arXiv:0704.0798v1  [math.AT]  5 Apr 2007
NONIMMERSIONS OF RP n IMPLIED BY tmf, REVISITED
DONALD M. DAVIS AND MARK MAHOWALD
Abstract. In a 2002 paper, the authors and Bruner used the new
spectrum tmf to 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∗(RP∞ ×
RP∞) and tmf∗(CP∞ × CP∞) in positive dimensions.
1. Introduction
In [6], the authors and Bruner described a proof of the following theorem, along
with some additional nonimmersion results.
Theorem 1.1. ([6, 1.1]) Assume that M is divisible by the smallest 2-power greater
than or equal to h.
• If α(M) = 4h − 1, then P 8M+8h+2 cannot be immersed in ( 6⊆)
16M−8h+10.
• If α(M) = 4h− 2, then P 8M+8h 6⊆ R16M−8h+12.
Here and throughout, α(M) denotes the number of 1’s in the binary expansion of M ,
and P n denotes 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.
http://arxiv.org/abs/0704.0798v1
2 DONALD M. DAVIS AND MARK MAHOWALD
In [6], the theorem is followed by a comment that this is new provided α(M) ≥ 6,
i.e., h ≥ 2, and the first new result occurs for P 1536. In this note, we point out
that 1.1 is valid when h = 1, and these results are new when M is even, including
new nonimmersions of P n for 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 incorrect.
Letting h = 1 in 1.1, we have the following result.
Corollary 1.2. a. If α(M) = 3, then P 8M+10 6⊆ R16M+2.
b. If α(M) = 2, then P 8M+8 6⊆ 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, immersions,
nonembeddings, and embeddings of P n is presented, arranged according to n = 2i+d
with 0 ≤ d < 2i and 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 n with n as small as 56. The result of 1.2.b with M = 2i + 1 was also
proved very recently by Kitchloo and Wilson in [15]. This result for P 2
k+16, 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 proof of the
general case in [6] requires some extremely elaborate arguments and calculations. 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 argument
there was that an immersion of RP n in Rn+k implies existence of an axial map P n×
f−→ Pm+k for an appropriate value of m, and obtains a contradiction for certain
n, m, and k by consideration of tmf∗(f). Here tmf is the spectrum of topological
modular forms, which was discussed in [6]. A class X ∈ tmf8(P n) was described,
along with X1 = X × 1 and X2 = 1 × X in tmf8(P n × Pm). 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 Xℓ = 0 but (X1+X2)
ℓ 6= 0. We recently realized that
it is conceivable that f ∗(X) might contain other terms coming from tmf8(P n ∧ Pm).
In Section 3 (see Theorem 3.7) we perform a complete calculation of tmf∗(P∞×P∞)
in positive gradings divisible by 8, and in Section 4 we use it to show that effectively
f ∗(X) = u(X1 + X2), where u is a unit in tmf
∗(P∞ × P∞), which enables us to
retrieve all the nonimmersions of [6].
In Section 5, we compute tmf∗(CP∞ × CP∞) 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 × CP ). This approach to retrieving the nonimmersions of [6] did not yield
the desired result, but the later approach given in Section 4 did. Nevertheless the nice
result for tmf∗(CP∞ × CP∞) 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 least to
[14]. If P 8M+10 ⊆ R16M+2, then gd((2L+3 − 8M − 11)ξ8M+10) ≤ 8M − 8, hence this
bundle has (2L+3−16M −3) linearly independent sections, and thus there is an axial
P 8M+10 × P 2L+3−16M−4 f−→P 2L+3−8M−12.
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 tmf
8(−) described
in [6] and also in Section 1. In Section 4, we will show that we may assume that
f ∗(X) = X1+X2, as was done in [6], since this is true up to multiplication by a unit.
Since tmf2
L+3−8M−8(P 2
L+3−8M−12) = 0, we have
0 = f ∗(0) = f ∗(X2
L−M−1) = (X1+X2)
2L−M−1 ∈ tmf2L+3−8M−8(P 8M+10×P 2L+3−16M−4).
Expanding, we obtain
2L−M−1
XM+11 X
2L−2M−2
2L−M−1
XM1 X
2L−2M−1
2 as the only
terms which are possibly nonzero. Next we note that, with all u’s representing odd
integers,
2L−M−1
= 2α(M)−ν(M+1)u2 = 2
3−ν(M+1)u2,
4 DONALD M. DAVIS AND MARK MAHOWALD
where we have used α(M) = 3 at the last step. Here and throughout, ν(2eu) = e.
Similarly,
2L−M−1
= 2α(M)u4 = 2
3u4. Thus an immersion implies that in
L+3−8M−8(P 8M+10 × P 2L+3−16M−4), we have
23−ν(M+1)u2X
2L−2M−2
2 + 2
2L−2M−1
2 = 0.
(2.1)
We recall [6, 2.6], which states that there is an equivalence of spectra P k+8b+8 ∧
tmf ≃ Σ8P kb ∧ tmf. Combining this with duality, we obtain tmf
8M+8(P 8M+10) ≈
tmf−1(P−3) ≈ Z/8, and so 8XM+11 X2
L−2M−2
2 = 0. Here and throughout, Pn = P
RP∞/RP n−1. Similarly tmf2
L+3−16M−8(P 2
L+3−16M−4) ≈ tmf7(P3) ≈ Z/16, and hence
16XM1 X
2L−2M−1
2 = 0. Duality also implies
L+3−8M−8(P 8M+10 × P 2L+3−16M−4) ≈ tmf14(P−3 ∧ P3).
Calculations such as E2(tmf∗(P−3∧P3)), the E2-term of the Adams spectral sequence
(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
0 3 7 11 15
r r r r r�
r r r
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 paragraph of page
54 of [6], the class in (15, 0) corresponds to an easily-constructed nontrivial map. The
monomials XM+11 X
2L−2M−2
2 and X
2L−2M−1
2 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 filtration
0 is Z/8 ⊕ Z/16, we conclude that 8 and 16, respectively, are the precise orders of
the monomials. In particular, the order of XM1 X
2L−2M−1
2 is 16, and hence the class
in (2.1) is nonzero since it has a term 8uXM1 X
2L−2M−1
2 , and so (2.1) contradicts the
hypothesized immersion.
Part b of 1.2 is proved similarly. If P 8M+8 immerses in R16M+4, then there is an
axial map
P 8M+8 × P 2L+3−16M−6 f−→P 2L+3−8M−10,
and hence, up to odd multiples,
22−ν(M+1)XM+11 X
2L−2M−2
2 + 2
2XM1 X
2L−2M−1
2 (2.3)
= 0 ∈ tmf2L+3−8M−8(P 8M+8 ∧ P 2L+3−16M−6),
since α(M) = 2. We have tmf8M+8(P 8M+8) ≈ tmf−1(P−1) ≈ Z/2, and
L+3−16M−8(P 2
L+3−16M−6) ≈ 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 filtration).
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 classes, we
conclude that the orders of the monomials in (2.3) are precisely 2 and 8, respectively,
and so the term 4XM1 X
2L−2M−1
2 in (2.3) is nonzero, contradicting the immersion.
3. tmf-cohomology of P∞ × P∞
In this section, we compute tmf∗(P∞) and tmf8∗(P∞ × P∞) in positive gradings.
These will be used in the next section in studying the axial class in tmf-cohomology.
There is an element c4 ∈ π8(tmf) which reduces to v41 ∈ π8(bo); it has Adams
filtration 4. It acts on tmf∗(X) with degree −8. Recall also that π∗(bo) = bo∗ is as
depicted in 5.1. We denote bo∗ = bo−∗. We use P1 and P
∞ interchangeably.
6 DONALD M. DAVIS AND MARK MAHOWALD
Theorem 3.1. There is an element X ∈ tmf8(P1) of Adams filtration 0, described
in [6], such that, in positive dimensions divisible by 8, tmf∗(P1) is isomorphic as an
algebra over Z(2)[c4] to Z(2)[c4][X ]. In particular, each tmf
8i(P1) with i > 0 is a free
abelian group with basis {cj4X i+j : j ≥ 0}. There is a class L ∈ t0(P1) such that
• tmf0(P1) is a free abelian group with basis {L, cj4Xj : j ≥ 1},
• L2 = 2L and LX = 2X.
