 1 j 3. Our E2 = E3 term is something new: v2s2ffku, v2s2u{2,3}, and v2s+12ffku3. Comparing our spectral sequence with those for RP 6=RP 4and RP 1 we can com- pute our d3 to get E4 = E5: v{0,4}2u{1-3} v{2,6}2u{2,3} v{3,7}2u3. For purely degree reasons, there are no d5 differentials. Since u must be a tar* *get for d7 the d7 differential is what we stated. The d7 on the rest is solved by compa* *rison again with RP 6=RP 4but we don't need that in the statement of the theorem. 13. The Bockstein spectral sequence for ER(2)*(RP 2n) We want to compute the Bockstein spectral sequence for ER(2)*(RP 2n). It isn't really that hard to do except that it breaks up into 8 distinct cases depending* * on n modulo 8. For now we want to assume that n > 3. Keep in mind that we have an even degree spectral sequence so all d2r are zero because they are odd degre* *e. We only have d{1,3,5,7}to consider. We have already computed n = 1 and n = 3 (n = 2 isn't hard). d1 does not depend on n. E1 ' vi2ffkuj 0 i < 8 0 k 0 < j n d1(v2s-52ffkuj) = 2v2s2ffkuj for j < n. These elements represent the x1-torsion elements ffiffkuj, j < n. We know that 2ffkuj = ffk+1uj+1 modulo higher powers of u so we have, for E2 = E3: v2s2ffku, v2s2uj, v2s+12ffkun, 0 k, 1 < j n, 0 k, 0 s < 4. We know that ffku and uj are infinite cycles because they are in the image from ER(2)*(RP 1) so we can compute d3 on the first two terms just using d3(v{2,6}2)* * = 20 NITU KITCHLOO AND W. STEPHEN WILSON ffv{4,0}2. We use the fact that E3 is a vector space over Z=(2). That reduces o* *ur relation to 0 = ffu2 +F u4. Modulo higher powers of u, this is just ffu2 = u4. * *So, modulo higher powers of u we have: d3(v{6,2}2ffku) = v{0,4}2ffk+1u d3(v{6,2}2uj) = v{0,4}2ffuj = v{0,4}2uj+2 1 < j n - 2 The E1 of the Bockstein spectral sequence for ER(2)*(RP 2n=RP 2n-2), i.e. E(2)*(RP 2n=RP 2n-2), injects to that for ER(2)*(RP 2n). The map is given by: (13.1) z2n = z16K+2j- ! v5j2u8K+j z2n-18= z16K+2j-18-! v5j+32u8K+j where 0 < j 8. In particular, when j = 3, 4, 7 or 8, either v72or v32times u8* *K+j is in the image and can therefore have no differential. The usual d3(v22) = ff* *v42 determines the differentials: d3(v{2,6}2v72u8K+j) = v{4,0}2v72ffu8K+j. Similarly when j = 1, 2, 5 or 6 we have v52or v2 times u8K+j in the image and we get: d3(v{2,6}2v52u8K+j) = v{4,0}2v52ffu8K+j. Combining all of our computations for d3 we have E4 = E5: v{0,4}2u{1-3} v{6,2}2u{n-1,n} v{2s+1,2s+5}2un where s = 0 if n = 1, 2, 5 or 6 mod 8, and s = 1 if n = 3, 4, 7 or 8 mod 8. We have computed ER(2)*(RP 6) and shown that the elements u{1-3}are all x7 torsion and d7(v42u{1-3}) = u{1-3}. By naturality, the elements u{1-3} 2 ER(2)*(RP 2n) must also be x7 torsion with the same differential. The only el- ements we have left to worry about in our spectral sequence are: v{6,2}2u{n-1,n} v{2s+1,2s+5}2un where s = 0 or 1 as above. We collect what we know so far in the following preliminary result: Theorem 13.2. Let n > 3, the Bockstein spectral sequence for ER(2)*(RP 2n) begins as follows (with differentials modulo higher powers of u): E1 = E(2)*(RP 2n) is represented by vi2ffkuj 0 i < 8 0 k 0 < j n. d1(v2s-52ffkuj) = 2v2s2ffkuj = v2s2ffk+1uj+1 j < n E2 = E3 is given by: v2s2ffku0 k, v2s2uj1 < j n, v2s+12ffkun0 k d3(v{6,2}2ffku) = v{0,4}2ffk+1u d3(v{6,2}2uj) = v{0,4}2ffuj = v{0,4}2uj+2 1 < j n - 2 d3(v{2s+1,2s+5}2ffkun) = v{2s+3,2s+7}2ffk+1un THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 21 where s = 0 for n = 3, 4, 7 and 8 modulo 8 and s = 1 for n = 1, 2, 5 and 6 modu* *lo 8. E4 = E5 is given by v{0,4}2u{1-3},v{6,2}2u{n-1,n},v{2s+1,2s+5}2un where s = 0 if n = 1, 2, 5 or 6 mod 8, and s = 1 if n = 3, 4, 7 or 8 mod 8. d7(v42u{1-3}) = u{1-3}. The only remaining undetermined part of the Bockstein spectral sequence is in E* *5: v{6,2}2u{n-1,n} v{2s+1,2s+5}2un where s is 0 or 1 as above. We now have to start working our way through the 8 cases. There can be significant variation on what happens. We only have 6 elements here in our basis and we must kill them all off with d5 and d7. For purely degree reasons, if the* *re is a d5 it must be d5(v{6,2}2un-1) = v{5,1}2un. Of course, if those last elements * *aren't there, d5 must be zero. We collect the remaining differentials for all 8 cases in one place: Theorem 13.3. The remaining differentials for the Bockstein spectral sequence f* *or ER(2)*(RP 2n), n > 3,together with a little of the map ER(2)*(RP 2n=RP 2n-2) ! ER(2)*(RP 2n) are as follows: ER(2)*(RP 16K+2) 4v12u8K+14 hd5hhhhhhh hhhh u v22u8K__________//_v22u8K+1____ ____________________ __d7______________________________________* *_____ __________________________ __________________________ * 5 8K+1__________________________z16K+2qoo_ d5 hhh4v2u4h _______________________ hhhhhhhhu ""_______________________ v62u8K__________//_v62u8K+1 ER(2)*(RP 16K+4) 3v12u8K+23 hhd5hhhhhhh hhh u q* v22u8K+1__________//_v22u8K+2oo______`z16K+4` __________________ __d7_____________________________________* *______ ____________________ _____q*________________________________* *__ 3v52u8K+2oo________z16K-143_____________________* *_______ hhd5hhhhhhh ____________________________ hhh u _______________ v62u8K+1__________//_v62u8K+2 22 NITU KITCHLOO AND W. STEPHEN WILSON ER(2)*(RP 16K+6) q* v22u8K+2_____u____//_v22u8K+3oo_______z16K-12>>__>>__ ____________________________________________________________ _____________________________________________________ ____________________v32u8K+3______________________________________* *___________ d7_________________________d__________________________7_____________* *____________ __________________________________________________________________* *____________ _________________________________________________________________* *_______________________________________ ____________u ________d7___________________________________* *____________________ v62u8K+2__________//_v62u8__________________________K+3 _________________ ""_________________q* v72u8K+3oo_________z16K+6 ER(2)*(RP 16K+8) v22u8K+3_____u____//_v22u8K+4>>__>>__ ____________________________________________________________ _____________________________________________________ ____________________v32u8K+4_____________________________________* *____________ d7_________________________d__________________________7____________* *_____________ _________________________________________________________________* *_____________ ________________________________________________________________* *________________________________________ ____________u ________d7__________________________________* *_____________________ v62u8K+3__________//_v62u8__________________________K+4 _________________ ""_________________q* v72u8K+4oo_________z16K-8 ER(2)*(RP 16K+10) q* 3v12u8K+5oo________z16K+103 hhd5hhhhhhh hhh u v22u8K+4__________//_v22u8K+5``____ __________________________ __________________________ __________________________ d7_____________________________________* *___ 3v52u8K+53____________________________ hhd5hhhhhhh ____________________________ hhh u _______________ v62u8K+4__________//_v62u8K+5 ER(2)*(RP 16K+12) q* 3v12u8K+6oo________3z16K-6 hhd5hhhhhhh hhh u v22u8K+5__________//_v22u8K+6____ __________________________ __________________________ __________________________ _d7_____________________________________* *_____________ v52u8K+__________________________6 d5hhhh33hh _________________ hhhhhhu ""_________________q* v62u8K+5__________//_v62u8K+6oo______z16K+12 THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 23 ER(2)*(RP 16K+14) v22u8K+6_____u____//_v22u8K+7 ___________________________________________ ____________________________________________________ * * * _________________________________________________________________v* *3u8K+7z16K+14qoo_ d7___________________________________________________d7_____________* *_________________________2``______________________ _____________________________________________________d7___________* *_______________ _________________ _______________________________________________* *___________ ______________________________________uq*_______________________* *_______________ v62u8K+6__________//_v62u8K+7oo_______z16K-4_____________________* *_______________ ________________ __________________________ v72u8K+7 ER(2)*(RP 16K+16) v22u8K+7_____u____//_v22u8K+8 ___________________________________________ ____________________________________________________ * * * _________________________________________________________________* *v3u8K+8z16K-2qoo_ d7___________________________________________________d7____________* *__________________________2``______________________ _________________________________________________________________* *______________ _________________ ______________________________________________* *____________ ______________________________________ud7______________________* *________________ v62u8K+7__________//_v62u8K+8____________________________________ ________________ __________________________ v72u8K+8 Proof.As already discussed, for degree reasons, there can be no d5 for the Bock- stein spectral sequence for ER(2)*(RP 2n) when n = 3, 4, 7 or 8 mod 8. The two * *d7 differentials for n = 3 and 7 mod 8 follow from the map RP 2n! RP 2n=RP 2n-2. One of the d7 differentials for n = 4 and 8 mod 8 follows from the map RP 2n! RP 2n=RP 2n-2and the other follows from the map RP 2n-2! RP 2n. This com- pletes the four cases n = 3, 4, 7 and 8 mod 8. The other four cases all have a non-trivial d5. We begin by looking at the n = 2 mod 8 case. The map to the n = 3 mod 8 case takes care of d7(v62u8K+2) = d7(v22u8K+2). If there is no d5 it would a* *lso give the d7 on v62u8K+1 and we would have a generator, represented by v22u8K+1, that was not in the image of the n = 3 mod 8 case. We now use the cofibration RP 2n-2! RP 2n! RP 2n=RP 2n-2where n = 3 mod 8. We have a complete de- scription of ER(2)*(RP 2n=RP 2n-2). All of the elements associated with z2n inj* *ect, i.e. x{0-6}x2n and x{0-2}wfflffkz2n. We also have x{0-6}x2n-18and wfflffkz2n-18* *in- jecting. The only possible elements left for the kernel are x{1,2}wfflffkz2n-18* *, where ffl + k > 0. Thus the boundary on the element represented by v22u8K+1 must hit * *one of these elements. The boundary homomorphism increases degree by 1 so, modulo 8, the degree of the image is -3. However, the degrees, modulo 8, of the elemen* *ts x{1,2}wfflffkz2n-18 are -5 and -6 (remember, n = 3 mod 8 here). There must be a d5 to prevent this impossibility. A similar argument works for n = 6 mod 8 comparing it with n = 7 mod 8. We work on the n = 1 mod 8 case now using the cofibration RP 2n-2! RP 2n! RP 2n=RP 2n-2for n = 2 mod 8. Here, all of the elements associated with z2n-18 inject with the possible exception of x{5,6}z2n-18. The other possible elements* * in the kernel are x{1,2}wfflffkz2n, ffl + k > 0. If there is no d5 then d7(v22u8K * *) = v62u8K is determined by comparison with the n = 0 mod 8 case. The element representing 24 NITU KITCHLOO AND W. STEPHEN WILSON v62u8K is not in the image and so must have boundary non-trivial in the above cofibration for n = 2 mod 8. The degree of the boundary of this element is 16K-* *35. Using the n = 2 mod 8 cofibration the degrees of x{1,2}wfflffkz2n mod 8 are -5 * *and -6. The degrees of x{5,6}z2n-18are -5x17+16K +4-18 = 16K -3 and 16K -4. There is nowhere for our element to go so there must be a d5. We still have to deal with the d7 because it is not induced by any of our map* *s. One of v{2,6}2u8K+1 must have a non-trivial boundary homomorphism on it. The degree of the boundary image will be 16K - 16 - 12 + 1 = 16K - 27 (for v22u8K+1) or 16K - 3 (for v62u8K+1). Thus we must have d7(v22u8K+1) = v62u8K+1 and the boundary of v62u8K+1 must hit x5z2n-18, a fact sure to be useful sometime. A similar argument works for n = 5 mod 8 comparing it to n = 6 mod 8. For our applications, what we really need to know is ER(2)16*(RP 2n) and how these elements sit in ER(2)*(RP 2n). The simple version of this is stated in t* *he Introduction as Theorem 1.6. Theorem 13.4. For all n there is a short exact sequence: (13.5) ER(2)16*(RP 2n-2)oo___ER(2)16*(RP 