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 2power grea*
*ter
than or equal to h.
oIf ff(M) = 4h  1, then P 8M+8h+2cannot be immersed in (6 )
R16M8h+10.
oIf ff(M) = 4h  2, then P 8M+8h6 R16M8h+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 selfcontained 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+316M4f 2L+38M12
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+38M8 2L+38M12
Since tmf2 (P ) = 0, we have
LM1 2LM1 2L+38M8 8M+10 2L+316M4
0 = f*(0) = f*(X2 ) = (X1+X2) 2 tmf (P xP ).
i L j L i L j L
Expanding, we obtain 2MM1+1XM+11X222M2+ 2 M1MXM1X222M1 as the only
terms which are possibly nonzero. Next we note that, with all u's representing *
*odd
integers,
i2LM1j 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 M1M= u3 2MM = 2ff(M)u4 = 23u4. Thus an immersion implies that in
L+38M8 8M+10 2L+316M4
tmf2 (P x P ), we have
L2M2 3 M 2L2M1
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)
L2M2 1
tmf1(P3) Z=8, and so 8XM+11X22 = 0. Here and throughout, Pn = Pn =
L+316M8 2L+316M4
RP 1=RP n1. Similarly tmf2 (P ) tmf7(P3) Z=16, and hence
L2M1
16XM1X22 = 0. Duality also implies
L+38M8 8M+10 2L+316M4
tmf2 (P x P ) tmf14(P3 ^ P3).
Calculations such as E2(tmf*(P3^P3)), the E2term of the Adams spectral sequen*
*ce
(ASS), were made by Bruner's minimalresolution 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*(P3 ^ P3)), * 15
r


rr r
 
r r r r
  
r r rrr
 
r r rrrr
  
r r r r r r r r rrrr
    
r rr r rr r r r r r r rrr
     
___________________________________________________rrrrrrr*
*rrrrrrr
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 easilyconstructed nontrivial*
* map. The
L2M2 M 2L2M1
monomials XM+11X22 and X1 X2 are detected in mod2 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(P3^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
L2M1
the monomials. In particular, the order of XM1X22 is 16, and hence the cla*
*ss
L2M1
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+316M6f 2L+38M10
P 8M+8x P 2 ! P ,
and hence, up to odd multiples,
L2M2 2 M 2L2M1
22 (M+1)XM+11X22 + 2 X1 X2 (2.3)
L+38M8 8M+8 2L+316M6
= 0 2 tmf2 (P ^ P ),
since ff(M) = 2. We have tmf8M+8(P 8M+8) tmf1(P1) Z=2, and
L+316M8 2L+316M6
tmf 2 (P ) tmf1(P3) 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(P1 ^ P3). A minimal
resolution calculation easier than the one in Diagram 2.2 shows that tmf6(P1^ *
*P3)
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 filtration0 cla*
*sses, we
conclude that the orders of the monomials in (2.3) are precisely 2 and 8, respe*
*ctively,
L2M1
and so the term 4XM1X22 in (2.3) is nonzero, contradicting the immersion.
3. tmfcohomology 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 tmfcohom*
*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 E2term of the ASS converging to tmf*(P1), 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(P2n1) = tmf*1(P21).
(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(P21) from the exact sequence
q* s,t 1
! Es1,t2(P11) ! Es,t2(P21) ! Es,t2(P11) ! E2 (P1) ! .
(3.4)
It was proved in [17] that
M
ExtA2(P11, Z2) ExtA1( 8i1Z2, Z2).
i2Z
Here we have initiated a notation that Pmn:= H*(Pnm). A complete calculation of
ExtA2(P11, Z2) was performed in [10], but all we need here are the first few g*
*roups.
We can now form a chart for E2(P21) from (3.4), as in Diagram 3.5, where O in*
*dicate
elements of ExtA2(P11, 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*(P21), 17 * 2
    
66 66 66 66 66
    
 r  r 
    
rr r r rr r r rr r
    
rrr r rrr r rrr
    
rr r rr r rr b
     @
. . . r r r r r b b@
  
r r r r rb r b
  
rr r r b rr b
  @ 
_____________________________________________________________rrr@
17 9 1
Dualizing, we obtain Diagram 3.6 for the desired tmf*(P11).
Diagram 3.6. tmf*(P11), * 2
    