Moreover, in positive dimensions tmf∗(P1) is isomorphic as a graded abelian group 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 with 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 isomorphisms
tmf∗(P1) ≈ lim← tmf
∗(P n1 ) ≈ lim← tmf−∗−1(P
−n−1) = tmf−∗−1(P
(3.3)
Since H∗(tmf;Z2) ≈ A//A2, there is a spectral sequence converging to tmf∗(X) with
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
−∞) from the exact sequence
→ Es−1,t2 (P∞−1)→ E
−∞)→ E
q∗−→ Es,t2 (P∞−1)→ .
(3.4)
It was proved in [17] that
ExtA2(P
−∞,Z2) ≈
ExtA1(Σ
Z2,Z2).
Here we have initiated a notation that Pmn := H
∗(Pmn ). A complete calculation of
ExtA2(P
−1,Z2) was performed in [10], but all we need here are the first few groups.
We can now form a chart for E2(P
−∞) from (3.4), as in Diagram 3.5, where ◦ indicate
elements of ExtA2(P
−1,Z2) suitably positioned, and lines of negative slope correspond
to cases of q∗ 6= 0 in (3.4).
NONIMMERSIONS IMPLIED BY TMF, REVISITED 7
Diagram 3.5. tmf∗(P
−∞), −17 ≤ ∗ ≤ 2
−17 −9 −1
· · ·
✻ ✻ ✻
r r r
r r r
✻ ✻ ✻
r r r
r r r
r r r
r r r
r r r
Dualizing, we obtain Diagram 3.6 for the desired tmf∗(P∞1 ).
Diagram 3.6. tmf∗(P∞1 ), ∗ ≥ −2
0 8 16
✻ ✻ ✻✻ ✻ ✻
r r r
r r r
· · ·
Naming of the generators X i is clear since X has filtration 0. The free action of c4
is also clear. The class L is (up to sign) the composite P1
λ−→ S0 → tmf, where λ is
the well-known Kahn-Priddy map. Thus L is the image of a class L̂ ∈ π0(P1). Lin’s
theorem ([16]) says that π0(P1) ≈ Z∧2 , generated by L̂. Since π0(P1)→ ko0(P1) is an
isomorphism, and, since (1 − ξ)2 = 2(1 − ξ) for a generator (1 − ξ) of ko0(P1), we
obtain L̂2 = 2L̂, and hence also for L. We chose the generator to be (1 − ξ) rather
than (ξ − 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 = p
2, from which we
conclude p = 2.
In tmf∗(P1 × P1), for i = 1, 2, let Li and Xi denote the classes L and X in the ith
factor. Note that there is an isomorphism as tmf∗-modules, but not as rings,
tmf∗(P1 × P1) ≈ tmf∗(P1 ∧ P1)⊕ tmf∗(P1 × ∗)⊕ tmf∗(∗ × 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 X i1X
2 with i, j > 0
direct sum with a free Z[c4]-module with basis {L1X i2, X i1L2 : i ≥ 1}. The product
and Z[c4]-module structure is determined from 3.1 and
c4(X1X2) = (c4X1)X2 = X1(c4X2) =
4(L1X
1 L2),
for certain integers γi with γ0 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 similar
to (3.4). But to get started, we need ExtA2(P ⊗ P,Z2). Here we have begun to
abbreviate P := P∞−∞. We begin with a simple lemma. Throughout this section, x1
and x2 denote nonzero elements coming from the factors in H
1(RP × 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 x
0, which is an A-submodule.
A splitting morphism P⊗P g−→Z2 ⊗P is defined by g(xi1 ⊗ x
2) = x
1 ⊗ x
2 . This is
A-linear since
g(Sqk(xi1⊗x
2)) =
x01⊗x
i+j+k
x01⊗x
i+j+k
2 = Sq
k g(xi1⊗x
The following result is more substantial. We will prove it at the end of this section.
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) ≈ Σ8pA2/ Sq2, p ∈ Z,
and B has a filtration with
Fp(B)/Fp−1(B) ≈
Z copies
Σ4p−2A2/ Sq
1, p ∈ Z.
The generator of Fp(C)/Fp−1(C) is x
2 ; a basis over Z2 for C is
{x21xi+22 +x41xi2, x41xi2+x81xi−42 , i ∈ Z}∪{x11xi−12 +x21xi−22 , i 6≡ 0 (8)}∪{x11x
2 , p ∈ Z}.
A minimal set of generators as an A2-module for the filtration quotients of B is
{x8i−11 x
2 : i, j ∈ Z}.
Corollary 3.10. A chart for Ext
(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 classes is
repeated with vertical period 4. Thus, for example, in 8p−1 there is an infinite tower
emanating from filtration 4i for each i ≥ 0.
10 DONALD M. DAVIS AND MARK MAHOWALD
Diagram 3.11. Ext
(P⊗P,Z2) in 8p− 3 ≤ t− s ≤ 8p+ 4
8p+ −2 0 2 4
✻✻✻✻✻✻✻✻✻✻✻ ✻✻✻✻✻✻✻✻✻✻✻✻ ✻✻
✻✻ ✻✻
Proof of Corollary 3.10. We first note that ExtA2(P,Z2) is identical to the left portion
of Diagram 3.5 extended periodically in both directions. Also, ExtA2(A2/ Sq
1,Z2) ≈
ExtA0(Z2,Z2) is just an infinite tower, and
ExtA2(A2/ Sq
2,Z2) ≈ ExtA1(A1/ Sq2,Z2)
is given as in Diagram 3.14. We will show at the end of this proof that
ExtA2(C,Z2) ≈
ExtA2(Σ
8pA2/ Sq
2,Z2) (3.12)
and similarly
ExtA2(B,Z2) ≈
ExtA2(Σ
4p−2A2/ Sq
1,Z2).
These would follow by induction on p once you get started, but since p ranges over
all integers, that is not automatic.
Thus ExtA2(P⊗P,Z2) is formed from
ExtA2(P,Z2)⊕
ExtA2(Σ
8pA2/ Sq
2,Z2)⊕
ExtA2(Σ
4p−2A2/ Sq
1,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 there are
no possible boundary morphisms in the Ext sequence of 3.9, yielding the claim of 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, Ext
(Fp0(C),Z2) = 0. Actually a
much sharper lower vanishing line can be established, but this is good enough for our
purposes. Thus, for this (s, t),
(Fp1(C),Z2) ≈
(Σ8p−2A2/ Sq
2) (3.13)
for p1 ≤ p0, as both are 0. Let p1 be minimal such that (3.13) does not hold. Then
comparison of exact sequences implies that
s−1,t
(Fp1−1(C),Z2)→ Ext
(Fp1(C)/Fp1−1(C),Z2)
must be nonzero. But one or the other of these groups is always 0,1 as both charts
(Fp1−1(C),Z2) and Ext
(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, and hence
(3.12) holds. A similar proof works when C is replaced by B.
Diagram 3.14. ExtA2(A2/ Sq
2,Z2)
· · ·
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 this
result is pictured in Diagram 3.18.
Proposition 3.15. In dimensions t− s ≡ 2 mod 4 with t− s ≤ −10, ExtA2(P−2−∞⊗
P−2−∞,Z2) consists of i infinite towers emanating from filtration 0 in dimensions −8i−
1Actually this is not quite true; for one family of elements we need to use h0-
naturality.
12 DONALD M. DAVIS AND MARK MAHOWALD
6 and −8i − 10, together with the relevant portion of two copies of ExtA1(Z2,Z2)
beginning in filtration 1 in each dimension −8i − 2. The generators of the towers in
−8i− 10 correspond to cohomology classes x−91 x−8i−12 , . . . , x−8i−11 x−92 . The generators
of the two copies of ExtA1(Z2,Z2) in −8i−2 arise from h0 times classes corresponding
to x−11 x
2 and x
−8i−1
Proof. Using exact sequences like (3.4) on each factor, we build Ext
(P−2−∞⊗P−2−∞,Z2)
fromA := Ext
(P⊗P,Z2), B := Ext∗−1,∗A2 (P
−1⊗P,Z2), C := Ext
∗−1,∗
(P⊗P∞−1,Z2),
and D := Ext
∗−2,∗
(P∞−1 ⊗ P∞−1,Z2), with possible d1-differential from A and into D.
In the range of concern, t− s ≤ −9, the D-part will not be present, and the part 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, together
with B and C, is pictured in Diagram 3.16.