2n)oo___ER(2)16*(RP 2n=RP 2n-2). We have elements ffkuj 2 ER(2)16*(RP 2n), 0 k, 0 < j n that reduce to elements of the same name in E(2)16*(RP 2n). Depending on n modulo 8 there are other elements in ER(2)16*(RP 2n). For n = 8K + 8 and 8K + 7 there are no other elements. For n = 8K +6 there is an x5-torsion element, z16K-30, that reduces to v52u8K* *+6 in the Bockstein spectral sequence such that x2ffkz16K-30 = ffku8K+7. For n = 8K +5 there is an x5-torsion element, z16K-14, that reduces to v52u8K* *+5 in the Bockstein spectral sequence such that x2ffkz16K-14 = ffku8K+6 and an x7-torsion element, z16K+4 that reduces to v22u8K+5 in the Bockstein spe* *ctral sequence such that x2uz16K-14 = x4z16K+4 = u8K+7. For n = 8K + 4 there are x7-torsion elements, z16K-12, and z16K-10 that reduce to v22u8K+3 and v72u8K+4 respectively in the Bockstein spectral sequence such t* *hat x4z16K-12 = u8K+5 x4uz16K-12 = u8K+6 and x4u2z16K-12 = x6z16K-10 = u8K+7. For n = 8K + 3 there are x7-torsion elements, z16K+4, and z16K-42 that reduce to v22u8K+2 and v72u8K+3 respectively in the Bockstein spectral sequence such t* *hat x4z16K+4 = u8K+4 x4uz16K+4 = u8K+5 and x4u2z16K+4 = x6z16K-42 = u8K+6. THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 25 For n = 8K +2 there is an x5-torsion element, z16K-14, that reduces to v52u8K* *+2 in the Bockstein spectral sequence such that x2ffkz16K-14 = ffku8K+3 and an x7-torsion element, z16K+4 that reduces to v22u8K+2 in the Bockstein spe* *ctral sequence such that x2uz16K-14 = x4z16K+4 = u8K+4. For n = 8K + 1 there is an x5-torsion element, z16K+2, that reduces to v52u8K* *+1 in the Bockstein spectral sequence such that x2ffkz16K+2 = ffku8K+2. Proof.We have computed the Bockstein spectral sequence for all of the spaces RP 2n-2, RP 2n, and RP 2n=RP 2n-2. From this we can just read off the elements * *in degree 16*. In every case the x1-torsion elements ff0ffkuj for j < n - 1 corres* *pond using the map induced by RP 2n-2! RP 2n. Likewise for the elements ffku, u{1-3}, and uj, j < n so we will ignore these elements. First note that ff0ffkun-1 = 2ffkun-1 = ffk+1un. For n = 8 mod 8, there is nothing else in ER(2)16*(RP 2n-2). All that is left* * of 13.5 is ffkz2n ! ffkun. For n = 7 mod 8, ER(2)16*(RP 2n=RP 2n-2) = 0. We must have ffkun ! x2ffkun-1v52. [Technically, we need to worry that perhaps un goes to x2ff3kun-1* *v52 for some k. If this is the case then the boundary homomorphism on x2un-1v52must be non-trivial but we can check that there is nowhere for it to go. Consequently we will ignore this kind of possibility in the rest of this proof.] For n = 6 mod 8 things are a little more complicated. The only elements in ER(2)16*(RP 2n=RP 2n-2) are x2wffkz2n-18and we can compute directly that they go to x2ffk+1unv52. ffkun must go to x2ffkun-1v52. The only possibility left is* * for x2unv52to go to x4un-1v22. Recall from above that this last element is un+1. For n = 5 mod 8, we compute the map to ER(2)16*(RP 2n) directly and we have wffkz2n-18- ! ffk+1un x2wffkz2n -! x2ffk+1unv52. Keep in mind that this last represents ffk+1un+1. We then have un+1 = x2unv52 maps to x4un-1v22. We must have un map to x4un-2v22and x4unv22(which repre- sents un+2) map to x6un-1v72. For n = 4 mod 8 we compute x6z2n-18 ! x6unv72= un+3 and wffkz2n ! ffk+1un. That leaves un ! x4un-2v22, x4un-1v22= un+1 ! x4un-1v22, and x4unv22= un+2 ! x6un-1v72. For n = 3 mod 8 we compute x4z2n-18! x4unv22= un+2 and x6z2n ! x6unv72= un+3. That leaves x4un-1v22= un+1 ! x4un-1v22and ffkun ! x2ffkun-1v52. For n = 2 mod 8 we compute x2ffkz2n-18! x2ffkunv52= ffkun+1 and x4z2n ! x4unv22= un+2. All that is left is ffkun ! x2ffkun-1v52. The n = 1 mod 8 case is simple again with ffkz2n-18! ffkun and x2ffkz2n ! x2ffkunv52= ffkun+1. 26 NITU KITCHLOO AND W. STEPHEN WILSON 14. Beginning with products For use with our Bockstein spectral sequence we need descriptions of E(2)*(-) for various products. We always use reduced cohomology. We start with a result proven by modifying techniques of [JW85 ]: Theorem 14.1 ([GW ]). Let m < n, then BP *(RP 2m^ RP 2n) ' BP *(RP 2m) BP* BP *(RP 2n) 2n-1BP *(RP 2m) Remark 14.2. It is important to note, because we use it later, that this is nat* *ural in the obvious way for the RP 2mwhen m < n. It is enough to prove this using BP <2>, where BP <2>* ' Z(2)[v1, v21], becau* *se v2 multiplication is injective and so it determines the Brown-Peterson cohomology.* * We can now invert v2 to get the to get E(2)*(-) and the same theorem holds. Because there is no v2-torsion, BP <2>*(RP 2m^ RP 2n) injects into E(2)*(RP 2m^ RP 2n). This is important because we rely on Don Davis's computations. He does his in BP <2>*(-) but this shows they just as well could have been done in E(2)*(-). We do not use the standard notation because we need to be compatible with ER(2)*(-). Above, the bottom class in the suspension is 2n-1x2. We shift this using the unit v32raised to the n-th power, i.e. we shift the suspension down b* *y -18n so our bottom class is now -16n-1x2 but we also replace x2 with our u = v32x2. Our bottom class is now in degree -16n - 1 + 2 - 18 = -16n - 17. We give it the name z-16n-17. The result for our 48-periodic theory that we use is as foll* *ows where we also include the more detailed description from [GW ]. Much of this * *is well known. Theorem 14.3. Let m < n, then E(2)*(RP 2m^RP 2n) ' E(2)*(RP 2m) E(2)*E(2)*(RP 2n) -16n-1E(2)*(RP 2m) represented by vs2ffkui1u2 0 k 0 < i m 0 s < 8 vs2ui1uj2 0 < i m 1 < j n 0 s < 8 and vs2ffkujz-16n-17 0 k 0 j < m 0 s < 8. E(2)*(RP 2n^ RP 1) ' E(2)*(RP 2n) E(2)*E(2)*(RP 1) represented by vs2ffkui1u2 0 k 0 < i n 0 s < 8 and vs2ui1uj2 0 < i n 1 < j 0 s < 8. E(2)*(RP 2n^ RP 1=RP 2m) ' E(2)*(RP 2n) E(2)*E(2)*(RP 1=RP 2m) represented by vs2ffkui1um+12 0 k 0 < i n 0 s < 8 vs2ui1uj2 0 < i n m + 1 < j 0 s < 8. THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 27 15. A review of our relation We need a bit more detail aboutXour relation: 0 = [2](u) = asus+1 = 2u +F ffu2 +F u4. s 0 The degree of our as is 16s and the degree of the relation is -16. Lemma 15.1. 