NONIMMERSIONS OF RP nIMPLIED BY tmf, REVISITED DONALD M. DAVIS AND MARK MAHOWALD Abstract.In a 2002 paper, the authors and Bruner used the new spectrum tmfto obtain some new nonimmersions of real projective spaces. In this note, we complete/correct two oversights in that paper. The first is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for RPn with n as small as 48, and yet it was stated there that the first new result occurred when n = 1536. Here we give a simple proof of those overlooked results. Secondly, we fill in a gap in the proof of the 2002 paper. There it was claimed that an axial map f must satisfy f*(X) = X1+X2. We realized recently that this is not clear. However, here we show that it is true up multiplication by a unit in the appropriate ring, and so we retrieve all the nonimmersion results claimed in [6]. Finally, we present a complete determination of tmf8*(RP1 x RP1 ) and tmf*(CP1 x CP1 ) in positive dimensions. 1.Introduction In [6], the authors and Bruner described a proof of the following theorem, al* *ong with some additional nonimmersion results. Theorem 1.1. ([6, 1.1]) Assume that M is divisible by the smallest 2-power grea* *ter than or equal to h. oIf ff(M) = 4h - 1, then P 8M+8h+2cannot be immersed in (6 ) R16M-8h+10. oIf ff(M) = 4h - 2, then P 8M+8h6 R16M-8h+12. Here and throughout, ff(M) denotes the number of 1's in the binary expansion of* * M, and P ndenotes real projective space. __________ Date: April 5, 2007. 2000 Mathematics Subject Classification. 57R42, 55N20. Key words and phrases. immersion, projective space, elliptic cohomology. We thank Steve Wilson for causing us to take a look at these matters. 1 2 DONALD M. DAVIS AND MARK MAHOWALD In [6], the theorem is followed by a comment that this is new provided ff(M) * * 6, i.e., h 2, and the first new result occurs for P 1536. In this note, we poin* *t out that 1.1 is valid when h = 1, and these results are new when M is even, includi* *ng new nonimmersions of P nfor n as small as 56. A remark in [6, p.66] that the nonimmersions when h = 1 were implied by earlier work of the authors was incorr* *ect. Letting h = 1 in 1.1, we have the following result. Corollary 1.2. a. If ff(M) = 3, then P 8M+106 R16M+2. b. If ff(M) = 2, then P 8M+86 R16M+4. Part (a) is new when M is even. It is 2 better than the previous best result,* * proved in [4], and the nonembedding result that it implies is also new, 1 better than * *the previous best, proved in [3]. In [7], a table of known nonimmersions, immersio* *ns, nonembeddings, and embeddings of P nis presented, arranged according to n = 2i+d with 0 d < 2iand d < 64. Part (a) enters the table with a new result for d = * *58, applying first to P 122. If M is even, 1.2.b is new, 1 better than the previous best result, of [12], * *and the nonembedding result implied is also new. It enters [7] at d = 24 and 40, with a* * new result for P nwith n as small as 56. The result of 1.2.b with M = 2i+ 1 was also k+16 proved very recently by Kitchloo and Wilson in [15]. This result for P 2 , 2 * *better than the previous result of [4] and also new as a nonembedding, enters [7] at d* * = 16, and applies for n as small as 48. In Section 2, we present a self-contained proof of Corollary 1.2. The primary* * reason for doing this, which amounts to a reproof of part of [6, 1.1], is that the pro* *of of the general case in [6] requires some extremely elaborate arguments and calculation* *s. Our proof here, which is just for the case h = 1, is much more comprehensible. The proof in [6] contained an oversight which we shall correct here. The argu* *ment there was that an immersion of RP nin Rn+k implies existence of an axial map P * *nx f m+k P m -!P for an appropriate value of m, and obtains a contradiction for certa* *in n, m, and k by consideration of tmf*(f). Here tmf is the spectrum of topologic* *al modular forms, which was discussed in [6]. A class X 2 tmf8(P n) was described, along with X1 = X x 1 and X2 = 1 x X in tmf8(P nx P m). It was asserted that f*(X) = X1+X2, and a contradiction obtained by showing that, for certain values* * of NONIMMERSIONS IMPLIED BY TMF, REVISITED 3 the parameters, we might have X` = 0 but (X1+ X2)`6= 0. We recently realized th* *at it is conceivable that f*(X) might contain other terms coming from tmf8(P n^ P * *m). In Section 3 (see Theorem 3.7) we perform a complete calculation of tmf*(P 1x* *P 1) in positive gradings divisible by 8, and in Section 4 we use it to show that ef* *fectively f*(X) = u(X1 + X2), where u is a unit in tmf*(P 1 x P 1), which enables us to retrieve all the nonimmersions of [6]. In Section 5, we compute tmf*(CP 1 x CP 1) in positive gradings. The original purpose of doing this was, prior to our obtaining the argument of Section 4, to* * see whether we might mimic the argument of [2] and [8] to conclude that if f is an axial map, then f*(X) might necessarily equal u(X1 - X2), where u is a unit in tmf*(CP x CP ). This approach to retrieving the nonimmersions of [6] did not yi* *eld the desired result, but the later approach given in Section 4 did. Nevertheless* * the nice result for tmf*(CP 1 x CP 1) obtained in Theorem 5.19 should be of independent interest. 2. Proof of Corollary 1.2 We begin by proving 1.2.a. The following standard reduction goes back at leas* *t to [14]. If P 8M+10 R16M+2, then gd((2L+3 - 8M - 11),8M+10) 8M - 8, hence this bundle has (2L+3- 16M - 3) linearly independent sections, and thus there is an * *axial map L+3-16M-4f 2L+3-8M-12 P 8M+10x P 2 -! P . The bundle here is the stable normal bundle, L is a sufficiently large integer,* * and gd refers to geometric dimension. Let X, X1, and X2 be elements of tmf8(-) describ* *ed in [6] and also in Section 1. In Section 4, we will show that we may assume th* *at f*(X) = X1+ X2, as was done in [6], since this is true up to multiplication by * *a unit. L+3-8M-8 2L+3-8M-12 Since tmf2 (P ) = 0, we have L-M-1 2L-M-1 2L+3-8M-8 8M+10 2L+3-16M-4 0 = f*(0) = f*(X2 ) = (X1+X2) 2 tmf (P xP ). i L j L i L j L Expanding, we obtain 2M-M-1+1XM+11X22-2M-2+ 2 -M-1MXM1X22-2M-1 as the only terms which are possibly nonzero. Next we note that, with all u's representing * *odd integers, i2L-M-1j i2M+1j ff(M)- (M+1) 3- (M+1) M+1 = u1 M+1 = 2 u2 = 2 u2, 4 DONALD M. DAVIS AND MARK MAHOWALD where we have used ff(M) = 3 at the last step. Here and throughout, (2eu) = e. i L j i j Similarly, 2 -M-1M= u3 2MM = 2ff(M)u4 = 23u4. Thus an immersion implies that in L+3-8M-8 8M+10 2L+3-16M-4 tmf2 (P x P ), we have L-2M-2 3 M 2L-2M-1 23- (M+1)u2XM+11X22 + 2 u4X1 X2 = 0. (2.1) We recall [6, 2.6], which states that there is an equivalence of spectra Pbk+* *8+8^ tmf ' 8Pbk^ tmf. Combining this with duality, we obtain tmf8M+8(P 8M+10) L-2M-2 1 tmf-1(P-3) Z=8, and so 8XM+11X22 = 0. Here and throughout, Pn = Pn = L+3-16M-8 2L+3-16M-4 RP 1=RP n-1. Similarly tmf2 (P ) tmf7(P3) Z=16, and hence L-2M-1 16XM1X22 = 0. Duality also implies L+3-8M-8 8M+10 2L+3-16M-4 tmf2 (P x P ) tmf14(P-3 ^ P3). Calculations such as E2(tmf*(P-3^P3)), the E2-term of the Adams spectral sequen* *ce (ASS), were made by Bruner's minimal-resolution computer programs in our work on [6]. This one is in a small enough range to actually do by hand. The result is * *given in Diagram 2.2. Diagram 2.2. E2(tmf*(P-3 ^ P3)), * 15 |r | | rr |r | || r r|| r |r | | | r r| r||r||r | || r r rr|r||r || || |||| r r| r r r r r r |rrr||r | | | || | r r||r r rr r|| r r ||r r r ||r||r||r | | | | || | ___________________________________________________rrrrr||r|r||* *rrrr|rr|r 0 3 7 11 15 The Z=8 Z=16 arising from filtration 0 in grading 14 in 2.2 is not hit by a differential from the class in (15, 0) because, as explained in the last paragr* *aph of page 54 of [6], the class in (15, 0) corresponds to an easily-constructed nontrivial* * map. The L-2M-2 M 2L-2M-1 monomials XM+11X22 and X1 X2 are detected in mod-2 cohomology, NONIMMERSIONS IMPLIED BY TMF, REVISITED 5 and so their duals emanate from filtration 0. We saw in the previous paragraph * *that 8 and 16, respectively, annihilate these monomials, and hence also their duals.