Diagram 3.16. Portion of A+B+C
✻✻✻✻✻✻✻✻✻✻ ✻✻✻✻✻✻✻✻✻✻ ✻✻✻✻✻✻✻✻✻✻
−2 2 68p+
rr rrr
In dimension 8p−2, the towers in A arise from all cohomology classes x−8i−11 x
−8j−1
with i+ j = −p, while in dimension 8p+ 2, they arise from x8i−11 x
2 ∼ x8i+31 x
The finite towers in B arise from x4i−11 x
2 with i ≥ 0, and those from C from
x8i−11 x
2 with j ≥ 0. The homomorphism
Ext0A2(P⊗P,Z2)→ Ext
(P∞−1 ⊗P,Z2)⊕ Ext0A2(P⊗P
−1,Z2),
NONIMMERSIONS IMPLIED BY TMF, REVISITED 13
which is equivalent to the d1-differential mentioned above, sends classes to those with
the same name. In dimension ≤ −10, this is surjective, with kernel spanned by classes
with both components < −1. In dimension −8i−6 and −8i−10, there will be i such
classes. We illustrate by listing the classes in the first few gradings:
−14 : x−91 x−52 ∼ x−51 x−92
−18 : x−91 x−92
−22 : x−171 x−52 ∼ x−131 x−92 , x−91 x−132 ∼ x−51 x−172
−26 : x−171 x−92 , x−91 x−172 .
These kernel classes yield infinite towers emanating from filtration 0.
For each p < 0, the towers arising from x
2 , j ≥ 0, in A combine with those
in the p-summand of
ExtA1(Σ
8p−1P∞−1,Z2)
as in Diagram 3.17 to yield one of the copies of ExtA1(Z2,Z2) arising from filtration
1. An identical picture results when the factors are reversed.
Diagram 3.17. Part of ExtA2(P
−∞ ⊗P−2−∞,Z2)
✻ ✻ ✻
Putting things together, we obtain that in dimensions less than −8, ExtA2(P−2−∞⊗
P−2−∞,Z2) consists of a chart described in Proposition 3.15 and partially illustrated
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
−26 −18 −10
✻✻✻✻✻✻ ✻ ✻✻✻
✻✻✻✻ ✻✻
The only possible differentials in the Adams spectral sequence of P−2−∞∧P−2−∞∧ 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]. Similarly to
(3.3), we have
tmf∗(P1 ∧ P1) ≈ tmf−∗−2(P−2−∞ ∧ P−2−∞),
and so we obtain a turned-around version of Diagram 3.18, of the same general sort
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 X i1X
2 described 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). Simi-
larly the classes L1X
2 and X
1L2 have Adams filtration 1, and so one would anticipate
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 spectral
sequence; however, the Atiyah-Hirzebruch spectral sequence shows this quite clearly.
The class X i1 is detected by H
8i(P1; π0(tmf)), while L is detected by H
1(P1; π1(tmf)).
NONIMMERSIONS IMPLIED BY TMF, REVISITED 15
Under the pairing, their product is detected in H8i+1(P1; π1(tmf)), clearly of Adams
filtration 1.
The last part of Theorem 3.7 deals with the action of c4 on the monomials X
Since tmf is a commutative ring spectrum, tmf∗(P1 ∧ P1) is a graded commutative
algebra over tmf∗. The action c4(X1X2) must be of the form
i≥0 γic
4(L1X
as these are the only elements in tmf8(P1∧P1), and the class must be invariant under
reversing factors. The divisibility of γ0 by 8 follows since c4 has Adams filtration 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 x11x
2 , p ∈ Z. Note that Sq2(x11x
2 ) = Sq
4 Sq6(x11x
2 ). Thus a basis
of A2/ Sq
2 acting on all x11x
2 spans C. The 24 elements in a basis of A/ Sq
acting on x11x
2 yield x
2 + x
2 + x
2 + x
2 , x
2 + x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
and x41x
2 + x
2 . These classes with second components shifted by all multiples 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 more
elaborate. For the 32 elements θ in a basis of A2/ Sq
1, we list θ(x−11 x
2 ) and θ(x
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 elements
θ(x8i−11 x
2 ) and θ(x
2 ). We will show Q = 0. This will complete the proof of
Proposition 3.9, implying in particular that Sq1(x8i−11 x
2 ) and Sq
1(x8i−11 x
2 ) 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 x
i depends only on i mod 8. We illustrate with
the case in which degree ≡ 0 mod 8. The other 7 congruences are handled similarly,
although some are a bit more complicated.
A basis of A2/ Sq
1 in degree ≡ 2 mod 8 acting on x−11 x−12 yields the following
elements: x−11 x
2 + x
2 + x
2 , x
2 + x
2 + x
2 + x
2 + x
2 + x
16 DONALD M. DAVIS AND MARK MAHOWALD
and x41x
2 + x
2. A basis of A2/ Sq
1 in degree ≡ 6 mod 8 acting on x−11 x32 yields
the following elements: x21x
2 + x
2 + x
2 + x
2 + x
2 + x
2 + x
2 + x
2, and x
2 + x
2. Because we allow both components 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−11 x
2 + x
2 + x
2 become 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) are 0 in
Q. First, R8 implies that all classes X8i are congruent to one another. Since X0 is
0 in the quotient due to P/Z2, we conclude that all classes X8i are 0 in Q. Next,
R4 implies that all X8i+4 are congruent to one another. Since X4 +X8 ∈ C, and we
have just shown that X8 ≡ 0 in Q, we deduce that all X8i+4 are 0 in Q. Now we
use R2 + R7 to see that all X8i+2 + X8i+4 are congruent to one another, then that
X2 +X4 ∈ 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+1 are congruent to one
another, and since X1 ∈ 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 careful consid-
eration of the possible “other terms” in the axial class discussed in the Introduction.
We show that, at least as far as the monomials cX i1X
2 in its powers are concerned,
NONIMMERSIONS IMPLIED BY TMF, REVISITED 17
the axial class equals u(X1+X2), where u is a unit in tmf
0(RP∞×RP∞). Thus the
ℓth power of the axial class is nonzero in tmf8ℓ(RP n×RPm) if and only if (X1+X2)ℓ
is nonzero there, and the latter is the condition which yielded the nonimmersions of
[6, 1.1]. Thus we have a complete proof of [6, 1.1].
If P n × Pm f−→Pm+k is an axial map, then there is a commutative diagram
P n × Pm f−−−→ Pm+k
P∞ × P∞ g−−−→ P∞,
where g is the standard multiplication of P∞, since P∞ = K(Z2, 1). Since X ∈
tmf8(Pm+k) has been chosen to extend over P∞, we obtain that f ∗(X) is the restric-
tion of g∗(X). By Theorem 3.7 and the symmetry of g, we must have
g∗(X) = X1 +X2 +
4(L1X
1 L2), (4.1)
for some integers κi. This is what we call the “axial class.” Then g
∗(Xℓ) equals the
ℓth power of (4.1). Using the formulas for L2i , LiXi, and c4(X1X2) in 3.1 and 3.7 and
the binomial theorem, this ℓth power can be written in terms of the basis described
in 3.7. If some κi’s are nonzero, the coefficients of X
2 in g
∗(Xℓ) will not equal
, 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 like 2
when multiplying by Xi, and if, for example, L1 is present without X1, then the terms
ci4L1X
2 cannot cancel our X
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 throughout
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 − 80X42 is the same as that in
(X1 +X2)
4, and L1X
2 is a separate basis element.
With this simplification, the axial class in (4.1) becomes
X1 +X2 + 2
2 ) (4.2)
18 DONALD M. DAVIS AND MARK MAHOWALD
for some integers κi. There was another term 2κ0(X1+X2), but it can be incorporated
into the leading (X1 +X2). The odd multiple that it can create is not important.
From Theorem 3.7, we have
c4(X1X2) = 16(X1 +X2) + 2
(4.3)
for some integers γk. The 16 comes from γ0 = 8 and Li = 2. Actually we don’t really
know that γ0 = 8, even just up to multiplication by a unit, but it is divisible by 8
and the possibility of equality must be allowed for. This gives
2 ) = 16(X
2 ) + 2
i+k+1
j+k+1
(4.4)
Here we use that in a graded tmf∗-algebra tmf
∗(X) with even-degree elements, c(xy) =
cx · y, for c ∈ tmf∗ and x, y ∈ tmf∗(X).
There is an iterative nature to the action of c4 in (4.4), but the leading coefficient
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 equals
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 Ai such that
24+iAi
Remark 4.6. This formula will be evaluated on (i.e. multiplied by) monomialsXk1X
One might worry that the negative powers of X1 or X2 in 4.5 will cause nonsensical
negative powers in c4X
2. This will, in fact, not occur because the monomials on
which we act always have total degree greater than the dimension of either factor.