0 = 2u +F ffu2 +F u4 = 2u + ffu2 + u4 + 2u3za(u) + ffu6zb(u). Proof.The proof follows immediately from the fact that F (y, 0) = y. The za(u) and zb(u) are power series in u and are not determined uniquely because many elements are divisible by both 2 and ff. Definition 15.2. We need a filtration on our elements ffkui1u2 and ui1uj2in the tensor product part of our description of E(2)*(RP 2m^ RP 2n). We say ua1ub2is * *of higher filtration than ui1uj2if a + b > i + j or, if a + b = i + j and we have * *a > i. Remark 15.3 (The Algorithm). In our tensor product description of E(2)*(RP 2m^ RP 2n) with m < n we use no elements with a 2 or an ffu22. We need an algorithm that shows how any element can be reduced to those in our description, i.e. ffk* *ui1u2 and ui1uj2, j > 1. It is enough if our algorithm increases filtration as that * *will eventually lead to terms in our description. If we have a 2 we use our relation: X j 2ui1uj2= (2u1)ui-11uj2= -( akuk+11)ui-11u2 k>0 All of these terms have higher filtration. If 2 does not divide and if j = 1 th* *en we are done. So, we are left with the case where ffu22divides our element. In this* * case, modulo higher filtrations, we have: ffui1uj2= ui1uj-22(ffu22) = ui1uj-22(-2u2) = -2ui1uj-12. and we use the first reduction on this to get, modulo higher filtration, ffui+1* *1uj-12. Even this term is of higher filtration than we need. If neither 2 nor ff is pre* *sent then we are done. However, there is one last step. Since we are using our 2-adic representation for everything we only want 0 and 1 for coefficients. Whenever we have a -z we can replace it by z - 2z and use the algorithm on -2z. This shows that -z = z modulo higher filtration. The algorithm ends after a finite number of steps when the power of u1 is gre* *ater than m, the power of u2 is greater than n, or the power of u2 = 1 and there are* * no more 2's left. Lemma 15.4. There is an element z with filtration greater than u1u22such that 2(u1u22+ z) = u21u42 modulo filtrations higher than that of u21u42. Proof.We compute with 2(u1u22- u21u2 - u21u32za(u2) + u31za(u1)u22) = -(ff + 2u1za(u1) + u21+ ffu41zb(u1))u21u22 +(ff + 2u1za(u1) + u21+ ffu41zb(u1))u31u2 -2u21u32za(u2) + 2u31za(u1)u22 28 NITU KITCHLOO AND W. STEPHEN WILSON The very first term, -ffu21u22, is, using the algorithm and ignoring higher ter* *ms: -ffu31u2 - u51u2 - 2u41za(u1)u2 + u21u42+ 2u21u32za(u2). Most terms now cancel out and we are left with, modulo the higher filtration te* *rms, u21u42. We are getting nearer what we really need. Lemma 15.5. For ui1uj2with j > 1 there is a z in E(2)*(RP 2m^ RP 2n) with m < n having higher filtration than ui1uj2such that 2(ui1uj2+ z) = ui+11uj+22 modulo the terms ffkuc1u2 with c i + j + 2, uc1u22with c i + j + 1 and uc1u* *32with c i + j. Proof.We do this by downward induction on the filtration of the target term. Th* *ere is nothing to prove if i + 1 + j + 2 > m + n + 3 because both ui1uj2and the tar* *get are zero. Assume we know this for all elements in higher filtration than ui+11u* *j+22. We know, from the previous lemma, that 2(ui1uj2+ ui-11zuj-22) = ui+11uj+22 modulo elements of higher filtration. By our induction we can take care of all * *of the elements of higher filtration except those listed that we are working modul* *o. We can only handle elements with the power of u2 greater than or equal to 4. This lemma is one of our goals in this section and we get our other goal as an immediate corollary. Corollary 15.6. If n > m there is an element b1,n-1= u1un-12+ u21z with2b1,n-1= 0 and the filtration of u21z is higher than that of u1un-12. Proof.From the lemma there is a z of higher filtration than u1un-12such that 2(u1un-12+z) = u21un+12= 0. Since 2u1un2= 0 we need not have any un2in any part of z. So, to have higher filtration than u1un-12we must have u21dividing z. Remark 15.7. A tactical mistake was made while trying to understand these com- putations. The "simple" test case that was studied at length was RP 16x RP 16. This "easiest" case turned out to be significantly harder because 2b1,7= u81u32* *. The shift to m < n simplified things a lot. 16.The Bockstein spectral sequence for ER(2)*(RP 2n^ RP 1) ER(2)*(RP 2n^ RP 1) depends on n, but, as with ER(2)*(RP 2n), d1 doesn't. Everything is still even degree so we only have to worry about the 4 odd differ* *entials. Theorem 16.1. In the Bockstein spectral sequence for ER(2)*(RP 2n^RP 1) where 3 < n we have (with differentials all modulo higher filtrations): E1 is vs2ffkui1u2 0 s < 8 0 k 0 < i n and vs2ui1uj2 0 s < 8 0 < i n 1 < j. d1(v2s-52ffkui1u2) = v2s2ffk+1ui+11u2 for0 i < n THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 29 d1(v2s-52ui1uj2) = v2s2ui+11uj+22 for 0 < i < n and 1 < j. E2 = E3 is: v2s2ffku1u2 k 0 v2s2ui1u{1,2,3}21 < i n v2s2u1uj2 1 < j v2s+12ffkun1u2 v2s+12un1uj2 1 < j d3(v{2,6}2ffku1u2) = v{4,0}2ffk+1u1u2 d3(v{2,6}2ui1u{1,2,3}2) = v{4,0}2ui+21u{1,2,3}2 1 < i < n - 1 d3(v{2,6}2u1uj2) = v{4,0}2u1uj+22 For n = 1, 2, 5 or 6 mod 8: d3(v{3,7}2ffkun1u2) = v{5,1}2ffk+1un1u2. d3(v{3,7}2un1uj2) = v{5,1}2un1uj+22 1 < j. For n = 3, 4, 7 or 8 mod 8: d3(v{1,5}2ffkun1u2) = v{3,7}2ffk+1un1u2. d3(v{1,5}2un1uj2) = v{3,7}2un1uj+22 1 < j. E4 = E5 is: v{0,4}2ui1uj2 0 < i < 4 0 < j < 4 v{2,6}2u{n-1,n}1u{1,2,3}2 For n = 1, 2, 5 or 6 mod 8: v{5,1}2un1u{1,2,3}2 For n = 3, 4, 7 or 8 mod 8: v{3,7}2un1u{1,2,3}2 For n = 1, 2, 5 or 6 mod 8: d5(v{2,6}2un-11u{1,2,3}2) = v{1,5}2un1u{1,2,3}2 For n = 1, 2, 5 or 6 mod 8: E6 = E7 is: v{0,4}2ui1uj2 0 < i < 4 0 < j < 4 v{2,6}2un1u{1,2,3}2 d7(v42ui1uj2) = ui1uj2 For n = 1 or 6 modulo 8: d7(v22un1u{1,2,3}2) = v62un1u{1,2,3}2 and for n = 2 or 5 modulo 8: d7(v62un1u{1,2,3}2) = v22un1u{1,2,3}2. For n = 3, 4, 7 or 8 mod 8: E5 = E6 = E7. d7(v42ui1uj2) = ui1uj2 30 NITU KITCHLOO AND W. STEPHEN WILSON For n = 3 or 4 mod 8 d7(v62u{n-1,n}1u{1,2,3}2) = v22u{n-1,n}1u{1,2,3}2 d7(v32un1u{1,2,3}2) = v72unu{1,2,3}2. For n = 7 or 8 mod 8 d7(v22u{n-1,n}1u{1,2,3}2) = v62u{n-1,n}1u{1,2,3}2 d7(v72un1u{1,2,3}2) = v32un1u{1,2,3}2. Proof.The computation of d1 is made possible by Lemma 15.5. The higher differ- entials all come from products where the differential on RP 2nis the one used. Corollary 16.2. Let m = 8K and 3 < n. In the Bockstein spectral sequence for ER(2)*(RP 2n^ RP 1=RP 2m) we have the same result as above, just multiply everything by um2. 