* * Since the chart shows that the subgroup of tmf14(P-3^P3) generated by classes of filt* *ration 0 is Z=8 Z=16, we conclude that 8 and 16, respectively, are the precise order* *s of L-2M-1 the monomials. In particular, the order of XM1X22 is 16, and hence the cla* *ss L-2M-1 in (2.1) is nonzero since it has a term 8uXM1X22 , and so (2.1) contradicts* * the hypothesized immersion. Part b of 1.2 is proved similarly. If P 8M+8immerses in R16M+4, then there is* * an axial map L+3-16M-6f 2L+3-8M-10 P 8M+8x P 2 -! P , and hence, up to odd multiples, L-2M-2 2 M 2L-2M-1 22- (M+1)XM+11X22 + 2 X1 X2 (2.3) L+3-8M-8 8M+8 2L+3-16M-6 = 0 2 tmf2 (P ^ P ), since ff(M) = 2. We have tmf8M+8(P 8M+8) tmf-1(P-1) Z=2, and L+3-16M-8 2L+3-16M-6 tmf 2 (P ) tmf-1(P-3) Z=8. Thus the two monomials in (2.3) have order at most 2 and 8, respectively. On the other hand, the group in (2.3) is isomorphic to tmf6(P-1 ^ P-3). A minimal resolution calculation easier than the one in Diagram 2.2 shows that tmf6(P-1^ * *P-3) has Z=2 Z=8 emanating from filtration 0 (and another Z=2 Z=8 in higher filtrati* *on). The monomials of (2.3) are generated in filtration 0, and since the above upper* * bound for their orders equals the order of the subgroup generated by filtration-0 cla* *sses, we conclude that the orders of the monomials in (2.3) are precisely 2 and 8, respe* *ctively, L-2M-1 and so the term 4XM1X22 in (2.3) is nonzero, contradicting the immersion. 3. tmf-cohomology of P 1x P 1 In this section, we compute tmf*(P 1) and tmf8*(P 1x P 1) in positive grading* *s. These will be used in the next section in studying the axial class in tmf-cohom* *ology. There is an element c4 2 ss8(tmf) which reduces to v412 ss8(bo); it has Adams filtration 4. It acts on tmf*(X) with degree -8. Recall also that ss*(bo) = bo** * is as depicted in 5.1. We denote bo* = bo-*. We use P1 and P 1 interchangeably. 6 DONALD M. DAVIS AND MARK MAHOWALD Theorem 3.1. There is an element X 2 tmf8(P1) of Adams filtration 0, described in [6], such that, in positive dimensions divisible by 8, tmf*(P1) is isomorphi* *c as an algebra over Z(2)[c4] to Z(2)[c4][X]. In particular, each tmf8i(P1) with i > 0 * *is a free abelian group with basis {cj4Xi+j: j 0}. There is a class L 2 t0(P1) such that otmf0(P1) is a free abelian group with basis {L, cj4Xj : j 1}, and oL2 = 2L and LX = 2X. Moreover, in positive dimensions tmf*(P1) is isomorphic as a graded abelian gro* *up to bo*[X], and is depicted in Diagram 3.6. Remark 3.2. A complete description of tmf*(P1) as a graded abelian group could probably be obtained using the analysis in the proof which follows, together wi* *th the computation of the E2-term of the ASS converging to tmf*(P-1), which was given * *in [10]. However, this is quite complicated and unnecessary for this paper, and so* * will be omitted. Proof.We begin with the structure as graded abelian group. There are isomorphis* *ms tmf *(P1) limtmf*(P1n) limtmf-*-1(P--2n-1) = tmf-*-1(P--21). (3.3) Since H*(tmf; Z2) A==A2, there is a spectral sequence converging to tmf*(X) w* *ith E2(X) = ExtA2(H*X, Z2). Here A2 is the subalgebra of the mod 2 Steenrod algebra A generated by Sq1, Sq2, and Sq4. Also Z2 = Z=2. We compute E2(P--21) from the exact sequence q* s,t 1 ! Es-1,t2(P-11) ! Es,t2(P--21) ! Es,t2(P-11) -! E2 (P-1) ! . (3.4) It was proved in [17] that M ExtA2(P1-1, Z2) ExtA1( 8i-1Z2, Z2). i2Z Here we have initiated a notation that Pmn:= H*(Pnm). A complete calculation of ExtA2(P1-1, Z2) was performed in [10], but all we need here are the first few g* *roups. We can now form a chart for E2(P--21) from (3.4), as in Diagram 3.5, where O in* *dicate elements of ExtA2(P1-1, Z2) suitably positioned, and lines of negative slope co* *rrespond to cases of q* 6= 0 in (3.4). NONIMMERSIONS IMPLIED BY TMF, REVISITED 7 Diagram 3.5. tmf*(P--21), -17 * 2 || | || | || |6|6 |6|6 |6|6 |6|6 |6|6 || || || || || || ||r || ||r || || | || | || r||r r |r r||r r |r r||r r || | || | || r||rr |r r||rr |r r||rr || | || | || r||r |r r||r |r r||r b | | | | | @ . . . r| |r r| |r r| b b|@ | | | r| r r| r r|b r b| | | | r|r r| r b r|r b| | | @ | _____________________________________________________________rrr|||@ -17 -9 -1 Dualizing, we obtain Diagram 3.6 for the desired tmf*(P11). Diagram 3.6. tmf*(P11), * -2 || | || | || ||66 |6|6 |6|6 |6|6 ||66 || || || || || || ||r || ||r || || | || | || r |r|r |r r r||r |r r |r|r @ || | @ || | @ || r |r|r |r r r||r |r r |r|r @ || | @ || | @ || @ |r|r |r @ r||r |r @|r|r | | | | | r r| |r r| |r r| . . . | | | r@r@ r| r r| r r| | | | @@r@ r| @ @r r| @ @r r| | | __________________________________________________________rr@@|| 0 8 16 Naming of the generators Xi is clear since X has filtration 0. The free actio* *n of c4 is also clear. The class L is (up to sign) the composite P1 ~-!S0 ! tmf, where * *~ is the well-known Kahn-Priddy map. Thus L is the image of a class ^L2 ss0(P1). Lin* *'s theorem ([16]) says that ss0(P1) Z^2, generated by ^L. Since ss0(P1) ! ko0(P1* *) is an isomorphism, and, since (1 - ,)2 = 2(1 - ,) for a generator (1 - ,) of ko0(P1),* * we obtain ^L2= 2^L, and hence also for L. We chose the generator to be (1 - ,) rat* *her than (, - 1) to avoid minus signs later in the paper. 8 DONALD M. DAVIS AND MARK MAHOWALD To prove the claim about LX, first note that, by the structure of tmf8(P1), we must have LX = p(c4X)X for some polynomial p. Multiply both sides by L and apply the result about L2 to get 2LX = p(c4X)LX, hence 2p = p2, from which we conclude p = 2. || In tmf*(P1x P1), for i = 1, 2, let Liand Xidenote the classes L and X in the * *ith factor. Note that there is an isomorphism as tmf*-modules, but not as rings, tmf*(P1x P1) tmf*(P1^ P1) tmf*(P1x *) tmf*(* x P1). Theorem 3.7. In positive dimensions divisible by 8, tmf*(P1 ^ P1) is isomorphic as a graded abelian group to a free abelian group on monomials Xi1Xj2with i, j * *> 0 direct sum with a free Z[c4]-module with basis {L1Xi2, Xi1L2 : i 1}. The prod* *uct and Z[c4]-module structure is determined from 3.1 and X c4(X1X2) = (c4X1)X2 = X1(c4X2) = flici4(L1Xi+12+ Xi+11L2), i 0 for certain integers fliwith fl0 divisible by 8. The proof of this theorem involves a number of subsidiary results. They and * *it occupy the remainder of this section. We will use duality and exact sequences s* *imilar to (3.4). But to get started, we need ExtA2(P P, Z2). Here we have begun to abbreviate P := P1-1. We begin with a simple lemma. Throughout this section, x1 and x2 denote nonzero elements coming from the factors in H1(RP x RP ; Z2). Lemma 3.8. ([9]) There is a split short exact sequence of A-modules 0 ! Z2 P ! P P ! (P=Z2) P ! 0. Proof.The Z2 is, of course, the subgroup generated by x0, which is an A-submodu* *le. g i j 0 i+j A splitting morphism P P -!Z2 P is defined by g(x1 x2) = x1 x2 . This is A-linear since X i ijijj i+j+k ii+jj i+j+k k j g(Sqk(xi1 xj2)) = ` k-`x01 x2 = k x01 x2 = Sq g(xi1 x2). ` || The following result is more substantial. We will prove it at the end of this s* *ection. NONIMMERSIONS IMPLIED BY TMF, REVISITED 9 Proposition 3.9. There is a short exact sequence of A2-modules 0 ! C ! (P=Z2) P ! B ! 0, where C has a filtration with Fp(C)=Fp-1(C) 8pA2= Sq2, p 2 Z, and B has a filtration with M 1 Fp(B)=Fp-1(B) 4p-2A2= Sq, p 2 Z. Z copies The generator of Fp(C)=Fp-1(C) is x11x8p-12; a basis over Z2 for C is {x21xi+22+x41xi2, x41xi2+x81xi-42, i 2 Z}[{x11xi-12+x21xi-22, i 6 0 (8)}[{x11x* *8p-12, p 2 Z}. A minimal set of generators as an A2-module for the filtration quotients of B is {x8i-11x4j-12: i, j 2 Z}. Corollary 3.10. A chart for Exts,tA2(P P, Z2) in 8p - 3 t - s 8p + 4 is as suggested in Diagram 3.11, for all integers p. The big batch of towers in each * *grading 2 (4) represents an infinite family of towers. The pattern of the other clas* *ses is repeated with vertical period 4. Thus, for example, in 8p - 1 there is an infin* *ite tower emanating from filtration 4i for each i 0. 