Thus if, after multiplication by c4, a term with negative exponent of Xi appears, then
the accompanying X
3−i-term will be 0 for dimensional reasons.
Proof of Theorem 4.5. The defining equation (4.3) may be written as, with θ =
X1X2 and z =
X1/X2,
θ = 16(z + z−1) +
i(zi+1 + z−(i+1)). (4.7)
NONIMMERSIONS IMPLIED BY TMF, REVISITED 19
Let pi = z
i + z−i. We will show that
24+iAip2i+1 (4.8)
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
k = pΣiei + L,
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 γi which occur in (4.7). This is allowable
if we agree that when collecting terms, we only make crude estimates about their
2-divisibility. We have
θ = 16p1 + 2θp2 + 2θ
2p3 + 2θ
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 factor.
Now let T1 = 16p1 and, for i ≥ 2, let Ti = 2θi−1pi. Each term in the expansion of
θ 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 θi−1. For each of these Tj with j > 1,
choose j − 1 additional factors, and continue this procedure. This builds a tree, and
we don’t get an explicit product term until every branch ends with T1. Each selected
factor Tj with j > 1 contributes a factor 2pj. There will also be binomial coefficients
and the omitted γi’s occurring as additional factors.
For example, Diagram 4.9 illustrates the choices leading to one term in the expan-
sion of θ. This yields the term 2p2 · 2p4 · 16p1 · 2p2 · 16p1 · 2p3 · 16p1 · 2p2 · 16p1, which
equals 221(p17 +L), where L is a sum of pi with i < 17 and i odd. By induction, one
sees in general that the sum of the subscripts emanating from any node, including
the subscript of the node itself, is odd.
20 DONALD M. DAVIS AND MARK MAHOWALD
Diagram 4.9. A possible choice of terms
T2 T4 T2 T1
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 2
k+4(p2k+1+L).
Note that a term 2k+4p2i+1 with i < k obtained from L will be more 2-divisible than
the 2i+4p2i+1 term 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 then
T2 then T1. Incorporating γi’s may make terms even more divisible, but the claim of
(4.8) is only that p2i+1 occurs with coefficient divisible by 2
Now we incorporate 4.5 into (4.2) to obtain the following key result, which we prove
at the end of the section.
Theorem 4.10. The monomials ciX
2 in the nth power of the axial class in
tmf8n(RP∞ × RP∞) are equal to those in the nth power of
(X1 +X2)
24+iαi
, (4.11)
where u is an odd 2-adic integer and αi are 2-adic integers.
The factor which accompanies (X1+X2) in (4.11) is a unit in tmf
∗(RP∞×RP∞);
we referred to it earlier as u. Indeed, its inverse is a series of the same form, obtained
by solving a sequence of equations. This justifies the claim in the first paragraph of
this section regarding retrieval of the nonimmersions of [6, 1.1].
We must also observe that restriction to tmf8ℓ(RP n × RPm) of the non-X i1Xℓ−i2
parts of the basis of tmf8ℓ(RP∞ × RP∞) cannot cancel the X i1Xℓ−i2 terms essential
for the nonimmersion. This is proved by noting that these elements such as L1X
NONIMMERSIONS IMPLIED BY TMF, REVISITED 21
and ci4L1X
2 will restrict to a class of the same name in tmf
8ℓ(RP n × RPm), and
will be 0 there for dimensional reasons, since 8ℓ > 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
We let z =
X1/X2 and pj = z
j + z−j as in the proof of 4.5.
The summand with i = 2t becomes
2κi(X1 +X2)
X t1X
2jAjp2j+1
= 2κi(X1 +X2)(p2t + L)24i
k(p2k+i + L).
Here k is a sum of j-values taken from the various factors in the ith power. Also, in
pj + L, L denotes a combination of pt’s with t < j. Noting (p2t + L)(p2k+i + L) =
p2k+2i + L, this becomes
2(X1 +X2)2
k(p2k+2i + L). (4.12)
The argument when i = 2t + 1 is similar but slightly more complicated because
(X i+11 +X
2 ) is not divisible by (X1 +X2). We obtain
X i+11 +X
X1X2)2t+1
2jAjp2j+1
For one of the factors of the ith power, say the first, we treat p2j+1 as
X1+X2√
(p2j +L).
The expression then becomes
2(X1 +X2)pi+12
k(p2k+i−1 + L),
where k is obtained as in the previous case. We again obtain (4.12).
Thus when g∗(X) − (X1 +X2) is written as (X1 + X2)
βjp2j , the coefficient βj
satisfies ν(βj) ≥ (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 all along.
This yields the claim of (4.11).
22 DONALD M. DAVIS AND MARK MAHOWALD
5. tmf-cohomology of CP∞ × CP∞
In [2], [4], and [8], it was noted, first by Astey, that the axial class using BP (or
BP 〈2〉) was u(X2 − X1), where u is a unit in BP ∗(P∞ ∧ P∞). 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 section
unnecessary. To do this, we calculate tmf∗(CP∞) and tmf∗(CP∞×CP∞) 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∞ and CP = CP∞
dR−−−→ RP × RP mR−−−→ RP
dC−−−→ CP × CP CP
1×(−1)
CP × CP mC−−−→ CP
The generator XR ∈ BP 2(RP ) satisfies XR = h∗(X). We also have that mC ◦ (1 ×
(−1))◦dC is null-homotopic. The key fact, which will fail for tmf, is BP ∗(CP×CP ) ≈
BP ∗[X1, X2].
The axial class is m∗R(XR). It equals (h× h)∗(1× (−1))∗m∗C(X). But
(1× (−1))∗m∗C(X) ∈ ker(d∗C).
By the above “key fact,” d∗C is the projection BP
∗[X1, X2]→ BP ∗[X ] in which each
Xi 7→ X . The kernel of this projection is the ideal (X2 −X1). To see this, just note
that in grading 2n a kernel element must be
2 with
ci = 0, and hence is
2 −Xn1 ) =
1(X2 −X1)
n−i−1−j
Thus (1 × (−1))∗m∗C(X) = (X2 − X1)u for some u ∈ BP ∗(CP × CP ). This u is
a unit by consideration of its reduction to H∗(−;Z), as in [2]. Since h∗(u) will then
be a unit in BP ∗(RP ×RP ) and h∗(Xi) = XRi, we obtain the claim about the axial
class being a unit times XR2 −XR1.
NONIMMERSIONS IMPLIED BY TMF, REVISITED 23
In order to see if there is any chance of adapting this to tmf, we compute tmf∗(CP∞)
and tmf∗(CP∞ × CP∞) in positive gradings. We begin with the relevant Ext calcu-
lations.
Let bo = Ext
(Z2,Z2). Recall that a chart for this is given as in Diagram 5.1,
extended with period (t− s, s) = (8, 4).
Diagram 5.1. Ext
(Z2,Z2)
0 4 8
· · ·
Let M10 denote the A2-module 〈1, Sq4, Sq2 Sq4, Sq4 Sq2 Sq4〉.
Lemma 5.2. There is an additive isomorphism
(M10,Z2) ≈ bo[v2],
where v2 ∈ Ext1,7(−).
Thus the chart for Ext
(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→ Σ7M10 → A2//A1 → M10 → 0.
This yields a spectral sequence which builds Ext
(M10,Z2) from
∗−i,∗−7i
(A2//A1,Z2).
Since Ext
(A2//A1,Z2) ≈ bo, one easily checks that there are no possible differen-
tials in this spectral sequence.
Let Cmn = H
∗(CPmn ;Z2).
24 DONALD M. DAVIS AND MARK MAHOWALD
Theorem 5.3. There is an additive isomorphism
(C∞−∞,Z2) ≈
Σ8p−2bo[v2].
Of course Σ applied to a module or an Ext group just means to increase the t-grading
by 1.
Proof. There is a filtration of C∞−∞ with Fp/Fp−1 ≈ Σ8p−2M10 for p ∈ Z. We have
Sq2 ι8p−2 = Sq
4 Sq2 Sq4 ι8p−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-isomorphism
claimed in the theorem.
Corollary 5.4. In gradings (t− s) less than −1,
(C−2−∞,Z2) ≈
Σ8p−2bo[v2].
Proof. There is an exact sequence
→ Exts−1,tA2 (C
−1,Z2)→ Ext
(C−2−∞,Z2)→ Ext
(C∞−∞,Z2)
q∗−→ Exts,tA2(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(C
−1,Z2).