17. The Bockstein spectral sequence for ER(2)*(RP 1 ^ RP 1) We know from Theorem 3.4 that ER(2)*(RP 1 ^ RP 1) ' ER(2)*(RP 1) ^ER(2)*ER(2)*(RP 1). We can write down the entire Bockstein spectral sequence for this as a Corollary to the previous section just by letting n go off to infinity. We also want to s* *ee the elements which represent things in our spectral sequence. Theorem 17.1. In the Bockstein spectral sequence for ER(2)*(RP 1 ^ RP 1) we have, where everything is modulo higher filtrations: E1 is vs2ffkui1u2 0 s < 8 0 k 0 < i vs2ui1uj2 0 s < 8 0 < i 1 < j. d1(v2s-52ffkui1u2) = 2v2s2ffkui1u2 = v2s2ffk+1ui+11u2 d1(v2s-52ui1uj2) = 2v2s2ui1uj2= v2s2ui+11uj+22 0 < i 1 < j. E2 = E3 is v2s2ffku1u2 0 k v2s2ui1u{1,2,3}21 < i v2s2u1uj2 1 < j d3(v{2,6}2ffku1u2) = v{4,0}2ffk+1u1u2 d3(v{2,6}2ui1u{1,2,3}2) = v{4,0}2ui+21u{1,2,3}2 1 < i d3(v{2,6}2u1uj2) = v{4,0}2u1uj+22 1 < j E4 = E5 = E6 = E7 is v{0,4}2ui1uj2 0 < i < 4 0 < j < 4 d7(v42ui1uj2) = ui1uj2 The x1-torsion is given by ffsffkui1u2 -! 2v2s2ffkui1u2 = v2s2ffk+1ui+11u2 ffsui1uj2-! 2v2s2ui1uj2= v2s2ui+11uj+22 0 < i 1 < j. The x3-torsion is given by ffku1u2 -! ffku1u2 0 < k wffku1u2 -! v42ffk+1u1u2 0 k THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 31 ui+21u{1,2,3}2-! ui+21u{1,2,3}2 1 < i wui1u{1,2,3}2-! v42ui+21u2 1 < i u1uj2-! u1uj2 3 < j wu1uj2-! v42u1uj+22 1 < j. The x7-torsion is given by ui1uj2-! ui1uj2 i < 4 j < 4. Proof.The differentials follow from the previous section. The elements describe* *d in ER(2)*(RP 1^RP 1) have the appropriate torsion and map to the correct elements in E(2)*(RP 1 ^ RP 1). Remark 17.2. Note that we have no elements divisible by wu21u42. u42can be repl* *aced using 1.3 and this can be rewritten in terms of other elements. Corollary 17.3. The map ER(2)*(-) ! E(2)*(-) induces an isomorphism ER(2)16*(RP 1 ^ RP 1) -! E(2)16*(RP 1 ^ RP 1). Proof.E(2)16*(RP 1 ^ RP 1) has, for a 2-adic basis, ffiui1u2 and ui1uj2for j > * *1. Since ff, u1 and u2 all come from ER(2)*(RP 1^RP 1) we have a surjection. From the Bockstein spectral sequence for ER(2)*(RP 1^RP 1) we can just read off all * *of the elements in degree 16*. From the x1-torsion we have, modulo higher filtrati* *ons, ff0ffkui1u2 = ffk+1ui+11u2 ff0ui1uj2= ui+11uj+22 0 < i 1 < j. From the x3-torsion we have ffku1u2 0 < k ui+21u{1,2,3}21 < i u1uj2 3 < j. Finally, from the x7-torsion we have ui1uj2 i < 4 j < 4. Combining all of these elements we get exactly what we need. Remark 17.4. In the next paper, we will need the slightly more delicate fact th* *at ER(2)16*+8(RP 1 ^ RP 1) injects into E(2)16*+8(RP 1 ^ RP 1). 18.A special element To extract the information we need from the Bockstein spectral sequence for ER(2)*(RP 2n^RP 2m) we need to deal with odd degree elements for the first time. Our approach to this will be to use the long exact sequence coming from: RP 2n^ RP 2m-! RP 2n^ RP 1 -! RP 2n^ RP 1=RP 2m. From Section 16 we know ER(2)*(-) for the two terms on the right and we will compute a special element in the kernel. Many thanks to Jesus Gonz'alez for his work with the second author on BP *(RP 2n^ RP 2m). Ideas from there translated nicely to this situation and saved us from many a contorted filtration. Recall X [2](u) = akuk+1 k 0 in degree -16. 32 NITU KITCHLOO AND W. STEPHEN WILSON Definition 18.1. Let ffl(n) be 0 for n = 7 or 0 mod 8, 1 for n = 1 or 6 mod 8, * *2 for n = 2 or 5 mod 8, and 3 for n = 3 or 4 mod 8. These are just the numbers such that 0 6= un+ffl(n)2 ER(2)*(RP 2n) and 0 = un+ffl(n)+1. Theorem 18.2. Let m = 8K, and m < n. Define the degree -16(n + 1) element m-1X 1X g0 = un-m+1+i1 akum-i+k2. i=0 k=i+1 The element uffl(n)1g0 is in the kernel of the map ER(2)*(RP 2n^ RP 1=RP 2m) -! ER(2)*(RP 2n^ RP 1). The elements ui1g0, 0 i < m, are non-zero, not divisible by x and x2ui1g0 6= * *0. For n = 1, 2, 5 and 6 mod 8, ffkum1g0 6= 0. For n = 2, 3, 4 and 5 mod 8, um-1+ffl(n* *)1g0 6= 0. Proof.The map to ER(2)*(RP 2n^ RP 1) takes g0 to an element with the same notation. To see that uffl(n)1g0 is in the kernel we will add 0 to it in the fo* *rm of uffl(n)1 times m-1X Xi g1 = un-m+1+i1 akum-i+k2. i=0 k=0 Fix q = m - i + k, 0 < q m. Then i = m - q + k and we look at the coefficient of uq2in uffl(n)1g1: m-1X akun+1-q+k+ffl(n)1. k=0 This is zero because it is the relation in ER(2)*(RP 2n). Adding, we have: m-1X X1 g0 + g1 = un-m+1+i1 akum-i+k2 i=0 k=0 where the sum 1 X akum-i+k2= 0. k=0 This shows that uffl(n)1g0 is in the kernel. Although the image of g0, when ad* *ded to g1 is zero, g1 isn't zero until it has been multiplied by uffl(n)1so g0 is n* *ot in the kernel until it too has been multiplied by uffl(n)1. Multiply g0 by um-11to get m-1X X1 um-11g0 = un+i1 akum-i+k2. i=0 k=i+1 Since un+11is divisible by x, if we reduce modulo x all we have left is: 1X um-11g0 = un1 akum+k2. k=1 We need to show that this element is not divisible by x and that x2 times it is non-zero. We use the algorithm in Remark 15.3. The first term in the sum, a1un1um+12= ffun1um+12, represents an x1-torsion generator in the spectral sequ* *ence of Corollary 16.2. a3 gives un1um+32, an x3-torsion element. Any ak divisible b* *y 2, THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 33 such as a2, has that 2 applied to un1and the element becomes divisible by x. All we have left to consider are elements ak that are powers of ff. In this case we* * know that k > 3. Since we can work mod 2 the algorithm just uses ffu22= u42modulo higher powers of u2. All such elements end up as un1uj2with j > 3 and as such a* *re x1-torsion elements. We can conclude that our element is not divisible by x and that x2 times it is non-zero. Next we deal with the n = 3 and 4 mod 8 cases when we know un+316= 0. Multiply g0 by um+21to get: 1X um+21g0 = un+31 akum+k2. k=1 We know that