10 DONALD M. DAVIS AND MARK MAHOWALD Diagram 3.11. Exts,tA2(P P, Z2) in 8p - 3 t - s 8p + 4 |||||||||||| |||||||||||| |||||||||||||| ||||||||||||| |||||||||||||| ||||||||||||||| |6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6 |||||||||||||| ||||||||||||||| |||||||||||||| |||||||||||||||r |||||||||||||| |||||||||||||| |||||||||||||| |||||||||||||| |||||||||||||| |||||||||||||| |||||||||||||| r ||||||||||||||r |||||||||||||| ||||||||||||| |||||||||||||| ||||||||||||| |||||||||||||| ||||||||||||| ||||||||||||||r |||||||||||r|| |||||||||||||| ||||||||||||| |||||||||||||| ||||||||||||| |||||||||||||| ||||||||||||| ||||||||||||||rr ||||||||||||| ||||||||||||| ||||||||||||| ||||||||||||| ||||||||||||| ||||||||||||| ||||||||||||| |||||||||||||r |||||||||||||r |||||||||||| |||||||||||| |||||||||||| |||||||||||| |||||||||||| |||||||||||| |||||||||||| r ||||||||||||r |||||||||||| ||||||||||| |||||||||||| ||||||||||| |||||||||||| ||||||||||| ||||||||||||r |||||||||||r |||||||||||| ||||||||||| |||||||||||| ||||||||||| |||||||||||| ||||||||||| _________________________________________|||||||||||||||||||||||rr 8p+ -2 0 2 4 Proof of Corollary 3.10.We first note that ExtA2(P, Z2) is identical to the lef* *t portion of Diagram 3.5 extended periodically in both directions. Also, ExtA2(A2= Sq1, Z* *2) ExtA0(Z2, Z2) is just an infinite tower, and ExtA2(A2= Sq2, Z2) ExtA1(A1= Sq2, Z2) is given as in Diagram 3.14. We will show at the end of this proof that M 2 Ext A2(C, Z2) ExtA2( 8pA2= Sq, Z2) (3.12) p2Z and similarly M M 1 ExtA2(B, Z2) ExtA2( 4p-2A2= Sq, Z2). p Z These would follow by induction on p once you get started, but since p ranges o* *ver all integers, that is not automatic. Thus ExtA2(P P, Z2) is formed from M 2 M 1 ExtA2(P, Z2) ExtA2( 8pA2= Sq, Z2) ExtA2( 4p-2A2= Sq, Z2), NONIMMERSIONS IMPLIED BY TMF, REVISITED 11 using the sequences in 3.8 and 3.9. The Ext sequence of 3.8 must split, and the* *re are no possible boundary morphisms in the Ext sequence of 3.9, yielding the claim o* *f the corollary. To prove (3.12), let (s, t) be given, and choose p0 so that 8p0 < t - 23s + 2* *. Since the highest degree element in A2 is in degree 23, Exts,tA2(Fp0(C), Z2) = 0. Act* *ually a much sharper lower vanishing line can be established, but this is good enough f* *or our purposes. Thus, for this (s, t), M s,t 2 Exts,tA2(Fp1(C), Z2) ExtA2( 8p-2A2= Sq)(3.13) p p1 for p1 p0, as both are 0. Let p1 be minimal such that (3.13) does not hold. T* *hen comparison of exact sequences implies that Exts-1,tA2(Fp1-1(C), Z2) ! Exts,tA2(Fp1(C)=Fp1-1(C), Z2) must be nonzero. But one or the other of these groups is always 0,1 as both cha* *rts Ext*,*A2(Fp1-1(C), Z2) and Ext*,*A2(Fp1(C)=Fp1-1(C), Z2) are copies of Diagram * *3.14 dis- placed by 4 vertical units from one another. Thus (3.13) is true for all p1, an* *d hence (3.12) holds. A similar proof works when C is replaced by B. || Diagram 3.14. ExtA2(A2= Sq2, Z2) . . . |6 |6 | |r r |r |rr | | |r |r |r r _________________r 0 Now we can prove a result which will, after dualizing, yield Theorem 3.7. The groups ExtA1(Z2, Z2) to which it alludes are depicted in 5.1. The content of t* *his result is pictured in Diagram 3.18. Proposition 3.15. In dimensions t - s 2 mod 4 with t - s -10, ExtA2(P-2-1 P-2-1, Z2) consists of i infinite towers emanating from filtration 0 in dimensi* *ons -8i- 6 and -8i - 10, together with the relevant portion of two copies of ExtA1(Z2, Z* *2) __________ 1Actually this is not quite true; for one family of elements we need to use h* *0- naturality. 12 DONALD M. DAVIS AND MARK MAHOWALD beginning in filtration 1 in each dimension -8i - 2. The generators of the towe* *rs in -8i - 10 correspond to cohomology classes x-91x-8i-12, . .,.x-8i-11x-92. The ge* *nerators of the two copies of ExtA1(Z2, Z2) in -8i-2 arise from h0 times classes corresp* *onding to x-11x8i-12and x-8i-11x-12. Proof.Using exact sequences like (3.4) on each factor, we build Ext*,*A2(P-2-1 * *P-2-1, Z2) from A := Ext*,*A2(P P, Z2), B := Ext*-1,*A2(P1-1 P, Z2), C := Ext*-1,*A2(P P1-* *1, Z2), and D := Ext*-2,*A2(P-11 P-11, Z2), with possible d1-differential from A and i* *nto D. In the range of concern, t - s -9, the D-part will not be present, and the pa* *rt of Diagram 3.11 in dimension 6 2 mod 4 will not be involved in d1. Using [17] for* * B and C, the relevant part, namely the portion of A in dimension 2 mod 4, toget* *her with B and C, is pictured in Diagram 3.16. Diagram 3.16. Portion of|A|+|B|+|C|||||||||||||||||||||||||||||||||||||||||||||* *|||||||||| |||||||||| |||||||||| |||||||||| |||6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6|6* *|6|6|| rr|||||||||||| |||||||||| rr|||||||||||| |||||||||||| |||||||||| |||||||||||| |||||||||||||| |||||||||| |||||||||||| rr||||||||||||r||||||||||rrr|||||||||||||| |||||||||||||| |||||||||| |||||||||||| rr||||||||||||r||||||||||rrr||||||||||||rr|| |||||||||||||| |||||||||| |||||||||||| |||||||||||||| |||||||||| |||||||||||| rr||||||||||||r||||||||||rrr||||||||||||rr|| |||||||||||||| |||||||||| |||||||||||| |||||||||||||| |||||||||| |||||||||||| rr||||||||||||rr||||||||||rr||||||||||||rr||||rr|| |||||||||||||||||||||||||||||||||||||||| rr||||||||||||rr||||||||||rr||||||||||||rr||||rrrr|| |||||||||||||||||||||||||||||||||||||||||| |||||||||||||||||||||||||||||||||||||||||| rr||||||||||||rr||||||||||rr||||||||||||rr||||rrrr||rr|| |||||||||||||||||||||||||||||||||||||||||| |||||||||||||||||||||||||||||||||||||||||| rr||||||||||||rr||||||||||rr||||||||||||rr||||rrrr||rr|| |||||||||||||||||||||||||||||||||||||||||| rrr||||||||||r||||||||||||rrr||||||||||r||rrrr||||rr||rr|| |||||||||| |||||||||| |||||||||| |||||||||| |||||||||| |||||||||| ________________________________|||||||||||||||||||||||||||||| 8p+ -2 2 6 In dimension 8p-2, the towers in A arise from all cohomology classes x-8i-11x* *-8j-12 with i + j = -p, while in dimension 8p + 2, they arise from x8i-11x8j+32~ x8i+3* *1x8j-12. The finite towers in B arise from x4i-11x8j-12with i 0, and those from C from x8i-11x4j-12with j 0. The homomorphism Ext0A2(P P, Z2) ! Ext0A2(P1-1 P, Z2) Ext0A2(P P1-1, Z2), NONIMMERSIONS IMPLIED BY TMF, REVISITED 13 which is equivalent to the d1-differential mentioned above, sends classes to th* *ose with the same name. In dimension -10, this is surjective, with kernel spanned by c* *lasses with both components < -1. In dimension -8i - 6 and -8i - 10, there will be i s* *uch classes. We illustrate by listing the classes in the first few gradings: -14 : x-91x-52~ x-51x-92 -18 : x-91x-92 -22 : x-171x-52~ x-131x-92, x-91x-132~ x-51x-172 -26 : x-171x-92, x-91x-172. These kernel classes yield infinite towers emanating from filtration 0. For each p < 0, the towers arising from x4j-11x8p-12, j 0, in A combine wit* *h those in the p-summand of M B ExtA1( 8p-1P-11, Z2) p2Z as in Diagram 3.17 to yield one of the copies of ExtA1(Z2, Z2) arising from fil* *tration 1. An identical picture results when the factors are reversed. || Diagram 3.17. Part of ExtA2(P-2-1 P-2-1, Z2) | | | | |6 |6 |6 |6 |6 |6 r|| ||r ||r r r| |r |r r | | | r| |r |r r r| |r |rr | | | | | | r| |r r |r r| |r |r | | @|| | | r| r |r r@|r| =) r| |r | @|| @|| | r| r r@|r| r@|r| r| r | @|| @|| | r|r r@|r| r@|r| r| r | @|| @|| | r r| r@|r| r@|r| r| @ | @ | @ | ____________________________r|||rr@@@ ____________________________ Putting things together, we obtain that in dimensions less than -8, ExtA2(P-2* *-1 P-2-1, Z2) consists of a chart described in Proposition 3.15 and partially illu* *strated in Diagram 3.18 together with the classes in Diagram 3.11 which are not part of* * the infinite sums of towers in dimension 2 mod 4. 14 DONALD M. DAVIS AND MARK MAHOWALD Diagram|3.18.