66 66 66 66 66
    
 r  r 
    
r rr r r rr r r rr
@   @   @ 
r rr r r rr r r rr
@   @   @ 
@ rr r @ rr r @rr
    
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 wellknown KahnPriddy 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 := P11. 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 Amodules
0 ! Z2 P ! P P ! (P=Z2) P ! 0.
Proof.The Z2 is, of course, the subgroup generated by x0, which is an Asubmodu*
*le.
g i j 0 i+j
A splitting morphism P P !Z2 P is defined by g(x1 x2) = x1 x2 . This is
Alinear 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 A2modules
0 ! C ! (P=Z2) P ! B ! 0,
where C has a filtration with
Fp(C)=Fp1(C) 8pA2= Sq2, p 2 Z,
and B has a filtration with
M 1
Fp(B)=Fp1(B) 4p2A2= Sq, p 2 Z.
Z copies
The generator of Fp(C)=Fp1(C) is x11x8p12; a basis over Z2 for C is
{x21xi+22+x41xi2, x41xi2+x81xi42, i 2 Z}[{x11xi12+x21xi22, i 6 0 (8)}[{x11x*
*8p12, p 2 Z}.
A minimal set of generators as an A2module for the filtration quotients of B is
{x8i11x4j12: 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
 
 
 
66666666666666666666666666666
 
 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( 4p2A2= 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( 4p2A2= 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( 8p2A2= 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
Exts1,tA2(Fp11(C), Z2) ! Exts,tA2(Fp1(C)=Fp11(C), Z2)
must be nonzero. But one or the other of these groups is always 0,1 as both cha*
*rts
Ext*,*A2(Fp11(C), Z2) and Ext*,*A2(Fp1(C)=Fp11(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(P21
P21, 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 x91x8i12, . .,.x8i11x92. The ge*
*nerators
of the two copies of ExtA1(Z2, Z2) in 8i2 arise from h0 times classes corresp*
*onding
to x11x8i12and x8i11x12.
Proof.Using exact sequences like (3.4) on each factor, we build Ext*,*A2(P21 *
*P21, Z2)
from A := Ext*,*A2(P P, Z2), B := Ext*1,*A2(P11 P, Z2), C := Ext*1,*A2(P P1*
*1, Z2),
and D := Ext*2,*A2(P11 P11, Z2), with possible d1differential from A and i*
*nto D.
In the range of concern, t  s 9, the Dpart 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 ofA+B+C*
*
  
6666666666666666666666666666*
*66
rr  rr
  
  
rrrrrr
  
rrrrrrrr
  
  
rrrrrrrr
  
  
rrrrrrrrrr

rrrrrrrrrrrr


rrrrrrrrrrrrrr


rrrrrrrrrrrrrr

rrrrrrrrrrrrrrrr
  
  
________________________________
8p+ 2 2 6
In dimension 8p2, the towers in A arise from all cohomology classes x8i11x*
*8j12
with i + j = p, while in dimension 8p + 2, they arise from x8i11x8j+32~ x8i+3*
*1x8j12.
The finite towers in B arise from x4i11x8j12with i 0, and those from C from
x8i11x4j12with j 0. The homomorphism
Ext0A2(P P, Z2) ! Ext0A2(P11 P, Z2) Ext0A2(P P11, Z2),
NONIMMERSIONS IMPLIED BY TMF, REVISITED 13
which is equivalent to the d1differential 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 : x91x52~ x51x92
18 : x91x92
22 : x171x52~ x131x92, x91x132~ x51x172
26 : x171x92, x91x172.
These kernel classes yield infinite towers emanating from filtration 0.
For each p < 0, the towers arising from x4j11x8p12, j 0, in A combine wit*
*h those
in the psummand of
M
B ExtA1( 8p1P11, 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(P21 P21, 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
 @ @ 
rr r@r r@r r r
 @ @ 
r r r@r r@r r
@  @  @ 
____________________________rrr@@@ ____________________________
Putting things together, we obtain that in dimensions less than 8, ExtA2(P2*
*1
P21, 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
Diagram3.18.IllustrationofProposition3.15 
    
666666 666666 66666 66666 6666
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
   