un+31is divisible by x6 so both 2 and ff times it are zero. The on* *ly ak without a 2 or an ff is a3 so this reduces to un+31um+32. This is represente* *d by x6v72un1um+32in the spectral sequence and is non-zero. For all other n, un+31is zero. We look now at Let n = 2 or 5 mod 8. We know that un+21is non-zero and is x4 times the eleme* *nt in the Bockstein spectral sequence for ER(2)*(RP 2n) represented by v22un1. We * *also know that un+31= 0. Now multiply g0 by um+11to get 1X um+11g0 = un+21 akum+k2. k=1 Since our un+212 ER(2)*(RP 2n) is divisible by x4, both 2 and ff times un+21give zero. Recall also that every ak has a 2 or ff in it except for a3. Our formula * *is now just: um+11g0 = un+21um+32. In the Bockstein spectral sequence for ER(2)*(RP 2n^ RP 1=RP 2m) the element representing un+21um+32is x4 times v22un1um+32which is the target of a d7 so th* *is is non-zero. We want to do a bit more for um1g0 because we want ffjum1g0 when n = 1, 2, 5 and 6 mod 8. We know that un+21is divisible by x4 if it is non-zero so ff will * *kill it. So, for j > 0, ffjum1g0 is: 1X ffjun+11 akum+k2. k=1 Any 2 in ak will raise the power of u1 and give us x4 killing the ff, so, as in* * the previous cases, we are left with ffum+12+ um+32and higher powers of u2. Since we have an ff, the um+32also goes away and we are left with ffj+1un+11um+12. These elements are represented by x2v52ffj+1un1um+12in the spectral sequence and are * *all non-zero. 19.Starting the Bockstein spectral sequence for ER(2)*(RP 2n^ RP 2m) In the previous section we found a special element g0 2 ER(2)-16(n+1)(RP 2n^ RP 1=RP 2m), 34 NITU KITCHLOO AND W. STEPHEN WILSON where m = 8K, such that uffl(n)1g0 went to zero in ER(2)*(RP 2n^ RP 1). From the long exact sequence for the cofibration: (19.1) RP 2n^ RP 2m-! RP 2n^ RP 1 -! RP 2n^ RP 1=RP 2m we must have an element ^g0(n) 2 ER(2)-16(n+ffl(n))-17(RP 2n^ RP 2m) such that @(^g0(n)) = uffl(n)1g0. Because ui1g0 is not divisible by x for 0 i < m the s* *ame must be true of ui-ffl(n)1^g0(n) and that means these elements must reduce non-trivi* *ally to E(2)*(RP 2n^ RP 2m). The only elements in degree -1 mod 16 are ffkui1z-16n-17, with 0 i < m. The only elements in exactly degree -16(n + ffl(n)) - 17 with these ui-ffl(n)1non-zero are ff3kuffl(n)1z-16n-17 and so ^g0(n) must reduce to * *some combination of these elements. Theorem 19.2. The Bockstein spectral sequence for ER(2)*(RP 2m^ RP 2n) when m 8K < n: E1: E(2)*(RP 2m^RP 2n) ' E(2)*(RP 2m) E(2)*E(2)*(RP 2n) -16n-1E(2)*(RP 2m) is represented by vs2ffkui1u2 0 k 0 < i m s < 8 vs2ui1uj2 0 < i m 1 < j n s < 8 and vs2ffkui1z-16n-17 0 k 0 i < m s < 8 There is an element ^g0(n) 2 ER(2)-16(n+ffl(n))-17(RP 2m^RP 2n) ' ER(2)-16(n+ffl(n))-17(RP 2n^RP 2m) with 0 6= @(^g0(n)) 2 ER(2)-16(n+ffl(n))-16(RP 2n^ RP 1=RP 2m) such that ^g0(n) reduces to uffl(n)1z-16n-172 E(2)-16n-17(RP 2m^ RP 2n) and the ^g0(n) are compatible with the maps RP 2(m-1)! RP 2m. Modulo terms of higher filtration, d1 is d1(v2s-52ffkui1u2) = v2s2ffk+1ui+11u2 0 < i < m d1(v2s-52ui1uj2) = v2s2ui+11uj+22 0 < i < m 1 < j < n - 1 d1(v2s-52ffkui1z-16n-17) = v2s2ffk+1ui+11z-16n-17 0 i < m - 1 THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 35 E2 is: v2s2ffku1u2 0 k v2s2ui1u{1,2,3}2 1 < i < m v2s2u1uj2 1 < j n v2s+12ffkum1u2 0 < k v2s+12um1uj2 0 < j n v2s+12ui1un2 0 < i < m v2s+12ui1b1,n-1 0 i < m - 1 v2s2ffkz-16n-17 v2s2ui1z-16n-17 0 < i < m v2s+12ffkum-11z-16n-17 Proof.There are a couple of things to prove here. We must evaluate d1 and get E2 and then we must verify the reduction of ^g0(n) and prove its naturality. d1 is even degree so it acts independently on the even and odd degree parts of E1. d1 on the even degree part is induced from RP 1 ^ RP 1. It is only the third line, the differential on the odd degree elements, that we need to prove. If we* * can show that d1(z-16n-17) = 0 then the differential will follow from its behavior * *on the coefficients vs2. The cofibration 19.1 gives a long exact sequence in E(2)*(-). The two terms with RP 1 and RP 1=RP 2mare in even degrees so all of the even degree elements of E(2)*(RP 2n^ RP 2m) come from E(2)*(RP 2n^ RP 1) and all of the odd degree elements have boundary non-trivial and inject into E(2)*(RP 2n^ RP 1=RP 2m). The boundary is induced by the map RP 2n^ RP 1=RP 2m-! RP 2n^ RP 2m. The image of z-16n-17is in degree 0 mod 16 and so its representation must have a v02. All of d1 for RP 2n^ RP 1=RP 2mis on odd powers of v2 so since we have the odd degree elements injecting and d1 on the image of z-16n-17 equal to zero, we must have d1(z-16n-17) = 0. d1 follows as described above. We already know the reduction of ^g0(n) is of the form ff3kuffl(n)1z-16n-17so* * all we need to do is show that k = 0. Since we have computed d1 already, we know that ff3kum-11z-16n-17is in the image of d1 for k > 0. ff3kum-11z-16n-17represents t* *he element um-1-ffl(n)1^g0(n) and we know this has x2 on it non-zero. We just show* *ed that all of these elements with k > 0 are x1-torsion so we must have k = 0 and ^g0(n) maps to uffl(n)1z-16n-17, with, if necessary, a little redefinition of z* *-16n-17to avoid a sum. Consider the diagram: ER(2)*(RP 2m^ RP 2n)_____________//E(2)*(RP 2m^ RP 2n) fflffl| fflffl| ER(2)*(RP 2m-2^ RP 2n)___________//E(2)*(RP 2m-2^ RP 2n) By naturality, Remark 14.2, in E(2)*(-), z-16n-17 maps to the element of the same name on the right hand map. The element ^g0(n) in the upper left corner 36 NITU KITCHLOO AND W. STEPHEN WILSON must factor through a ^g0(n) in the lower left corner. It isn't obvious that @(* *^g0(n)) must be non-zero though. If it were zero then ^g0(n) would have to come from ER(2)-16(n+ffl(n))-17(RP 2n^ RP 1). We know that in here any odd degree elements are divisible by x but we also know that ^g0(n) is not divisible by x because it reduces to z-16n-17. It is an instructive exercise to apply the algorithm to see how the element g0 behaves under the map induced by RP 8K-8! RP 8K. Our goal with products all along has been to prove: Proposition 19.3. When n = 1, 2, 5 or 6 modulo 8, m 8K, and 