|Illustration|of|Proposition|3.15||||||||||||||||||| |||| |||||||| |||| ||||||| ||| |||||| |6|6|6|6|6|6 |6|6|6|6|6|6 |6|6|6|6|6 |6|6|6|6|6 |6|6|6|6 |||||| |||||| ||||| ||||| |||| |||||| |||||| ||||| ||||| |||| |||||| |||| ||||| ||| |||| |||||| |||| ||||| ||| |||| |||||| |||| ||||| ||| |||| |||||| |||| ||||| ||| |||| |||||| |||| ||||| ||| |||| |||||| |||| ||||| ||| |||| |||||| |||| ||||| ||| |||| |||||| |||| ||||| ||| |||| |||||| |||| ||||| ||| |||| |||| |||| ||| ||| || |||| |||| ||| ||| || |||| |||| ||| ||| || |||| |||| ||| ||| || |||| || ||| | || |||| || ||| | || |||| || ||| | || |||| || ||| | || |||| || ||| | || |||| || ||| | || |||| || ||| | || |||| || ||| | || |||| || ||| | || |||| || ||| | || || || | | || || | | __________________________________________________________________________|||* *||| -26 -18 -10 The only possible differentials in the Adams spectral sequence of P--21^ P--2* *1^ tmf involving the classes in dimensions 8p - 2 with p < 0 are from the towers in 8p* * - 1 in Diagram 3.11, but these differentials are shown to be 0 as in [6, p.54]. Sim* *ilarly to (3.3), we have tmf*(P1^ P1) tmf-*-2(P--21^ P--21), and so we obtain a turned-around version of Diagram 3.18, of the same general s* *ort as Diagram 3.6, as a depiction of a relevant portion of tmf*(P1^ P1), with the * *labeled columns in Diagram 3.18 corresponding to cohomology gradings 24, 16, and 8. The classes Xi1Xj2described in Theorem 3.7 are detected by the S-duals of the classes from which the filtration-0 towers in dimensions 8p - 2 in Diagram 3.18* * arise, and so they can be chosen to be the corresponding elements of tmf8*(P1^ P1). Si* *mi- larly the classes L1Xi2and Xi1L2 have Adams filtration 1, and so one would anti* *cipate that they represent the duals of the generators of the two towers in dimension * *8p - 2 with p < 0 in Diagram 3.18. This seems a bit harder to prove using the Adams sp* *ectral sequence; however, the Atiyah-Hirzebruch spectral sequence shows this quite cle* *arly. The class Xi1is detected by H8i(P1; ss0(tmf)), while L is detected by H1(P1; ss* *1(tmf)). NONIMMERSIONS IMPLIED BY TMF, REVISITED 15 Under the pairing, their product is detected in H8i+1(P1; ss1(tmf)), clearly of* * Adams filtration 1. The last part of Theorem 3.7 deals with the action of c4 on the monomials Xi1* *Xj2. Since tmf is a commutative ring spectrum, tmf*(P1 ^ P1) is a graded commutative P i i i algebra over tmf*. The action c4(X1X2) must be of the form i 0flic4(L1X2+ X1L* *2) as these are the only elements in tmf8(P1^P1), and the class must be invariant * *under reversing factors. The divisibility of fl0 by 8 follows since c4 has Adams filt* *ration 4. Having just completed the proof of Theorem 3.7, we conclude this section with* * the postponed proof of Proposition 3.9. Proof of Proposition 3.9.Let C denote the A2-submodule of (P=Z2) P generated by all x11x8p-12, p 2 Z. Note that Sq2(x11x8p-12) = Sq4Sq6(x11x8p-92). Thus a* * basis of A2= Sq2acting on all x11x8p-12spans C. The 24 elements in a basis of A= Sq2 acting on x11x72yield x11x72, x11x82+ x21x72, x21x92+ x41x72, x11x112+ x21x102,* * x11x122+ x21x112, x11x132+ x21x122, x11x142+ x21x132, x21x132+ x41x112, x41x112+ x81x72, x21x142+* * x41x122, x11x162+ x41x132, x11x172+x21x162, x21x162+x41x142, x11x182+x21x172, x21x172+x81x112, x21x182+x81* *x122, x11x202+x41x172, x41x172+x81x132, x21x202+x41x182, x41x182+x81x142, x41x202+x81x162, x11x242+x81* *x172, x21x242+x81x182, and x41x242+ x81x202. These classes with second components shifted by all multi* *ples of 8 exactly comprise the basis for C described in the proposition. The procedure to establish the structure of B = ((P=Z2) P)=C is similar but m* *ore elaborate. For the 32 elements ` in a basis of A2= Sq1, we list `(x-11x-12) and* * `(x-11x32). Then we show that these, with each component allowed to vary by multiples of 8, together with C, fill out all of (P=Z2) P. It is convenient to let Q denote the quotient of (P=Z2) P by C and all elem* *ents `(x8i-11x8j-12) and `(x8i-11x8j+32). We will show Q = 0. This will complete the* * proof of Proposition 3.9, implying in particular that Sq1(x8i-11x8j-12) and Sq1(x8i-11x8* *j+32) are decomposable over A2. A separate calculation is performed for each mod 8 value of the degree. Here * *we use repeatedly that the A2-action on xidepends only on i mod 8. We illustrate w* *ith the case in which degree 0 mod 8. The other 7 congruences are handled similar* *ly, although some are a bit more complicated. A basis of A2= Sq1in degree 2 mod 8 acting on x-11x-12yields the following elements: x-11x12+ x01x02+ x11x-12, x21x62+ x61x22, x-11x92+ x31x52+ x41x42+ x5* *1x32+ x91x-12, 16 DONALD M. DAVIS AND MARK MAHOWALD and x41x122+ x121x42. A basis of A2= Sq1in degree 6 mod 8 acting on x-11x32yi* *elds the following elements: x21x62+ x31x52+ x41x42+ x51x32, x-11x92+ x21x62+ x51x3* *2, x41x122+ x61x102+ x101x62+ x121x42, and x81x162+ x161x82. Because we allow both componen* *ts to vary by multiples of 8, we will list just the first component of the ordered pairs. * *These are considered as relations in Q. Thus the relation R1 below really means that * *all x8i-11x8j+12+ x8i1x8j2+ x8i+11x8j-12become 0 in Q. R1 : X-1 + X0+ X1, R2 : X2+ X6, R3 : X-1 + X3+ X4+ X5+ X9, R4 : X4+ X12, R5 : X2+ X3+ X4+ X5, R6 : X-1 + X2+ X5, R7 : X4+ X6+ X10+ X12, R8 : X8+ X16. We will use these relations to show that all classes (in degree 0 mod 8) ar* *e 0 in Q. First, R8 implies that all classes X8iare congruent to one another. Since X0* * is 0 in the quotient due to P=Z2, we conclude that all classes X8iare 0 in Q. Nex* *t, R4 implies that all X8i+4are congruent to one another. Since X4+ X8 2 C, and we have just shown that X8 0 in Q, we deduce that all X8i+4are 0 in Q. Now we use R2 + R7 to see that all X8i+2+ X8i+4are congruent to one another, then that X2+ X4 2 C to deduce all X8i+2+ X8i+4 0, and finally the result of the previous sentence to conclude all X8i+2 0. Then R2 implies all X8i+6 0. Now R1+R3+R5, together with relations previously obtained, implies all X8i+1are congruent to * *one another, and since X1 2 C, we conclude all X8i+1 0. Finally R1 implies X8i-1 * *0, R6 implies X8i+5 0, and then R3 implies X8i+3 0. || 4. Careful treatment of axial class In this section, we fill the gap in the proof in [6] of its Theorem 1.1 by ca* *reful consid- eration of the possible "other terms" in the axial class discussed in the Intro* *duction. We show that, at least as far as the monomials cXi1Xj2in its powers are concern* *ed, NONIMMERSIONS IMPLIED BY TMF, REVISITED 17 the axial class equals u(X1+ X2), where u is a unit in tmf0(RP 1x RP 1). Thus t* *he `th power of the axial class is nonzero in tmf8`(RP nx RP m) if and only if (X1* *+ X2)` is nonzero there, and the latter is the condition which yielded the nonimmersio* *ns of [6, 1.1]. Thus we have a complete proof of [6, 1.1]. f m+k If P nx P m-! P is an axial map, then there is a commutative diagram f m+k P nx P m ---! P ?? ? ?y ??y g 1 P 1x P 1 ---! P , where g is the standard multiplication of P 1, since P 1 = K(Z2, 1). Since X 2 tmf8(P m+k) has been chosen to extend over P 1, we obtain that f*(X) is the res* *tric- tion of g*(X). By Theorem 3.7 and the symmetry of g, we must have X g*(X) = X1+ X2+ ~ici4(L1Xi+12+ Xi+11L2), (4.1) i 0 for some integers ~i. This is what we call the "axial class." Then g*(X`) equal* *s the `th power of (4.1). Using the formulas for L2i, LiXi, and c4(X1X2) in 3.1 and 3* *.7 and the binomial theorem, this `th power can be written in terms of the basis descr* *ibed ini3.7. If some ~i's are nonzero, the coefficients of Xi1X`-i2in g*(X`) will no* *t equal `j i , as was claimed in [6]. We will study this possible deviation carefully. One simplification is to treat L1 and L2 as being just 2. Note that Li acts l* *ike 2 when multiplying by Xi, and if, for example, L1 is present without X1, then the* * terms ci4L1Xj2cannot cancel our Xk1X`2-classes because both are separate parts of