   
__________________________________________________________________________*
*
26 18 10
The only possible differentials in the Adams spectral sequence of P21^ P2*
*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(P21^ P21),
and so we obtain a turnedaround 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 Sduals of the
classes from which the filtration0 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 AtiyahHirzebruch 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 A2submodule of (P=Z2) P generated
by all x11x8p12, p 2 Z. Note that Sq2(x11x8p12) = Sq4Sq6(x11x8p92). Thus a*
* basis
of A2= Sq2acting on all x11x8p12spans 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 `(x11x12) and*
* `(x11x32).
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
`(x8i11x8j12) and `(x8i11x8j+32). We will show Q = 0. This will complete the*
* proof of
Proposition 3.9, implying in particular that Sq1(x8i11x8j12) and Sq1(x8i11x8*
*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 A2action 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 x11x12yields the following
elements: x11x12+ x01x02+ x11x12, x21x62+ x61x22, x11x92+ x31x52+ x41x42+ x5*
*1x32+ x91x12,
16 DONALD M. DAVIS AND MARK MAHOWALD
and x41x122+ x121x42. A basis of A2= Sq1in degree 6 mod 8 acting on x11x32yi*
*elds
the following elements: x21x62+ x31x52+ x41x42+ x51x32, x11x92+ 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
x8i11x8j+12+ x8i1x8j2+ x8i+11x8j12become 0 in Q.
R1 : X1 + X0+ X1,
R2 : X2+ X6,
R3 : X1 + X3+ X4+ X5+ X9,
R4 : X4+ X12,
R5 : X2+ X3+ X4+ X5,
R6 : X1 + 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 X8i1 *
*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`2classes 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 evendegree 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 2exponents 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 2divisible. 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 2adic 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 Xj3iterm 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 + z1) + 2fli`i(zi+1+ z(i+1)). (4.7)
i>0
NONIMMERSIONS IMPLIED BY TMF, REVISITED 19
Let pi= zi+ zi. We will show that
X
` = 24+iAip2i+1 (4.8)
i 0
for certain 2adic integers Ai, which interprets back to the claim of 4.5.
Note that pipj = pi+j+ pij, 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
2divisibility. 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`i1pi. Each term in the expansion*
* of
` involves a sequence of choices. First choose Ti for some i 1, and then if i*
* > 1
choose (i1) factors Tj, one from each factor of `i1. 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 2divisible 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 2divisible 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 ciXi1Xni2in 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 2adic integer and ffiare 2adic 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 nonXi1X`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+ zj as in the proof of 4.5.
The summand with i = 2t becomes
P 2tss _ ! 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 jvalues 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+i1+ 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.tmfcohomology 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 nullhomotopic. 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 ni P
that in grading 2n a kernel element must be ciX1X2 with ci= 0, and hence is
X X X j ni1j
ci(Xi1Xni2 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 A2modules
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(C11, Z2) 8p2bo[v2].
p2Z
24 DONALD M. DAVIS AND MARK MAHOWALD
Of course applied to a module or an Ext group just means to increase the tgr*
*ading
by 1.
Proof.There is a filtration of C11 with Fp=Fp1 8p2M10for p 2 Z. We have
Sq2'8p2= Sq4Sq2Sq4'8p10. The same argument used in the last paragraph of the
proof of Corollary 3.10 works to initiate an inductive proof of the Extisomorp*
*hism
claimed in the theorem. 
Corollary 5.4. In gradings (t  s) less than 1,
M
Ext*,*A2(C21, Z2) 8p2bo[v2].
p<0
Proof.There is an exact sequence
q* s,t 1
! Exts1,tA2(C11, Z2) ! Exts,tA2(C21, Z2) ! Exts,tA2(C11, Z2) ! ExtA2(C1,*
* Z2) ! .
The result is immediate from this and 5.3, since q* sends the initial tower in *
*F0=F1
isomorphically to the initial tower in ExtA2(C11, Z2). 
The Amodules C11and 2C21are dual. Thus, by [9, Prop 4],
Exts,tA2(Z2, C11) Exts,tA2( 2C21, 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 E2term 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 C21 C21. 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
A2morphism C21! Z2.
Lemma 5.6. Let
ae 2 10
K = ker(C21 C21!C1 Z2).
Let S denote the set of all classes x8i21x8j22with i 1 and j 2, togethe*
*r with
the classes x8i21x8j+22with i 1 and j 1. Then K is the direct sum of a f*
*ree
A2==E2module on S with a single relation Sq4Sq2Sq4(x101x62) = 0.
Proof.Since the generators of E2 have odd grading, A2==E2 acts on any element of
these evenlygraded modules. The action of A2==E2 on x21x22yields the additio*
*nal
elements x21x02+x01x22, x21x22+x01x02+x21x22, x21x42+x41x22, x01x22+x21x0*
*2, x01x42+x41x02,
x21x82+x21x42+x41x22+x81x22, and x01x82+x81x02. The action of A2==E2 on x21x*
*22yields the
additional elements x01x22+x21x42, x01x42+x21x22, x21x42+x41x22, x21x42+x21x8*
*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 C21 C21are obta*
*ined
exactly once from the described elements in K together with C21 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 xi1x28i2, we have:
(1)From generators in 28, we obtain just X10 in K. The class
X18is in C21 10Z2.
(2)From generators in 32, we obtain X8 + X6, X16+ X14,
and X24+ X22.
(3)From generators in 36, we obtain X8+X4 and X16+X12.
(4)From generators in 40, we obtain X4, X12+ X8, X20+
X16, and X24.
Note in (4) that X0 and X28do 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 X18, form a ba*
*sis for
the space spanned by X4, . .,.X24, 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 y212= X21X2 + X1X22. The monomials of the form
Xi1Xj2y12are acted on freely by Z2[v0, v1, v2]. Let Sn denote the Z2vector sp*
*ace with
basis the monomials Xi1Xni2, 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 8i4Z2[v0, v1, v2] and i copies of 8i16Z2[v0, v1, v2], *
*and also one
copy of 8i8bo[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(C21 C21, 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 A2morphisms 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 Xi1Xni2equals 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 A2resolution, with C1 = 1A2 2A2 4A2,
then h1Xi1Xni2is represented by the composite C1 ! C0 ! C11 C11 sending
'2 7! ' 7! Xi1Xni2, 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)
    