The A-modules C∞1 and Σ
2C−2−∞ are dual. Thus, by [9, Prop 4],
(Z2,C
1 ) ≈ Ext
(Σ2C−2−∞,Z2).
There is a ring structure on Ext
(Z2,C
1 ). We deduce the following result, which
is pictured in Diagram 5.12.
Corollary 5.5. In (t− s) gradings ≤ 0, there is a ring isomorphism
(Z2,C
1 ) ≈ bo[v2][X ],
where X ∈ 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,−8 equal the class
in Ext0,−8i for each i > 0.
NONIMMERSIONS IMPLIED BY TMF, REVISITED 25
The Ext groups computed here are the E2-term of the ASS converging to tmf
−∗(CP∞).
We will consider the differentials in this spectral sequence after performing the Ext
calculation relevant for tmf∗(CP∞ × CP∞).
Now we consider C−2−∞⊗C−2−∞. 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 nontrivial
A2-morphism C
ρ−→ΣjZ2.
Lemma 5.6. Let
K = ker(C−2−∞ ⊗C−2−∞
ρ−→C−2−∞ ⊗ Σ−10Z2).
Let S denote the set of all classes x8i−21 x
2 with i ≤ −1 and j ≤ −2, together with
the classes x8i−21 x
2 with i ≤ −1 and j ≤ −1. Then K is the direct sum of a free
A2//E2-module on S with a single relation Sq
4 Sq2 Sq4(x−101 x
2 ) = 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
2 yields the additional
elements x−21 x
2 , x
2 , x
2 , x
x−21 x
2 , and x
2. The action of A2//E2 on x
2 yields the
additional elements x01x
2 + x
2, and x
2 + x
2. Each exponent can be decreased by any multiple of 8.
One can easily check that in each grading all classes in C−2−∞ ⊗C−2−∞ are obtained
exactly once from the described elements in K together with C−2−∞ ⊗ Σ−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 that it
generalizes to all gradings ≡ 4 mod 8. Letting Xi denote xi1x−28−i2 , we have:
(1) From generators in −28, we obtain just X−10 in K. The class
X−18 is in C
−∞ ⊗ Σ−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.
26 DONALD M. DAVIS AND MARK MAHOWALD
Note in (4) that X0 and X−28 do not appear because each component must be ≤ −4
and the components sum to −28.
One easily checks that the 11 classes listed above, including X−18, form a basis for
the space spanned by X−4, . . . , X−24, in an orderly fashion that clearly generalizes 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,C
1 )⊗ ExtA2(Z2,C∞1 )→ ExtA2(Z2,C∞1 ⊗C∞1 ).
Let Xi denote the class in grading −8 coming from the ith factor. Then we obtain
Theorem 5.7. The algebra Ext
(Z2,C
1 ⊗ C∞1 ) in gradings ≤ −8 is isomorphic
to Z2[X1, X2]〈X1X2, y−12〉 with y2−12 = X21X2 + X1X22 . The monomials of the form
X i1X
2y−12 are acted on freely by Z2[v0, v1, v2]. Let Sn denote the Z2-vector space with
basis the monomials X i1X
2 , and define a homomorphism ǫ : Sn → Z2 by sending
each monomial to 1. Then Z2[v0, v1, v2] acts freely on ker(ǫ), while bo[v2] acts freely
on Sn/ ker(ǫ). Thus in dimensions t − s ≤ −8 Ext∗,∗A2 (Z2,C
1 ⊗ C∞1 ) has, for each
i > 0, i copies of Σ−8i−4Z2[v0, v1, v2] and i copies of Σ
−8i−16Z2[v0, v1, v2], and also one
copy of Σ−8i−8bo[v2].
Here Z2[X1, X2]〈X1X2, y−12〉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
(Z2,C
1 ⊗C∞1 ) ≈ Ext
∗,∗−4
(C−2−∞ ⊗C−2−∞,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 Σ−10 in the cokernel part of Lemma 5.6, and
that Theorem 5.5 was only valid in dimension ≤ 0. In the range under consideration,
the relation on the top class in Lemma 5.6 does not affect Ext.
The ring structure in filtration 0 comes from HomA2(Z2,C
1 ⊗C∞1 ) being isomorphic
to elements of C∞1 ⊗C∞1 annihilated by Sq2 and Sq4, which has as basis all elements
x4i1 ⊗ x
2 and (x
1 ⊗ x
2 )(x
1 ⊗ x22 + x21 + x42).
NONIMMERSIONS IMPLIED BY TMF, REVISITED 27
Now we show that Ext
1,−8n+2
(Z2,C
1 ⊗ C∞1 ) = Z2, and h1 times each mono-
mial in Ext
0,−8n
(Z2,C
1 ⊗ C∞1 ) equals the nonzero element here. An element in
1,−8n+2
(Z2,C
1 ⊗C∞1 ) = Z2 is an equivalence class of morphisms
Σ2A2 ⊕ Σ4A2
h−→C∞1 ⊗C∞1
which increase grading by 8n− 2, and yield a trivial composite when preceded by
Σ4A2 ⊕ Σ8A2
Sq2 Sq6
0 Sq4
−−−−−−−−−→ Σ2A2 ⊕ Σ4A2.
Morphisms h which can be factored as
Σ2A2 ⊕ Σ4A2
Sq2,Sq4−−−−→ A2
k−→C∞1 ⊗C∞1 (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 Σ2A2 or Σ
4A2 to any one of the following classes:
2 , x
2 , x
(5.9)
The classes are listed in this order because any two adjacent monomials are equivalent
using as k in (5.8) the morphism sending the generator to the indicated classes in
succession:
2 , x
For example, (Sq2, Sq4)(x11x
2 ) = (x
2 , x
2 ). Thus all classes in (5.9) are equiva-
lent to one another.
That h1 times any monomialX
2 equals this nonzero element of Ext
1,8n+2
(Z2,C
C∞1 ) follows from usual Yoneda product consideration. If 0 ← Z2 ← C0 ← C1 ←
is the beginning of a minimal A2-resolution, with C1 = Σ
1A2 ⊕ Σ2A2 ⊕ Σ4A2,
then h1X
2 is represented by the composite C1 → C0 → C∞1 ⊗ C∞1 sending
ι2 7→ ι 7→ X i1Xn−i2 , and this is equivalent to the element described in the previous
paragraph.
Here is a schematic way of picturing Theorem 5.7. We first list the generators 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
28 DONALD M. DAVIS AND MARK MAHOWALD
left half of Diagram 5.11 arises from one tower in dimensions −24 and −16, while the
Z2[v0, v1, v2]-structure in the right half of diagram 5.11 arises from the other towers
in Diagram 5.10.
Diagram 5.10. Generators of ExtA2(Z2,C
1 ⊗C∞1 )
−28 −24 −20 −16 −12
✻✻✻ ✻✻ ✻✻ ✻ ✻
Diagram 5.11. Structure on two types of generators
0 010 10
✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻✻ ✻ ✻
Now we consider the differentials in the ASS converging to tmf∗(CP∞) and then for
tmf∗(CP∞ ∧ CP∞). The gradings are negated when considered as tmf-cohomology
groups. Corollary 5.5 gives the E2-term converging to [Σ
∗CP∞1 , tmf] ≈ tmf
−∗(CP∞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 v41 and v2 acting on generators in
lower grading. The elements indicated by •’s are involved in differentials, as explained
later.
NONIMMERSIONS IMPLIED BY TMF, REVISITED 29
Diagram 5.12. A portion of E2 for [Σ
∗CP∞, tmf]
−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 [Σ∗CP∞, tmf],
∗ < 1, are given by
−2k+1) = hǫ+11 v
for ǫ = 0, 1, i, j ≥ 0, k ≥ 1.
Here h1, v
1, and v2 have the usual Ext
s,t gradings (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. The
elements indicated by •’s are involved in the differentials. The resulting picture is
nicer if the filtrations of all classes built on X−2k+1 are increased by 1. There is
a nontrivial extension (multiplication by 2) in dimension −6 due to the preceding
differential. This is equivalent to the way that bu∗ is formed from bo∗ and Σ
2bo∗.
We obtain Diagram 5.14 from Diagram 5.12 after the differentials, extensions, and
filtration shift are taken into account.