the* * basis. You have to carry the terms along, because they might get multiplied by an X1, * *and then it is as if L1 = 2. We will incorporate this important simplification thro* *ughout the remainder of this section. For example, one easily checks that, using L21= 2L1 and L1X1 = 2X1, we obtain (X1+ X2+ L1X2)4 = (X1+ 3X2)4- 80X42+ 40L1X42. The exponent of 2 in each monomial of (X1 + 3X2)4 - 80X42is the same as that in (X1+ X2)4, and L1X42is a separate basis element. With this simplification, the axial class in (4.1) becomes X X1+ X2+ 2 ~ici4(Xi+11+ Xi+12) (4.2) i>0 18 DONALD M. DAVIS AND MARK MAHOWALD for some integers ~i. There was another term 2~0(X1+X2), but it can be incorpor* *ated into the leading (X1+ X2). The odd multiple that it can create is not important. From Theorem 3.7, we have X c4(X1X2) = 16(X1+ X2) + 2 flkck4(Xk+11+ Xk+12), k>0 (4.3) for some integers flk. The 16 comes from fl0 = 8 and Li= 2. Actually we don't r* *eally know that fl0 = 8, even just up to multiplication by a unit, but it is divisibl* *e by 8 and the possibility of equality must be allowed for. This gives X j j+k+1 c4(Xi+11Xj+12) = 16(Xi+11Xj2+ Xi1Xj+12) + 2 flkck4(Xi+k+11X2 + Xi1X2 ). k>0 (4.4) Here we use that in a graded tmf*-algebra tmf*(X) with even-degree elements, c(* *xy) = cx . y, for c 2 tmf*and x, y 2 tmf*(X). There is an iterative nature to the action of c4 in (4.4), but the leading co* *efficient 16 enables us to keep track of 2-exponents of leading terms in the iteration. * *(As observed above, the leading coefficient might be an even multiple of 16, which * *would make the terms even more highly 2-divisible. We assume the worst, that it equa* *ls 16.) We obtain the following key result about the action of c4 on monomials in * *X1 and X2. Theorem 4.5. There are 2-adic integers Aisuch that _ ! X 1 iX2 ji 1i X1ji c4 = 24+iAi ___ ___ + ___ ___ . i 0 X1 X1 X2 X2 Remark 4.6. This formula will be evaluated on (i.e. multiplied by) monomials Xk* *1X`2. One might worry that the negative powers of X1 or X2 in 4.5 will cause nonsensi* *cal negative powers in c4Xk1X`2. This will, in fact, not occur because the monomial* *s on which we act always have total degree greater than the dimension of either fact* *or. Thus if, after multiplication by c4, a term with negative exponent of Xiappears* *, then the accompanying Xj3-i-term will be 0 for dimensional reasons. Proof of Theorem 4.5.The defining equation (4.3) may be written as, with ` = p _____ q______ c4 X1X2 and z = X1=X2 , X ` = 16(z + z-1) + 2fli`i(zi+1+ z-(i+1)). (4.7) i>0 NONIMMERSIONS IMPLIED BY TMF, REVISITED 19 Let pi= zi+ z-i. We will show that X ` = 24+iAip2i+1 (4.8) i 0 for certain 2-adic integers Ai, which interprets back to the claim of 4.5. Note that pipj = pi+j+ p|i-j|, and hence pe11. .p.ekk= p iei+ L, P P where L is a sum of integer multiples of pj with j < iei and j iei mod 2. We will ignore for awhile the coefficients fli which occur in (4.7). This is al* *lowable if we agree that when collecting terms, we only make crude estimates about their 2-divisibility. We have ` = 16p1+ 2`p2+ 2`2p3+ 2`3p4+ . . . = 16p1+ 2p2(16p1+ 2p2(16p1+ . .).+ 2p3(16p1+ . .).2+ . .). +2p3(16p1+ 2p2(16p1+ . .).+ . .).2+ . ... Note that the only terms that actually get evaluated must end with a 16p1 facto* *r. Now let T1 = 16p1 and, for i 2, let Ti= 2`i-1pi. Each term in the expansion* * of ` involves a sequence of choices. First choose Ti for some i 1, and then if i* * > 1 choose (i-1) factors Tj, one from each factor of `i-1. For each of these Tj wit* *h j > 1, choose j - 1 additional factors, and continue this procedure. This builds a tre* *e, and we don't get an explicit product term until every branch ends with T1. Each sel* *ected factor Tj with j > 1 contributes a factor 2pj. There will also be binomial coef* *ficients and the omitted fli's occurring as additional factors. For example, Diagram 4.9 illustrates the choices leading to one term in the e* *xpan- sion of `. This yields the term 2p2. 2p4. 16p1. 2p2. 16p1. 2p3. 16p1. 2p2. 16p1* *, which equals 221(p17+ L), where L is a sum of piwith i < 17 and i odd. By induction, * *one sees in general that the sum of the subscripts emanating from any node, includi* *ng the subscript of the node itself, is odd. 20 DONALD M. DAVIS AND MARK MAHOWALD Diagram 4.9. A possible choice of terms T1 T2___T4___T2___T1 @@ T1 T3HH T2___T1 The important terms are those in which T2 is chosen k times (k 0) and then * *T1 is chosen. These give (2p2)kp1 with no binomial coefficient. This term is 2k+4(p2k* *+1+L). Note that a term 2k+4p2i+1with i < k obtained from L will be more 2-divisible t* *han the 2i+4p2i+1term that was previously obtained. Thus it may be incorporated into the coefficient of that term. All other terms will be more highly 2-divisible than these. For example, the * *first would arise from choosing T3 then two copies of T1. This would give 2p3. 24p1. * *24p1 = 29p5+L, and the 29p5 can be combined with the 26p5 obtained from choosing T2 th* *en T2 then T1. Incorporating fli's may make terms even more divisible, but the cla* *im of (4.8) is only that p2i+1occurs with coefficient divisible by 24+i. || Now we incorporate 4.5 into (4.2) to obtain the following key result, which w* *e prove at the end of the section. Theorem 4.10. The monomials ciXi1Xn-i2in the nth power of the axial class in tmf8n(RP 1x RP 1) are equal to those in the nth power of _ ! X iiX1 ji iX2jij (X1+ X2) u + 24+iffi ___ + ___ , (4.11) i 1 X2 X1 where u is an odd 2-adic integer and ffiare 2-adic integers. The factor which accompanies (X1+ X2) in (4.11) is a unit in tmf*(RP 1x RP 1); we referred to it earlier as u. Indeed, its inverse is a series of the same for* *m, obtained by solving a sequence of equations. This justifies the claim in the first parag* *raph of this section regarding retrieval of the nonimmersions of [6, 1.1]. We must also observe that restriction to tmf8`(RP nx RP m) of the non-Xi1X`-i2 parts of the basis of tmf8`(RP 1x RP 1) cannot cancel the Xi1X`-i2terms essenti* *al for the nonimmersion. This is proved by noting that these elements such as L1X* *`2 NONIMMERSIONS IMPLIED BY TMF, REVISITED 21 and ci `+i 8` n m 4L1X2 will restrict to a class of the same name in tmf (RP x RP ), and will be 0 there for dimensional reasons, since 8` > n. Proof of Theorem 4.10.Let g*(X) denote the axial class as in (4.1). From (4.2) * *and 4.5, the difference g*(X) - (X1+ X2) equals _ _ !!i X X 1 iX2 jj 1i X1jj 2 ~i(Xi+11+ Xi+12)24i 2jAj ___ ___ + ___ ___ . i 1 j 0 X1 X1 X2 X2 q______ We let z = X1=X2 and pj = zj+ z-j as in the proof of 4.5. The summand with i = 2t becomes P 2t-ss _ ! i X1 X2 4i X j 2~i(X1+ X2)__s________tt2 2 Ajp2j+1 X1X2 j 0 X = 2~i(X1+ X2)(p2t+ L)24i ck2k(p2k+i+ L). k Here k is a sum of j-values taken from the various factors in the ith power. Al* *so, in pj+ L, L denotes a combination of pt's with t < j. Noting (p2t+ L)(p2k+i+ L) = p2k+2i+ L, this becomes X 2(X1+ X2)24i c0k2k(p2k+2i+ L). (4.12) The argument when i = 2t + 1 is similar but slightly more complicated because (Xi+11+ Xi+12) is not divisible by (X1+ X2). We obtain Xi+11+ Xi+12 4iiX j ji 2~i___________p_2 2 Ajp2j+1 . ( X1X2 )2t+1 j 0 For one of the factors of the ith power, say the first, we treat p2j+1as X1+X2_* *p_X1X2(p2j+ L). The expression then becomes X 2(X1+ X2)pi+124i ck2k(p2k+i-1+ L), where k is obtained as in the previous case. We again obtain (4.12). P Thus when g*(X) - (X1 + X2) is written as (X1 + X2) fijp2j, the coefficient* * fij satisfies (fij) (j - 1) + 4 + 1. Here the (j - 1) + 4 comes from the case i * *= 1, k = j - 1 in (4.12), and the extra +1 is the factor 2 which has been present al* *l along. This yields the claim of (4.11). || 22 DONALD M. DAVIS AND MARK MAHOWALD 5.tmf-cohomology of CP 1x CP 1 In [2], [4], and [8], it was noted, first by Astey, that the axial class usin* *g BP (or BP <2>) was u(X2 - X1), where u is a unit in BP *(P 1 ^ P 1). In this section, * *we review that argument and consider the possibility that it might be true when BP is replaced by tmf, which would render the considerations of the previous secti* *on unnecessary. To do this, we calculate tmf*(CP 1) and tmf*(CP 1xCP 1) in positive dimensions. (See Theorems 5.15 and 5.19.) Although our conclusion will be that Astey's BP -argument cannot be adapted to tmf, nevertheless these calculations * *may be of independent interest. We begin by reviewing Astey's argument. There is a commutative diagram, in which RP = RP 1 and CP = CP 1 RP -dR--!RP x RP -mR--!RP ?? ?? ?? h?y hxh?y h?y CP -dC--!CP x CP CP ?? ?? 