666 66 66 6 6
    
    
    
    
    
    
    
    
    
    
    
    
_____________________________________________
28 24 20 16 12
Diagram 5.11. Structure on two types of generators
        
6 6 6 6 6 6 6 6 66 66 66
          
          
          
          
        
        
      
      
    
    
 
______________________________ ______________________________
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 tmfcohomology
groups. Corollary 5.5 gives the E2term 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 E2term 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]

 666
  
 66 
  666 
 6   
 66    
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(hffl1v4i1vj2X2k+1) = hffl+11v4i1vj+12X2k
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 X2k+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

  666
 66 
 666  
 6   
66    
     
 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 degree1 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 AndoHopkinsRezk 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 d3differential 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 A2map 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*( 2CP21), which, as we have noted,
is isomorphic to that of [ *CP11, tmf]. The E2term was described in 5.4. Let
S16 ! 2CP21^ tmf correspond to the map in (5.16). Since E2(CP21^ tmf)
in negative dimensions is built from copies of ExtA2(M10, Z2), we deduce from t*
*he
previous paragraph that h21v41v2g16in the ASS for tmf*( 2CP21) must be hit b*
*y a
d2 or d3differential, since it is the image of a class hit by a d3. The only *
*possibility is
that it be d2 from h1v41g8, 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 v2g16 would be some dr with r > 2 hitting*
* an
element which is acted on nontrivially by h1. However h1v2g16 has become 0 in
E3 since it was hit by a d2differential. Thus a nonzero differential on v2g16*
* would
contradict naturality of differentials with respect to h1action. Hence there i*
*s a map
S10 ! 2CP21^ tmfhitting v2g16, and the argument of the previous paragraph
implies that d2(h1v41v2g8) = h21v41v22g16 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
h1naturality considerations. 
Remark 5.17. The proof of the key d3differential in the ASS of tmf from the 17
stem to the 16stem, 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 14dimensional 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. Portionof ASS of tmf 
______________________
8 r 
 BB 
 B r B 
 BB B 
Ar B rBB 
 A B A B 
5 rAr B rArB 
 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*(CP21^ CP21) are implied b*
*y the
same considerations that worked for CP21. The Z2[v0, v1, v2]parts in Theorem*
* 5.7
cannot support differentials by dimensionality and h1naturality. For the boli*
*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 Xi1Xni2for 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(hffl1v4i1vj2Z2k1) = 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 {Xi1Xni2: 1 i < n}, and ffl(Xi1X*
*ni2) = 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 Zi1Zni2should 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
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Lehigh University, Bethlehem, PA 18015, USA
Email address: dmd1@lehigh.edu
Northwestern University, Evanston, IL 60208, USA
Email address: mark@math.northwestern.edu