30 DONALD M. DAVIS AND MARK MAHOWALD
Diagram 5.14. Diagram 5.12 after differentials and filtration shift
−16 −8 0
The regular sequence of towers in the chart beginning in filtration 1 in dimension −10
is interpreted as vi1v2, i ≥ 0.
After negating dimensions to switch to cohomology indexing, we obtain the follow-
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∗(CP∞1 ) ≈ Z(2)[Z16](bo∗ ⊕ v2Z(2)[v1, v2]).
Here Z16 ∈ tmf16(CP∞1 ), and |v1| = −2 and |v2| = −6.
Recall that bo∗ = bo−∗ with bo∗ as suggested in 5.1. Much of the ring structure
of tmf∗(CP∞1 ) 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∞) 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 [Σ−16CP, 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 θ, this bundle
satisfies w2(θ) and p1(θ) = 0. Here we use from [18] that p1 generates the infinite
cyclic summand in H4(BSO;Z) and satisfies r∗(p1) = c
1 − 2c2 under BU
r−→BSO,
and ρ∗(p1) = 2e1 under BSpin
ρ−→ BSO, where H4(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(θ)) = (c1(9H −H3))2 − 2c2(9H −H3) = (6x)2 − 2 · 18x2 = 0.
Thus e1(θ) = 0, hence CP
∞ θ−→BSpin → K(Z, 4) is trivial, and so θ lifts to a map
CP∞ → BO[8]. Hence its Thom spectrum induces a degree-1 map T (θ) → MO[8].
Since ψ3(H) = H3 −H , by [19] θ is J(2)-equivalent to 8(H − 1), and hence its Thom
spectrum is T (8(H − 1)) = Σ−16CP∞8 . Using the Ando-Hopkins-Rezk orientation
([1]) MO[8]→ tmf, we obtain our desired class as the composite
Σ−16CP∞1
col−→ Σ−16CP∞8
T (θ)−−→MO[8]→ tmf . (5.16)
We will deduce our differentials from the d3-differential E
3 → E
3 in the ASS
converging to π∗(tmf). This can be seen in [13, p.537] or [11, Thm 2.2]. See Remark
5.17 for additional explanation. It is not difficult to show that, with M10 as in 5.2,
the morphism
(Z2,Z2)→ Exts,tA2(M10,Z2)
induced by the nontrivial A2-map M10 → Z2 sends the Z2 in Ext7,23A2 (Z2,Z2) which is
not part of the infinite tower to h21v
We prefer to think about the ASS for tmf∗(Σ
2CP−2−∞), which, as we have noted,
is isomorphic to that of [Σ∗CP∞1 , tmf]. The E2-term was described in 5.4. Let
S−16 → Σ2CP−2−∞ ∧ tmf correspond to the map in (5.16). Since E2(CP−2−∞ ∧ tmf)
in negative dimensions is built from copies of ExtA2(M10,Z2), we deduce from the
previous paragraph that h21v
1v2g−16 in the ASS for tmf∗(Σ
2CP−2−∞) must be hit by 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 h1v
1g−8, as indicated by the dotted line in Diagram 5.12. Naturality
32 DONALD M. DAVIS AND MARK MAHOWALD
of differentials with respect to h1 and v
1 implies the differentials of 5.13 for ǫ = 0, 1,
all i, j = 0, and k = 1. Using the diagonal map of CP∞1 and the multiplication
of tmf, powers of (5.16) give similar nontrivial elements in [Σ−16kCP∞1 , tmf] for 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 is a map
S−10 → Σ2CP−2−∞ ∧ tmf hitting v2g−16, and the argument of the previous paragraph
implies that d2(h1v
1v2g−8) = h
2g−16 and then other related differentials. This
now establishes the differentials of 5.13 when j = 1, and sets in motion an inductive
argument to establish these differentials for all j ≥ 1.
No further differentials in the spectral sequence are possible, by dimensional 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 merely
refers to “a homotopy argument.” The current authors cited Giambalvo’s result in
[11] without additional argument. We provide some more detail here regarding this
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,
∗(MO[8]),Z2) ≈ Ext∗,∗A2(Z2 ⊕ Σ
Z2,Z2).
One way of obtaining the differentials from 15 to 14, as in [13], is to note that 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 product
formula B = 〈A, h0, h1〉, where A and B are as in Diagram 5.18. This can be seen
as v41 times 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 η.
NONIMMERSIONS IMPLIED BY TMF, REVISITED 33
But only 0 can be divisible by η 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
14 16 18
The differentials in the ASS converging to tmf∗(CP
−∞∧CP−2−∞) are implied by the
same considerations that worked for CP−2−∞. The Z2[v0, v1, v2]-parts in Theorem 5.7
cannot support differentials by dimensionality and h1-naturality. For the bo-like part,
we prefer thinking about it as [Σ∗+4CP∞1 ∧ CP∞1 , tmf] ≈ tmf−∗−4(CP∞1 ∧ CP∞1 ),
where the product structure is more apparent.
Let Zn denote the nonzero element of Ext
0,−8n
(Z2,C
1 ⊗C∞1 )/ ker(h1). By Theorem
5.7, Zn can be represented by X
2 for 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 infinite
cycles. Hence by the proof of Theorem 5.13,
2Z2k−1) = h
2 Z2k
for ǫ = 0, 1, i, j ≥ 0, and k ≥ 2.
Finally, X1X2 is an infinite cycle since there is nothing that it can hit. Also,
h1v2X1X2 and h
1v2X1X2 are not hit by differentials since Ext
(Z2,C
1 ⊗C∞1 ) = 0
by Theorem 5.7. We obtain the following.
Theorem 5.19. In grading ≥ 10, there is an isomorphism of graded abelian groups
tmf∗(CP∞1 ∧CP∞1 ) ≈ yZ(2)[v1, v2, X1, X2]⊕
In·Z(2)[v1, v2]⊕Z(2)[Z](bo∗⊕v2Z(2)[v1, v2]),
34 DONALD M. DAVIS AND MARK MAHOWALD
where |y| = 12, |Xi| = 8, |Z| = 16, |v1| = −2, and |v2| = −6. Here In = ker(Fn
ǫ−→Z),
where Fn is a free abelian group with basis {X i1Xn−i2 : 1 ≤ i < n}, and ǫ(X i1Xn−i2 ) = 1.
Thus In consists of all polynomials of grading n with sum of coefficients equal to 0.
We could have extended the description in 5.19 down to grading 8, but the description
would have been slightly more complicated, since it would include h1v2Z and h
1v2Z.
The motivation for this section was to see if perhaps
ker(tmf∗(CP∞ × CP∞) 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∞ ∧ CP∞). To obtain tmf∗(CP∞ × CP∞), we
add on two copies of tmf∗(CP∞), which was described in 5.15. Denote by Z1 and
Z2 the generators in tmf
16(CP∞ × CP∞). Monomials Z i1Zn−i2 should equal Zn of
5.19 plus perhaps elements of I2n of 5.19. The class y of 5.19 plus perhaps a sum 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∞ × RP∞)
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 Z2 × 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-
[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

  In a 2002 paper, the authors and Bruner used the new spectrum tmf to 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 RP^n 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)=X_1+X_2. 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 the original paper.
  Finally, we present a complete determination of tmf^{8*}(RP^\infty\times
RP^\infty) and tmf^*(CP^\infty\times CP^\infty) in positive dimensions.

Introduction
In [6], the authors and Bruner described a proof of the following theorem, along
with some additional nonimmersion results.
Theorem 1.1. ([6, 1.1]) Assume that M is divisible by the smallest 2-power greater
than or equal to h.
• If α(M) = 4h − 1, then P 8M+8h+2 cannot be immersed in ( 6⊆)
16M−8h+10.
• If α(M) = 4h− 2, then P 8M+8h 6⊆ R16M−8h+12.
Here and throughout, α(M) denotes the number of 1’s in the binary expansion of M ,
and P n denotes 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.
http://arxiv.org/abs/0704.0798v1
2 DONALD M. DAVIS AND MARK MAHOWALD
In [6], the theorem is followed by a comment that this is new provided α(M) ≥ 6,
i.e., h ≥ 2, and the first new result occurs for P 1536. In this note, we point out
that 1.1 is valid when h = 1, and these results are new when M is even, including
new nonimmersions of P n for 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 incorrect.
Letting h = 1 in 1.1, we have the following result.