1x(-1)?y 1?y CP x CP -mC--!CP The generator XR 2 BP 2(RP ) satisfies XR = h*(X). We also have that mC O (1 x (-1))OdC is null-homotopic. The key fact, which will fail for tmf, is BP *(CP x* *CP ) BP *[X1, X2]. The axial class is m*R(XR). It equals (h x h)*(1 x (-1))*m*C(X). But (1 x (-1))*m*C(X) 2 ker(d*C). By the above "key fact," d*Cis the projection BP *[X1, X2] ! BP *[X] in which e* *ach Xi7! X. The kernel of this projection is the ideal (X2- X1). To see this, just * *note P i n-i P that in grading 2n a kernel element must be ciX1X2 with ci= 0, and hence is X X X j n-i-1-j ci(Xi1Xn-i2- Xn1) = ciXi1(X2- X1) X1X2 . i. Lemma 5.2. There is an additive isomorphism Ext*,*A2(M10, Z2) bo[v2], where v2 2 Ext1,7(-). Thus the chart for Ext*,*A2(M10, Z2) consists of a copy of bo shifted by (t -* * s, s) = (6i, i) units for each i 0. Proof.There is a short exact sequence of A2-modules 0 ! 7M10! A2==A1 ! M10! 0. This yields a spectral sequence which builds Ext*,*A2(M10, Z2) from M *-i,*-7i ExtA2 (A2==A1, Z2). i 0 Since Ext*,*A2(A2==A1, Z2) bo, one easily checks that there are no possible d* *ifferen- tials in this spectral sequence. || Let Cmn= H*(CPnm; Z2). Theorem 5.3. There is an additive isomorphism M Ext*,*A2(C1-1, Z2) 8p-2bo[v2]. p2Z 24 DONALD M. DAVIS AND MARK MAHOWALD Of course applied to a module or an Ext group just means to increase the t-gr* *ading by 1. Proof.There is a filtration of C1-1 with Fp=Fp-1 8p-2M10for p 2 Z. We have Sq2'8p-2= Sq4Sq2Sq4'8p-10. The same argument used in the last paragraph of the proof of Corollary 3.10 works to initiate an inductive proof of the Ext-isomorp* *hism claimed in the theorem. || Corollary 5.4. In gradings (t - s) less than -1, M Ext*,*A2(C-2-1, Z2) 8p-2bo[v2]. p<0 Proof.There is an exact sequence q* s,t 1 ! Exts-1,tA2(C1-1, Z2) ! Exts,tA2(C-2-1, Z2) ! Exts,tA2(C1-1, Z2) -! ExtA2(C-1,* * Z2) ! . The result is immediate from this and 5.3, since q* sends the initial tower in * *F0=F-1 isomorphically to the initial tower in ExtA2(C1-1, Z2). || The A-modules C11and 2C-2-1are dual. Thus, by [9, Prop 4], Exts,tA2(Z2, C11) Exts,tA2( 2C-2-1, Z2). There is a ring structure on Ext*,*A2(Z2, C11). We deduce the following resul* *t, which is pictured in Diagram 5.12. Corollary 5.5. In (t - s) gradings 0, there is a ring isomorphism Ext*,*A2(Z2, C11) bo[v2][X], where X 2 Ext0,-8. Proof.We apply the duality isomorphism to 5.4. The multiplicative structure is obtained from the observation that the powers of the class in Ext0,-8equal the * *class in Ext0,-8ifor each i > 0. || The Ext groups computed here are the E2-term of the ASS converging to tmf-*(C* *P 1). We will consider the differentials in this spectral sequence after performing t* *he Ext calculation relevant for tmf*(CP 1x CP 1). NONIMMERSIONS IMPLIED BY TMF, REVISITED 25 Now we consider C-2-1 C-2-1. Now x1 and x2 denote elements of H2(CP ; Z2). Let E2 denote the exterior subalgebra generated by the Milnor primitives of grading* * 1, 3, and 7. Note that A2==E2 has a basis with elements of grading 0, 2, 4, 6, 6, 8, * *10, and 12. Finally we note that for any j -2 mod 8 with j -10, there is a nontrivi* *al ae j A2-morphism C-2-1-! Z2. Lemma 5.6. Let ae -2 -10 K = ker(C-2-1 C-2-1-!C-1 Z2). Let S denote the set of all classes x8i-21x8j-22with i -1 and j -2, togethe* *r with the classes x8i-21x8j+22with i -1 and j -1. Then K is the direct sum of a f* *ree A2==E2-module on S with a single relation Sq4Sq2Sq4(x-101x-62) = 0. Proof.Since the generators of E2 have odd grading, A2==E2 acts on any element of these evenly-graded modules. The action of A2==E2 on x-21x-22yields the additio* *nal elements x-21x02+x01x-22, x-21x22+x01x02+x21x-22, x-21x42+x41x-22, x01x22+x21x0* *2, x01x42+x41x02, x-21x82+x21x42+x41x22+x81x-22, and x01x82+x81x02. The action of A2==E2 on x-21x* *22yields the additional elements x01x22+x-21x42, x01x42+x21x22, x21x42+x41x22, x21x42+x-21x8* *2, x01x82+x41x42, x21x82+ x81x22, and x41x82+ x81x42. Each exponent can be decreased by any multi* *ple of 8. One can easily check that in each grading all classes in C-2-1 C-2-1are obta* *ined exactly once from the described elements in K together with C-2-1 -10Z2. There are four cases, for the four even mod 8 values. We illustrate with the case of * *grading 4 mod 8. We will just consider the specific value -28, but it will be clear th* *at it generalizes to all gradings 4 mod 8. Letting Xidenote xi1x-28-i2, we have: (1)From generators in -28, we obtain just X-10 in K. The class X-18is in C-2-1 -10Z2. (2)From generators in -32, we obtain X-8 + X-6, X-16+ X-14, and X-24+ X-22. (3)From generators in -36, we obtain X-8+X-4 and X-16+X-12. (4)From generators in -40, we obtain X-4, X-12+ X-8, X-20+ X-16, and X-24. Note in (4) that X0 and X-28do not appear because each component must be -4 and the components sum to -28. 26 DONALD M. DAVIS AND MARK MAHOWALD One easily checks that the 11 classes listed above, including X-18, form a ba* *sis for the space spanned by X-4, . .,.X-24, in an orderly fashion that clearly general* *izes to any grading 4 mod 8. A similar argument works in the other three congruences. There are some minor variations in the top few dimensions. || Now we dualize. There is a pairing ExtA2(Z2, C11) ExtA2(Z2, C11) ! ExtA2(Z2, C11 C11). Let Xidenote the class in grading -8 coming from the ith factor. Then we obtain Theorem 5.7. The algebra Ext0,*A2(Z2, C11 C11) in gradings -8 is isomorphic to Z2[X1, X2] with y2-12= X21X2 + X1X22. The monomials of the form Xi1Xj2y-12are acted on freely by Z2[v0, v1, v2]. Let Sn denote the Z2-vector sp* *ace with basis the monomials Xi1Xn-i2, and define a homomorphism ffl : Sn ! Z2 by sending each monomial to 1. Then Z2[v0, v1, v2] acts freely on ker(ffl), while bo[v2] a* *cts freely on Sn= ker(ffl). Thus in dimensions t - s -8 Ext*,*A2(Z2, C11 C11) has, for * *each i > 0, i copies of -8i-4Z2[v0, v1, v2] and i copies of -8i-16Z2[v0, v1, v2], * *and also one copy of -8i-8bo[v2]. Here Z2[X1, X2] means a free Z2[X1, X2]-module on basis {X1X2, y-* *12} Proof.The structure as graded abelian group is straightforward from Lemma 5.6, Corollary 5.5, and the duality isomorphism Ext*,*A2(Z2, C11 C11) Ext*,*-4A2(C-2-1 C-2-1, Z2). We use that ExtA2(A2==E2, Z2) Z2[v0, v1, v2]. The reason that we only assert * *the structure in dimension -8 is due to the -10in the cokernel part of Lemma 5.6* *, and that Theorem 5.5 was only valid in dimension 0. In the range under considerat* *ion, the relation on the top class in Lemma 5.6 does not affect Ext. The ring structure in filtration 0 comes from Hom A2(Z2, C11 C11) being isomo* *rphic to elements of C11 C11annihilated by Sq2and Sq4, which has as basis all elemen* *ts x4i1 x4j2and (x4i1 x4j2)(x41 x22+ x21+ x42). Now we show that Ext1,-8n+2A2(Z2, C11 C11) = Z2, and h1 times each mono- mial in Ext0,-8nA2(Z2, C11 C11) equals the nonzero element here. An element in NONIMMERSIONS IMPLIED BY TMF, REVISITED 27 Ext1,-8n+2 1 1 A2 (Z2, C1 C1 ) = Z2 is an equivalence class of morphisms 2A2 4A2 h-!C11 C11 which increase grading by 8n - 2, and yield a trivial composite when preceded by _ ! Sq 2 Sq6 0 Sq4 4A2 8A2 ---------! 