8K + 8 < n, the element um1un+122 ER(2)*(RP 2m^ RP 2n) is non-zero. Proof.un+122 ER(2)*(RP 2n) is represented by x2 times the element represented in the spectral sequence by v52un2. So um1un+12is x2z where z reduces to um1un2* *v52in the Bockstein spectral sequence. um1un2v52survives to E2. For z to have x2z 6= * *0 it is enough that um1un2v52survives to E3, i.e. that it is not hit by a d2. (It ca* *nnot be the source of any differential because it is the product of the elements repres* *ented by um1and un2v52.) The differential d2 has degree 35 (-13). Our element um1un2v52has degree -16(* *m+ n) - 30 so the source that would have to hit it would have to have degree -16(m* * + n) - 17, in particular, it must be odd degree. The odd degree elements in the E2 term of our Bockstein spectral sequence are: v2s2ffkz-16n-17 v2s2ui1z-16n-17 0 < i < m and v2s+12ffkum-11z-16n-17 The only elements with degree equal to -1 modulo 16 are: ffkz-16n-17 and ui1z-16n-17 0 < i < m Since our differentials commute with multiplication by ff and u1, if such a dif* *ferential exists it has to be non-trivial on z-16n-17. Because uffl(n)1z-16n-17is in the * *image of ^g0(n) it must have all differentials on it trivial. Thus the target, d2(z-1* *6n-17) must be killed by uffl(n)1. If we do have a non-trivial differential for m = 8K* * + 8, by naturality, Remark 14.2, and the fact that the target is killed by uffl(n)1,* * the differential will be zero in m = 8K and for any m < 8K. Corollary 19.4. In ER(2)*(RP 2m^ RP 2n), m 8K, 8K + 8 < n, n = 1, 2, 5 and 6 modulo 8, the following elements are non-zero and independent in our 2-ad* *ic representation: ffkui1u2 k 0 i m and ui1uj2 i m j n + 1. Furthermore, ui1un+22= 0 when i 4. THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 37 Proof.The elements ffkui1u2, ui1uj2, i m, j n, reduce to E(2)*(RP 2m^ RP 2n* *). All we have left to worry about are the elements ui1un+12, i m. We know un+12= x2z, z ! un2v52so ui1un+12= x2ui1z with ui1z ! ui1un2v52. From Proposition 19.3* * we know that um1un+126= 0 and so ui1un+12must also be non-zero. The element un+22is zero when n = 1, 6, 7 or 8 mod 8. Otherwise it is divisib* *le by x4 so both 2 and ff times it are zero. We use 1.3 to replace u41in ui1un+22(* *i 4) with elements that all have either a 2 or an ff and so we have u41un+22= 0. 20. Non-immersions In this section we finish off the proofs of our non-immersion results. We sta* *rt with the first part of Theorem 1.8. Our goal is to show that the axial map K -2k-4 2K -2n-2 RP 2nx RP 2 -! RP does not exist for certain n and k. If n = 0 or 7 mod 8, K-1-n * 2K -2n-2 0 = u2 2 ER(2) (RP ). If we show that the image of this element in ER(2)*(RP 2nx RP 2K -2k-4) is non- zero then the axial map does not exist and RP 2ndoes not immerse in R2k+2. This computation is actually a coproduct because it can first be carried out * *for the map RP 1 RP 1x RP 1 and this last space has a K"unneth isomorphism for both our theories ER(2)*(-) and E(2)*(-). The first step, ER(2)*(RP 1) __________//_ER(2)*(RP 1) ^ER(2)*ER(2)*(RP 1) | | | | fflffl| fflffl| E(2)*(RP 1)_____________//_E(2)*(RP 1) ^E(2)*E(2)*(RP 1) is an isomorphism from the top row to the bottom in degrees 16* by Corollaries 8.3 and 17.3. The coproduct is therefore the same in both cases and comes from u -! m*(u) = u1 +F u2 = u1 + u2 + u1u2G where G is a power series. We are looking at m*(u)2K-1-n. If we write this out * *in our 2-adic basis it is: X X j ak,iffkui1u2 + bi,jui1u2 with j > 1 and the ak,iand bi,jeither 0 or 1. This is the same formula for eith* *er ER(2)*(-) or E(2)*(-). The way we do this reduction is to use our algorithm, 15.3. Our algorithm never lowers the sum of powers of u1 and u2, so i+j 2K-1 * *-n for example. 38 NITU KITCHLOO AND W. STEPHEN WILSON We continue the above map to E(2)*(RP 1)________________//E(2)*(RP 1) ^E(2)*E(2)*(RP 1) | | | | fflffl|K fflffl| K E(2)*(RP 2 -2n-2)_________//_E(2)*(RP 2n) ^E(2)*E(2)*(RP"2` -2k-2) | | fflffl|K E(2)*(RP 2nx RP 2 -2k-2) For the E(2)*(-) case, Don Davis, in [Dav84 ], showed that 0 = u2K-1-n mapped to non-zero when n = m + ff(m) - 1 and k = 2m - ff(m). The top map on the right goingKdown-takes1basis elements to zero or to basis elements. Since un+11= 0 = u22 -k our coproduct reduces to X X j ak,iffkui1u2 + bi,jui1u2 i n i n 1 < j 2K-1 - k - 1 and [Dav84 ] shows that this must be non-zero. We now do the same thing with ER(2)*(-). We use the diagram: ER(2)*(RP 1) _______________//_ER(2)*(RP 1) ^ER(2)*ER(2)*(RP 1) | | | | fflffl|K fflffl| K ER(2)*(RP 2 -2n-2)__________//ER(2)*(RP 2n) ^ER(2)*ER(2)*(RP 2 -2k-4) | | fflffl|K ER(2)*(RP 2nx RP 2 -2k-4) We assume that n = 0 or 7 mod 8, which gives us un+11= 0 = u2K-1-n, and -k-2 = 1, 2, 5 or 6 mod 8. Because of this restriction on k, ER(2)16*(RP 2K -2k* *-4) surjects to E(2)16*(RP 2K -2k-2). If we write out our coproduct in K -2k-4 ER(2)*(RP 2n) ER(2)*ER(2)*(RP 2 ) it is nearly the same as it was in E(2)*(RP 2n) E(2)*E(2)*(RP 2K -2k-2). We al* *ways K-1-k have uj2= 0 when j > 2K-1 - k. We could have elements ui1u22 . These elements are all zero by Corollary 19.4 when i 4 which we can get if we sneak* * in the inconsequential assumption that n+4 k. Consequently, our obstruction is exact* *ly the same linear combination for ER(2)*(-) as it was for E(2)*(-) and we have shown that these elements all map independently into ER(2)*(RP 2nxRP 2K -2k-4) by Corollary 19.4. As a result of the above discussion, Don Davis's obstructions work for us as * *well but with an improvement, in our special cases, of 2. To meet our conditions we must have (from Theorem 13.4) -k - 2 = {1, 2, 5, 6} mod 8 THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 39 and, from [Dav84 ], k = 2m - ff(m) and also n = m + ff(m) - 1 = {0, 7} mod 8. (from both). Our result in the Introduction follows once we get our pairs (m, f* *f(m)) from these equations. Our first is: -2m + ff(m) - 2 = {1, 2, 5, 6} mod 8 -2m + ff(m) = {3, 4, 7, 0} mod 8 2m - ff(m) = {5, 4, 1, 0} mod 8. The second: m + ff(m) - 1 = {0, 7} mod 8. m + ff(m) = {1, 0} mod 