Corollary 1.2. a. If α(M) = 3, then P 8M+10 6⊆ R16M+2.
b. If α(M) = 2, then P 8M+8 6⊆ 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, immersions,
nonembeddings, and embeddings of P n is presented, arranged according to n = 2i+d
with 0 ≤ d < 2i and 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 n with n as small as 56. The result of 1.2.b with M = 2i + 1 was also
proved very recently by Kitchloo and Wilson in [15]. This result for P 2
k+16, 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 proof of the
general case in [6] requires some extremely elaborate arguments and calculations. 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 argument
there was that an immersion of RP n in Rn+k implies existence of an axial map P n×
f−→ Pm+k for an appropriate value of m, and obtains a contradiction for certain
n, m, and k by consideration of tmf∗(f). Here tmf is the spectrum of topological
modular forms, which was discussed in [6]. A class X ∈ tmf8(P n) was described,
along with X1 = X × 1 and X2 = 1 × X in tmf8(P n × Pm). 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 Xℓ = 0 but (X1+X2)
ℓ 6= 0. We recently realized that
it is conceivable that f ∗(X) might contain other terms coming from tmf8(P n ∧ Pm).
In Section 3 (see Theorem 3.7) we perform a complete calculation of tmf∗(P∞×P∞)
in positive gradings divisible by 8, and in Section 4 we use it to show that effectively
f ∗(X) = u(X1 + X2), where u is a unit in tmf
∗(P∞ × P∞), which enables us to
retrieve all the nonimmersions of [6].
In Section 5, we compute tmf∗(CP∞ × CP∞) 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 × CP ). This approach to retrieving the nonimmersions of [6] did not yield
the desired result, but the later approach given in Section 4 did. Nevertheless the nice
result for tmf∗(CP∞ × CP∞) 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 least to
[14]. If P 8M+10 ⊆ R16M+2, then gd((2L+3 − 8M − 11)ξ8M+10) ≤ 8M − 8, hence this
bundle has (2L+3−16M −3) linearly independent sections, and thus there is an axial
P 8M+10 × P 2L+3−16M−4 f−→P 2L+3−8M−12.
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 tmf
8(−) described
in [6] and also in Section 1. In Section 4, we will show that we may assume that
f ∗(X) = X1+X2, as was done in [6], since this is true up to multiplication by a unit.
Since tmf2
L+3−8M−8(P 2
L+3−8M−12) = 0, we have
0 = f ∗(0) = f ∗(X2
L−M−1) = (X1+X2)
2L−M−1 ∈ tmf2L+3−8M−8(P 8M+10×P 2L+3−16M−4).
Expanding, we obtain
2L−M−1
XM+11 X
2L−2M−2
2L−M−1
XM1 X
2L−2M−1
2 as the only
terms which are possibly nonzero. Next we note that, with all u’s representing odd
integers,
2L−M−1
= 2α(M)−ν(M+1)u2 = 2
3−ν(M+1)u2,
4 DONALD M. DAVIS AND MARK MAHOWALD
where we have used α(M) = 3 at the last step. Here and throughout, ν(2eu) = e.
Similarly,
2L−M−1
= 2α(M)u4 = 2
3u4. Thus an immersion implies that in
L+3−8M−8(P 8M+10 × P 2L+3−16M−4), we have
23−ν(M+1)u2X
2L−2M−2
2 + 2
2L−2M−1
2 = 0.
(2.1)
We recall [6, 2.6], which states that there is an equivalence of spectra P k+8b+8 ∧
tmf ≃ Σ8P kb ∧ tmf. Combining this with duality, we obtain tmf
8M+8(P 8M+10) ≈
tmf−1(P−3) ≈ Z/8, and so 8XM+11 X2
L−2M−2
2 = 0. Here and throughout, Pn = P
RP∞/RP n−1. Similarly tmf2
L+3−16M−8(P 2
L+3−16M−4) ≈ tmf7(P3) ≈ Z/16, and hence
16XM1 X
2L−2M−1
2 = 0. Duality also implies
L+3−8M−8(P 8M+10 × P 2L+3−16M−4) ≈ tmf14(P−3 ∧ P3).
Calculations such as E2(tmf∗(P−3∧P3)), the E2-term of the Adams spectral sequence
(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
0 3 7 11 15
r r r r r�
r r r
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 paragraph of page
54 of [6], the class in (15, 0) corresponds to an easily-constructed nontrivial map. The
monomials XM+11 X
2L−2M−2
2 and X
2L−2M−1
2 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 filtration
0 is Z/8 ⊕ Z/16, we conclude that 8 and 16, respectively, are the precise orders of
the monomials. In particular, the order of XM1 X
2L−2M−1
2 is 16, and hence the class
in (2.1) is nonzero since it has a term 8uXM1 X
2L−2M−1
2 , and so (2.1) contradicts the
hypothesized immersion.
Part b of 1.2 is proved similarly. If P 8M+8 immerses in R16M+4, then there is an
axial map
P 8M+8 × P 2L+3−16M−6 f−→P 2L+3−8M−10,
and hence, up to odd multiples,
22−ν(M+1)XM+11 X
2L−2M−2
2 + 2
2XM1 X
2L−2M−1
2 (2.3)
= 0 ∈ tmf2L+3−8M−8(P 8M+8 ∧ P 2L+3−16M−6),
since α(M) = 2. We have tmf8M+8(P 8M+8) ≈ tmf−1(P−1) ≈ Z/2, and
L+3−16M−8(P 2
L+3−16M−6) ≈ 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 filtration).
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 classes, we
conclude that the orders of the monomials in (2.3) are precisely 2 and 8, respectively,
and so the term 4XM1 X
2L−2M−1
2 in (2.3) is nonzero, contradicting the immersion.
3. tmf-cohomology of P∞ × P∞
In this section, we compute tmf∗(P∞) and tmf8∗(P∞ × P∞) in positive gradings.
These will be used in the next section in studying the axial class in tmf-cohomology.
There is an element c4 ∈ π8(tmf) which reduces to v41 ∈ π8(bo); it has Adams
filtration 4. It acts on tmf∗(X) with degree −8. Recall also that π∗(bo) = bo∗ is as
depicted in 5.1. We denote bo∗ = bo−∗. We use P1 and P
∞ interchangeably.
6 DONALD M. DAVIS AND MARK MAHOWALD
Theorem 3.1. There is an element X ∈ tmf8(P1) of Adams filtration 0, described
in [6], such that, in positive dimensions divisible by 8, tmf∗(P1) is isomorphic as an
algebra over Z(2)[c4] to Z(2)[c4][X ]. In particular, each tmf
8i(P1) with i > 0 is a free
abelian group with basis {cj4X i+j : j ≥ 0}. There is a class L ∈ t0(P1) such that
• tmf0(P1) is a free abelian group with basis {L, cj4Xj : j ≥ 1},
• L2 = 2L and LX = 2X.
Moreover, in positive dimensions tmf∗(P1) is isomorphic as a graded abelian group 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 with 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 isomorphisms
tmf∗(P1) ≈ lim← tmf
∗(P n1 ) ≈ lim← tmf−∗−1(P
−n−1) = tmf−∗−1(P
(3.3)
Since H∗(tmf;Z2) ≈ A//A2, there is a spectral sequence converging to tmf∗(X) with
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
−∞) from the exact sequence
→ Es−1,t2 (P∞−1)→ E
−∞)→ E
q∗−→ Es,t2 (P∞−1)→ .
(3.4)
It was proved in [17] that
ExtA2(P
−∞,Z2) ≈
ExtA1(Σ
Z2,Z2).
Here we have initiated a notation that Pmn := H
∗(Pmn ). A complete calculation of
ExtA2(P
−1,Z2) was performed in [10], but all we need here are the first few groups.
We can now form a chart for E2(P
−∞) from (3.4), as in Diagram 3.5, where ◦ indicate
elements of ExtA2(P
−1,Z2) suitably positioned, and lines of negative slope correspond
to cases of q∗ 6= 0 in (3.4).
NONIMMERSIONS IMPLIED BY TMF, REVISITED 7
Diagram 3.5. tmf∗(P
−∞), −17 ≤ ∗ ≤ 2
−17 −9 −1
· · ·
✻ ✻ ✻
r r r
r r r
✻ ✻ ✻
r r r
r r r
r r r
r r r
r r r
Dualizing, we obtain Diagram 3.6 for the desired tmf∗(P∞1 ).