2A2 4A2. Morphisms h which can be factored as Sq2,Sq4 k 1 1 2A2 4A2 ----! A2 -!C1 C1 (5.8) are equivalent to 0 in Ext. We illustrate with the case n = 3. There are A2-morphisms increasing grading * *by 22 sending either 2A2 or 4A2 to any one of the following classes: x11x122, x21x102, x41x92, x41x82, x51x82, x61x62, x81x52, x81x42, x91x42, x101x* *22, x121x12. (5.9) The classes are listed in this order because any two adjacent monomials are equ* *ivalent using as k in (5.8) the morphism sending the generator to the indicated classes* * in succession: x11x102, x21x92, x41x72, x31x82, x51x62, x61x52, x81x32, x71x42, x91x22, x10* *1x12. For example, (Sq2, Sq4)(x11x102) = (x21x102, x11x122). Thus all classes in (5.9* *) are equiva- lent to one another. That h1times any monomial Xi1Xn-i2equals this nonzero element of Ext1,8n+2A2(* *Z2, C11 C11) follows from usual Yoneda product consideration. If 0 Z2 C0 C1 is the beginning of a minimal A2-resolution, with C1 = 1A2 2A2 4A2, then h1Xi1Xn-i2is represented by the composite C1 ! C0 ! C11 C11 sending '2 7! ' 7! Xi1Xn-i2, and this is equivalent to the element described in the pre* *vious paragraph. || Here is a schematic way of picturing Theorem 5.7. We first list the generator* *s in grading greater than -32. Then for each of the two types of generators, we list* * the structure arising from them in the first 10 dimensions. The bo[v2]-structure in* * the left half of Diagram 5.11 arises from one tower in dimensions -24 and -16, whil* *e the 28 DONALD M. DAVIS AND MARK MAHOWALD Z2[v0, v1, v2]-structure in the right half of diagram 5.11 arises from the othe* *r towers in Diagram 5.10. Diagram 5.10. Generators of ExtA2(Z2, C11 C11) ||| || || | | |6|6|6 |6|6 |6|6 |6 |6 ||| || || | | ||| || || | | ||| || || | | ||| || || | | ||| || || | | ||| || || | | ||| || || | | ||| || || | | ||| || || | | ||| || || | | ||| || || | | ||| || || | | _____________________________________________||||||||| -28 -24 -20 -16 -12 Diagram 5.11. Structure on two types of generators | | | | | | || | | |6 |6 |6 |6 |6 |6 |6 |6 |6|6 |6|6 |6|6 | | | | | | | | || || || | | | | | | | | || || || | | | | | | | | || || | | | | | | | | | || || | | | | | | | || | | | | | | | | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | ______________________________| ______________________________| 0 10 0 10 Now we consider the differentials in the ASS converging to tmf*(CP 1) and the* *n for tmf*(CP 1^ CP 1). The gradings are negated when considered as tmf-cohomology groups. Corollary 5.5 gives the E2-term converging to [ *CP11, tmf] tmf-*(CP1* *1). We will maintain the homotopy gradings until just before the end. In diagram 5.* *12, we depict a portion of the E2-term of this ASS in gradings -16 to 1. There are * *also classes in higher filtration arising from powers of v41and v2 acting on generat* *ors in lower grading. The elements indicated by o's are involved in differentials, as * *explained later. NONIMMERSIONS IMPLIED BY TMF, REVISITED 29 Diagram 5.12. A portion of E2 for [ *CP 1, tmf] || || |6|6|6 | || ||| ||| |6|6 ||| | || ||6|6||||6| |||| || |6 ||| || ||| | |6|6 | ||| || || |6 || | ||| || || | | || | || || ||r |6 | || | || || ||pp | | | || | || || r ||pp |6 | | || | || || ||pp | | | || | || || || pp | | | || | || || || r | | | || | || | | | | | || | || | |r | | | | || |r | | | | || | | | | r| || r | | | | | | | | | | | | | r | | r | | | | | | | r |r | | | | _____________________________________________________________||r -16 -8 0 We will prove the following key result about differentials in this ASS. Theorem 5.13. The nonzero differentials in the ASS converging to [ *CP 1, tmf], * < 1, are given by d2(hffl1v4i1vj2X-2k+1) = hffl+11v4i1vj+12X-2k for ffl = 0, 1, i, j 0, k 1. Here h1, v41, and v2 have the usual Exts,tgradings (s, t) = (1, 2), (4, 12), * *and (1, 7), respectively. Diagram 5.12 pictures the situation for k = 1 and small values of i and j. T* *he elements indicated by o's are involved in the differentials. The resulting pic* *ture is nicer if the filtrations of all classes built on X-2k+1 are increased by 1. Th* *ere is a nontrivial extension (multiplication by 2) in dimension -6 due to the precedi* *ng differential. This is equivalent to the way that bu* is formed from bo* and 2* *bo*. We obtain Diagram 5.14 from Diagram 5.12 after the differentials, extensions, a* *nd filtration shift are taken into account. 30 DONALD M. DAVIS AND MARK MAHOWALD Diagram 5.14. Diagram 5.12 after differentials and filtration shift ||| | || |6|6|6 || |6|6 ||| | |6|6|6 ||| |||| | ||6 ||| ||| |||| |6|6 | ||| || || | || | ||| || || | |6 || | ||| || || |6 | || | || || || | | || ||| | |||| || ||| |6 | | || | || || | | | | || | || || | | | | || | || || | | | | || | || | | | | | || | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | _____________________________________________________________| -16 -8 0 The regular sequence of towers in the chart beginning in filtration 1 in dimens* *ion -10 is interpreted as vi1v2, i 0. After negating dimensions to switch to cohomology indexing, we obtain the fol* *low- ing result, which is immediate from 5.13 after the extensions such as just seen* * are taken into account. Theorem 5.15. In positive gradings, there is an isomorphism of graded abelian groups tmf*(CP11) Z(2)[Z16](bo* v2Z(2)[v1, v2]). Here Z162 tmf16(CP11), and |v1| = -2 and |v2| = -6. Recall that bo* = bo-* with bo* as suggested in 5.1. Much of the ring struct* *ure of tmf*(CP11) is described in 5.15, since bo* and v2Z(2)[v1, v2] are rings, and* * it is quite clear how to multiply an element in bo* by one in v2Z(2)[v1, v2]. Because* * of the filtration shift that led to the identification of some of the classes in v2Z(2* *)[v1, v2], we hesitate to make any complete claims about the ring structure. A complete computation of tmf*(CP 1) was made in [5]. See there especially Theorem 7.1 and Diagram 7.1. At first glance, the two descriptions appear quite different, but they seem to be compatible. NONIMMERSIONS IMPLIED BY TMF, REVISITED 31 Proof of Theorem 5.15.We first prove that there is a nontrivial class in [ -16C* *P, tmf] detected in filtration 0. This is obtained using the virtual bundle 8(H -1)-(H3* *-H), where H denotes the complex Hopf bundle. Considered as a real bundle `, this bu* *ndle satisfies w2(`) and p1(`) = 0. Here we use from [18] that p1 generates the inf* *inite cyclic summand in H4(BSO; Z) and satisfies r*(p1) = c21- 2c2 under BU -r!BSO, ae 4 and ae*(p1) = 2e1 under BSpin -! BSO, where H (BSpin; Z) is an infinite cyclic group generated by e1. The total Chern class of 9H - H3 is (1 + x)9(1 + 3x)-1 = 1 + 6x + 18x2+ . .,. and hence r*(p1(`)) = (c1(9H - H3))2- 2c2(9H - H3) = (6x)2- 2 . 18x2 = 0. Thus e1(`) = 0, hence CP 1 `-!BSpin ! K(Z, 4) is trivial, and so ` lifts to a m* *ap CP 1 ! BO[8]. Hence its Thom spectrum induces a degree-1 map T (`) ! MO[8]. Since _3(H) = H3 - H, by [19] ` is J(2)-equivalent to 8(H - 1), and hence its T* *hom spectrum is T (8(H - 1)) = -16CP81. Using the Ando-Hopkins-Rezk orientation ([1]) MO[8] ! tmf, we obtain our desired class as the composite T(`) -16CP11 col-! -16CP81 --! MO[8] ! tmf. (5.16) We will deduce our differentials from the d3-differential E4,213! E7,233in th* *e ASS converging to ss*(tmf). This can be seen in [13, p.537] or [11, Thm 2.2]. See R* *emark 5.17 for additional explanation. It is not difficult to show that, with M10as i* *n 5.2, the morphism Exts,tA2(Z2, Z2) ! Exts,tA2(M10, Z2) induced by the nontrivial A2-map M10! Z2 sends the Z2 in Ext7,23A2(Z2, Z2) whic* *h is not part of the infinite tower to h21v41v2. We prefer to think about the ASS for tmf*( 2CP--21), which, as we have noted, is isomorphic to that of [ *CP11, tmf]. The E2-term was described in 5.4. Let S-16 ! 