8. Adding the two equations we get: 3m = {6, 5, 2, 1} or{5, 4, 1, 0} mod 8. Multiply by 3 (mod 8) to get: m = {2, 7, 6, 3} or{7, 4, 3, 0} mod 8. Substituting this into ff(m) = -m + {1, 0} mod 8 we get ff(m) = {7, 2, 3, 6} or{1, 4, 5, 0}. So, our result is as stated in the Theorem 1.8. For the second part of Theorem 1.8 we begin again with the main theorem of [Dav84 ], for n = m + ff(m) - 1 k = 2m - ff(m) there does not exist an axial map: K -2k-2 2n 2K -2n-2 RP 2 x RP -! RP and so RP 2n* R2k. This is proven by using the equivalent of E(2)*(-) and showing that the u2K-1-n = 0 on the right would have to go to a non-zero element on the left. That same element would prevent the existence of an axial map, K -2k-2 2n+2 2K -2n-2 RP 2 x RP -! RP and likewise K K RP 2 -2k-2x RP 2n+2-! RP 2 -2n-4. Furthermore, if u2K-1-n went to non-zero then we must also have u2K-1-n-1K=-1 0 also going to a non-zero element. If n + 1 = 7 mod 8 then u2 -n-1 = 0 for ER(2)*(-) and, if -k - 2 = {1, 2, 5, 6} mod 8 this must factor through the ER(2)*(-) cohomology of K -2k-4 2n+2 2K -2n-4 RP 2 x RP -! RP as above and we have that RP 2n+2* R2k+2, or, RP 2(m+ff(m))* R2(2m-ff(m)+1). We have to untangle some equations to get our (m, ff(m)) pairs for this. We have n + 1 = m + ff(m) = 7 mod 8 40 NITU KITCHLOO AND W. STEPHEN WILSON and -k - 2 = -2m + ff(m) - 2 = {1, 2, 5, 6} mod 8. The equation for k is the same as before so we have 2m - ff(m) = {5, 4, 1, 0} mod 8. The equation for n gives m + ff(m) = 7 mod 8. Adding, we have 3m = {4, 3, 0, 7} mod 8. Multiply by 3 to get m = {4, 1, 0, 5} mod 8. Substituting into ff(m) = -m + 7 mod 8 we get ff(m) = {3, 6, 7, 2} and our pairs are as in our Theorem 1.8. Remark 20.1. Don Davis does his work with the theory BP <2>*(-) with BP <2>* ' Z(2)[v1, v2]. For these spaces there is no v2 torsion so when v2 is inverted to* * create E(2)*(-), everything injects. The normal E(2) is 6 periodic but we can consider* * it 48 periodic just as well, it doesn't change anything. Davis does all of his com* *pu- tations with the standard 2-dimensional class, x2, but the computations all hol* *d if this is adjusted by a unit so we can use our u in degree -16. 21.The Atiyah-Hirzebruch spectral sequence approach The original computation of ER(2)*(RP 2n) was carried out using the Atiyah- Hirzebruch spectral sequence and we give a brief description of how that was do* *ne here. To begin we use the long exact sequence: (21.1) ER(2)*(X)c____x_____//ER(2)*(X)cH HHHH vvv @HHH --vaevvvv E(2)*(X) for X = RP 16and X = RP 1. Since we know E(2)*(RP 16) we can just look at the Atiyah-Hirzebruch spectral sequence for ER(2)*(RP 16) and see that what is there in the E2 term for ER(2)16*(RP 16) must map isomorphically to E(2)16*(RP 16) and so must also be E1 and cannot have any differentials entering or leaving. Using this isomorphism and the fact that E(2)*(RP 16) is even degree, the long exact sequence gives us ER(2)16*+1(RP 16) = 0 as it is trapped in: 0 ' E(2)16*-1(RP 16) -! ER(2)16*+17(RP 16) -! ER(2)16*(RP 16) ' E(2)16*(RP 16). It then follows that ER(2)16*+2(RP 16) = 0 from: ER(2)16*(RP 16) ' E(2)16*(RP 16) -! ER(2)16*+18(RP 16) -! ER(2)16*+1(RP 16) = 0. THE SECOND REAL JOHNSON-WILSON THEORY AND NON-IMMERSIONS OF RPn 41 We get one more, i.e. ER(2)16*+3(RP 16) = 0 from: 0 ' E(2)16*-15(RP 16) -! ER(2)16*+3(RP 16) -! ER(2)16*+2(RP 16) = 0. In order for the Atiyah-Hirzebruch spectral sequence for ER(2)*(RP 16) to end up with zero in these degrees we must have differentials, none of which can sta* *rt (or end) on ER(2)16*(RP 16). There is only one way for this to happen and it shows us what the dr are for r = 2, 3, 4, 5, 6 and 7. These differentials then work f* *or all ER(2)*(RP 2n). Elements can be identified using ER(2)*(RP 1) and the map to E(2)*(RP 1). This works quite well but breaks down when attempting products. References [BDM02]R. R. Bruner, D. M. Davis, and M. Mahowald. Nonimmersions of real projec* *tive spaces implied by tmf. In D. Davis, J. Morava, G. Nishida, W. S. Wilson, and N.* * Yagita, editors, Recent Progress in Homotopy Theory: Proceedings of a conference* * on Recent Progress in Homotopy Theory March 17-27, 2000, Johns Hopkins University,* * Balti- more, MD., volume 293 of Contemporary Mathematics, pages 45-68, Providen* *ce, Rhode Island, 2002. American Mathematical Society. [Dav] D. Davis. Table of immersions and embeddings of real projective* * spaces. http://www.lehigh.edu/~dmd1/immtable. [Dav84]D. M. Davis. A strong nonimmersion theorem for real projective spaces. A* *nnals of Mathematics, 120:517-528, 1984. [GW] J. Gonz'alez and W.S. Wilson. The BP-cohomology of two-fold products of * *real projec- tive spaces. In preparation. [HK01] P. Hu and I. Kriz. Real-oriented homotopy theory and an analogue of the * *Adams- Novikov spectral sequence. Topology, 40(2):317-399, 2001. [Hu01] P. Hu. The Ext0-term of the Real-oriented Adams-Novikov spectral sequenc* *e. In Homo- topy Methods in algebraic topology, volume 271 of Contemporary Mathemati* *cs, pages 141-153, Providence, Rhode Island, 2001. American Mathematical Society. [Jam63]I.M. James. On the immersion problem for real projective spaces. Bulleti* *n of the Amer- ican Mathematical Society, 69:231-238, 1963. [JW85] D. C. Johnson and W. S. Wilson. The Brown-Peterson homology of elementar* *y p-groups. American Journal of Mathematics, 107:427-454, 1985. [KW] N. Kitchloo and W.S. Wilson. On the Hopf ring for ER(n). Topology and it* *s Applica- tions. To appear. [KW07] N. Kitchloo and W.S. Wilson. On fibrations related to real spectra. In M* *. Ando, N. Mi- nami, J. Morava, and W.S. Wilson, editors, Proceedings of the Nishida Fe* *st (Kinosaki 2003), volume 10 of Geometry & Topology Monographs, pages 237-244, 2007. [Mas54]W.S. Massey. Products in exact couples. Annals of Mathematics, 59(3):558* *-569, 1954. Department of Mathematics, University of California, San Diego (UCSD), La Jol* *la, CA 92093-0112 E-mail address: nitu@math.ucsd.edu Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218 E-mail address: wsw@math.jhu.edu