2CP--21^ tmf correspond to the map in (5.16). Since E2(CP--21^ tmf) in negative dimensions is built from copies of ExtA2(M10, Z2), we deduce from t* *he previous paragraph that h21v41v2g-16in the ASS for tmf*( 2CP--21) must be hit b* *y a d2- or d3-differential, since it is the image of a class hit by a d3. The only * *possibility is that it be d2 from h1v41g-8, as indicated by the dotted line in Diagram 5.12. N* *aturality 32 DONALD M. DAVIS AND MARK MAHOWALD of differentials with respect to h1 and v41implies the differentials of 5.13 fo* *r ffl = 0, 1, all i, j = 0, and k = 1. Using the diagonal map of CP11 and the multiplication of tmf, powers of (5.16) give similar nontrivial elements in [ -16kCP11, tmf] f* *or all k 1, and by the argument just presented, we establish the differentials of 5.* *13 for all k (with j = 0 still). The only possible differentials on v2g-16 would be some dr with r > 2 hitting* * an element which is acted on nontrivially by h1. However h1v2g-16 has become 0 in E3 since it was hit by a d2-differential. Thus a nonzero differential on v2g-16* * would contradict naturality of differentials with respect to h1-action. Hence there i* *s a map S-10 ! 2CP--21^ tmfhitting v2g-16, and the argument of the previous paragraph implies that d2(h1v41v2g-8) = h21v41v22g-16 and then other related differential* *s. This now establishes the differentials of 5.13 when j = 1, and sets in motion an ind* *uctive argument to establish these differentials for all j 1. No further differentials in the spectral sequence are possible, by dimensiona* *l and h1-naturality considerations. || Remark 5.17. The proof of the key d3-differential in the ASS of tmf from the 17- stem to the 16-stem, which was cited above, has not had a thorough proof in the literature. Giambalvo's original argument was incorrect and his correction mer* *ely refers to "a homotopy argument." The current authors cited Giambalvo's result in [11] without additional argument. We provide some more detail here regarding th* *is differential. The relevant portion of the ASS of tmf appears in Diagram 5.18. In [13] and [* *11], this was pictured as the ASS of MO[8], but through dimension 18, Ext*,*A(H*(MO[8]), Z2) Ext*,*A2(Z2 16Z2, Z2). One way of obtaining the differentials from 15 to 14, as in [13], is to note th* *at the [8]- cobordism group of 14-dimensional manifolds is Z2, and so the top two elements * *must be killed by differentials. It is not difficult to compute in Ext the Massey p* *roduct formula B = , where A and B are as in Diagram 5.18. This can be seen as v41times a similar formula between classes in dimensions 6 and 8. Since A is* * 0 in homotopy, the associated Toda bracket formula says that B must be divisible by * *j. NONIMMERSIONS IMPLIED BY TMF, REVISITED 33 But only 0 can be divisible by j in dimension 16 here. Thus B must be killed by* * a differential, and the depicted way is the only way this can happen. Diagram 5.18. Portion|of ASS of tmf | ______________________|| 8|| r || || BB || || B r B || | BB B | |A|r| B r|BB || | |A B |A B | 5|| r||Ar| B r||ArB || | |AA | B | A | | | |AA | B| A | | | r| Ar| Br| Ar| | | A | | | A | | 3______Ar|_____________|||| 14 16 18 The differentials in the ASS converging to tmf*(CP--21^ CP--21) are implied b* *y the same considerations that worked for CP--21. The Z2[v0, v1, v2]-parts in Theorem* * 5.7 cannot support differentials by dimensionality and h1-naturality. For the bo-li* *ke part, we prefer thinking about it as [ *+4CP11 ^ CP11, tmf] tmf-*-4(CP11 ^ CP11), where the product structure is more apparent. Let Zn denote the nonzero element of Ext0,-8nA2(Z2, C11 C11)= ker(h1). By The* *orem 5.7, Zn can be represented by Xi1Xn-i2for any 1 i < n. If n is even and n 4, choosing i even, Zn is an infinite cycle because it is an external product of i* *nfinite cycles. Hence by the proof of Theorem 5.13, d2(hffl1v4i1vj2Z2k-1) = hffl+11v4i1vj+12Z2k for ffl = 0, 1, i, j 0, and k 2. Finally, X1X2 is an infinite cycle since there is nothing that it can hit. A* *lso, h1v2X1X2 and h21v2X1X2 are not hit by differentials since Ext0,-8A2(Z2, C11 C1* *1) = 0 by Theorem 5.7. We obtain the following. Theorem 5.19. In grading 10, there is an isomorphism of graded abelian groups M tmf*(CP11^CP11) yZ(2)[v1, v2, X1, X2] In.Z(2)[v1, v2] Z(2)[Z](bo* v2Z(2)[v* *1, v2]), n 3 34 DONALD M. DAVIS AND MARK MAHOWALD where |y| = 12, |Xi| = 8, |Z| = 16, |v1| = -2, and |v2| = -6. Here In = ker(Fn * *ffl-!Z), where Fn is a free abelian group with basis {Xi1Xn-i2: 1 i < n}, and ffl(Xi1X* *n-i2) = 1. Thus In consists of all polynomials of grading n with sum of coefficients equ* *al to 0. We could have extended the description in 5.19 down to grading 8, but the descr* *iption would have been slightly more complicated, since it would include h1v2Z and h21* *v2Z. The motivation for this section was to see if perhaps * * 1 ker(tmf*(CP 1x CP 1) d-!tmf (CP )) might be something nice like the I(X1 - X2) which was the case for BP *(-). In Theorem 5.19, we described tmf*(CP 1^ CP 1). To obtain tmf*(CP 1x CP 1), we add on two copies of tmf*(CP 1), which was described in 5.15. Denote by Z1 and Z2 the generators in tmf16(CP 1 x CP 1). Monomials Zi1Zn-i2should equal Zn of 5.19 plus perhaps elements of I2n of 5.19. The class y of 5.19 plus perhaps a s* *um of elements of higher filtration is in ker(d*) and not in the ideal generated by (* *Z1- Z2). Thus, as expected, ker(d*) does not have the nice form that it did for BP *(-),* * and so we cannot use this argument to show that the axial class in tmf*(RP 1x RP 1) is u(X1 - X2). However, we showed something like this by a completely different method in Theorem 4.10. We feel that the results obtained in Theorems 5.15 and 5.19 should be of independent interest. References [1]M. Ando, M. J. Hopkins, and C. Rezk, Multiplicative orientations of KO-theory and of the spectrum of topological modular forms, preprint, www.math.uiuc.edu/~mando/papers/koandtmf.pdf. [2]L. Astey, Geometric dimension of vector bundles over real projective spaces, Quar Jour Math Oxford 31 (1980) 139-155. [3]________, A cobordism obstruction to embedding manifolds, Ill Jour Math 31 (1987) 344-350. [4]L. Astey and D. M. Davis, Nonimmersions of real projective spaces implied by BP, Bol Soc Mat Mex 25 (1980) 15-22. [5]T. Bauer, Elliptic cohomology and projective spaces-a computation, preprint. wwwmath.uni-muenster.de/u/tbauer/cpinfty.pdf. [6]R. R. Bruner, D. M. Davis, and M. Mahowald, Nonimmersions of real projec- tive spaces implied by tmf, Contemp Math 293 (2002) 45-68. [7]D. M. Davis, Table of immersions and embeddings of real projective spaces, http://www.lehigh.edu/~dmd1/immtable. [8]________, A strong nonimmersion theorem for real projective spaces, Annals of Math 120 (1984) 517-528. NONIMMERSIONS IMPLIED BY TMF, REVISITED 35 [9]________, On the Segal Conjecture for Z2x Z2, Proc Amer Math Soc 83 (1981) 619-622. [10]D. M. Davis and M. Mahowald, Ext over the subalgebra A2 of the Steenrod algebra for stunted projective spaces, Can Math Soc Conf Proc 2 (1982) 297- 342. [11]________, A new spectrum related to 7-connected cobordism, Springer-Verlag Lecture Notes in Math 1370 (1989) 126-134. [12]D. M. Davis and V. Zelov, Some new embeddings and nonimmersions of real projective spaces, Proc Amer Math Soc 128 (2000) 3731-3740. [13]V. Giambalvo, On <8>-cobordism, Ill Jour Math 15 (1971) 533-541. Correction in Ill Jour Math 16 (1972) 704. [14]I. M. James, On the immersion problem for real projective spaces, Bull Amer Math Soc 69 (1963) 231-238. [15]N. Kitchloo and W. S. Wilson, The second real Johnson-Wilson theory and nonimmersions of RPn, preprint. [16]W. H. Lin, On conjectures of Mahowald, Segal, and Sullivan, Math Proc Camb Phil Soc 87 (1980) 449-458. [17]W. H. Lin, D. M. Davis, M. Mahowald, and J. F. Adams, Calculation of Lin's Ext groups, Math Proc Camb Phil Soc 87 (1980) 459-469. [18]J. Milnor and J. D. Stasheff, Characteristic classes, Princeton Univ Press (1974). [19]D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Annals of Math 100 (1974) 1-79. Lehigh University, Bethlehem, PA 18015, USA E-mail address: dmd1@lehigh.edu Northwestern University, Evanston, IL 60208, USA E-mail address: mark@math.northwestern.edu