Diagram 3.6. tmf∗(P∞1 ), ∗ ≥ −2
0 8 16
✻ ✻ ✻✻ ✻ ✻
r r r
r r r
· · ·
Naming of the generators X i is clear since X has filtration 0. The free action of c4
is also clear. The class L is (up to sign) the composite P1
λ−→ S0 → tmf, where λ is
the well-known Kahn-Priddy map. Thus L is the image of a class L̂ ∈ π0(P1). Lin’s
theorem ([16]) says that π0(P1) ≈ Z∧2 , generated by L̂. Since π0(P1)→ ko0(P1) is an
isomorphism, and, since (1 − ξ)2 = 2(1 − ξ) for a generator (1 − ξ) of ko0(P1), we
obtain L̂2 = 2L̂, and hence also for L. We chose the generator to be (1 − ξ) rather
than (ξ − 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 = p
2, from which we
conclude p = 2.
In tmf∗(P1 × P1), for i = 1, 2, let Li and Xi denote the classes L and X in the ith
factor. Note that there is an isomorphism as tmf∗-modules, but not as rings,
tmf∗(P1 × P1) ≈ tmf∗(P1 ∧ P1)⊕ tmf∗(P1 × ∗)⊕ tmf∗(∗ × 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 X i1X
2 with i, j > 0
direct sum with a free Z[c4]-module with basis {L1X i2, X i1L2 : i ≥ 1}. The product
and Z[c4]-module structure is determined from 3.1 and
c4(X1X2) = (c4X1)X2 = X1(c4X2) =
4(L1X
1 L2),
for certain integers γi with γ0 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 similar
to (3.4). But to get started, we need ExtA2(P ⊗ P,Z2). Here we have begun to
abbreviate P := P∞−∞. We begin with a simple lemma. Throughout this section, x1
and x2 denote nonzero elements coming from the factors in H
1(RP × 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 x
0, which is an A-submodule.
A splitting morphism P⊗P g−→Z2 ⊗P is defined by g(xi1 ⊗ x
2) = x
1 ⊗ x
2 . This is
A-linear since
g(Sqk(xi1⊗x
2)) =
x01⊗x
i+j+k
x01⊗x
i+j+k
2 = Sq
k g(xi1⊗x
The following result is more substantial. We will prove it at the end of this section.
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) ≈ Σ8pA2/ Sq2, p ∈ Z,
and B has a filtration with
Fp(B)/Fp−1(B) ≈
Z copies
Σ4p−2A2/ Sq
1, p ∈ Z.
The generator of Fp(C)/Fp−1(C) is x
2 ; a basis over Z2 for C is
{x21xi+22 +x41xi2, x41xi2+x81xi−42 , i ∈ Z}∪{x11xi−12 +x21xi−22 , i 6≡ 0 (8)}∪{x11x
2 , p ∈ Z}.
A minimal set of generators as an A2-module for the filtration quotients of B is
{x8i−11 x
2 : i, j ∈ Z}.
Corollary 3.10. A chart for Ext
(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 classes is
repeated with vertical period 4. Thus, for example, in 8p−1 there is an infinite tower
emanating from filtration 4i for each i ≥ 0.
10 DONALD M. DAVIS AND MARK MAHOWALD
Diagram 3.11. Ext
(P⊗P,Z2) in 8p− 3 ≤ t− s ≤ 8p+ 4
8p+ −2 0 2 4
✻✻✻✻✻✻✻✻✻✻✻ ✻✻✻✻✻✻✻✻✻✻✻✻ ✻✻
✻✻ ✻✻
Proof of Corollary 3.10. We first note that ExtA2(P,Z2) is identical to the left portion
of Diagram 3.5 extended periodically in both directions. Also, ExtA2(A2/ Sq
1,Z2) ≈
ExtA0(Z2,Z2) is just an infinite tower, and
ExtA2(A2/ Sq
2,Z2) ≈ ExtA1(A1/ Sq2,Z2)
is given as in Diagram 3.14. We will show at the end of this proof that
ExtA2(C,Z2) ≈
ExtA2(Σ
8pA2/ Sq
2,Z2) (3.12)
and similarly
ExtA2(B,Z2) ≈
ExtA2(Σ
4p−2A2/ Sq
1,Z2).
These would follow by induction on p once you get started, but since p ranges over
all integers, that is not automatic.
Thus ExtA2(P⊗P,Z2) is formed from
ExtA2(P,Z2)⊕
ExtA2(Σ
8pA2/ Sq
2,Z2)⊕
ExtA2(Σ
4p−2A2/ Sq
1,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 there are
no possible boundary morphisms in the Ext sequence of 3.9, yielding the claim of 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, Ext
(Fp0(C),Z2) = 0. Actually a
much sharper lower vanishing line can be established, but this is good enough for our
purposes. Thus, for this (s, t),
(Fp1(C),Z2) ≈
(Σ8p−2A2/ Sq
2) (3.13)
for p1 ≤ p0, as both are 0. Let p1 be minimal such that (3.13) does not hold. Then
comparison of exact sequences implies that
s−1,t
(Fp1−1(C),Z2)→ Ext
(Fp1(C)/Fp1−1(C),Z2)
must be nonzero. But one or the other of these groups is always 0,1 as both charts
(Fp1−1(C),Z2) and Ext
(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, and hence
(3.12) holds. A similar proof works when C is replaced by B.
Diagram 3.14. ExtA2(A2/ Sq
2,Z2)
· · ·
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 this
result is pictured in Diagram 3.18.
Proposition 3.15. In dimensions t− s ≡ 2 mod 4 with t− s ≤ −10, ExtA2(P−2−∞⊗
P−2−∞,Z2) consists of i infinite towers emanating from filtration 0 in dimensions −8i−
1Actually this is not quite true; for one family of elements we need to use h0-
naturality.
12 DONALD M. DAVIS AND MARK MAHOWALD
6 and −8i − 10, together with the relevant portion of two copies of ExtA1(Z2,Z2)
beginning in filtration 1 in each dimension −8i − 2. The generators of the towers in
−8i− 10 correspond to cohomology classes x−91 x−8i−12 , . . . , x−8i−11 x−92 . The generators
of the two copies of ExtA1(Z2,Z2) in −8i−2 arise from h0 times classes corresponding
to x−11 x
2 and x
−8i−1
Proof. Using exact sequences like (3.4) on each factor, we build Ext
(P−2−∞⊗P−2−∞,Z2)
fromA := Ext
(P⊗P,Z2), B := Ext∗−1,∗A2 (P
−1⊗P,Z2), C := Ext
∗−1,∗
(P⊗P∞−1,Z2),
and D := Ext
∗−2,∗
(P∞−1 ⊗ P∞−1,Z2), with possible d1-differential from A and into D.
In the range of concern, t− s ≤ −9, the D-part will not be present, and the part 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, together
with B and C, is pictured in Diagram 3.16.
Diagram 3.16. Portion of A+B+C
✻✻✻✻✻✻✻✻✻✻ ✻✻✻✻✻✻✻✻✻✻ ✻✻✻✻✻✻✻✻✻✻
−2 2 68p+
rr rrr
In dimension 8p−2, the towers in A arise from all cohomology classes x−8i−11 x
−8j−1
with i+ j = −p, while in dimension 8p+ 2, they arise from x8i−11 x
2 ∼ x8i+31 x
The finite towers in B arise from x4i−11 x
2 with i ≥ 0, and those from C from
x8i−11 x
2 with j ≥ 0. The homomorphism
Ext0A2(P⊗P,Z2)→ Ext
(P∞−1 ⊗P,Z2)⊕ Ext0A2(P⊗P
−1,Z2),
NONIMMERSIONS IMPLIED BY TMF, REVISITED 13
which is equivalent to the d1-differential mentioned above, sends classes to those with
the same name. In dimension ≤ −10, this is surjective, with kernel spanned by classes
with both components < −1. In dimension −8i−6 and −8i−10, there will be i such
classes. We illustrate by listing the classes in the first few gradings:
−14 : x−91 x−52 ∼ x−51 x−92
−18 : x−91 x−92
−22 : x−171 x−52 ∼ x−131 x−92 , x−91 x−132 ∼ x−51 x−172
−26 : x−171 x−92 , x−91 x−172 .
These kernel classes yield infinite towers emanating from filtration 0.
For each p < 0, the towers arising from x
2 , j ≥ 0, in A combine with those
in the p-summand of
ExtA1(Σ
8p−1P∞−1,Z2)
as in Diagram 3.17 to yield one of the copies of ExtA1(Z2,Z2) arising from filtration
1. An identical picture results when the factors are reversed.
Diagram 3.17. Part of ExtA2(P
−∞ ⊗P−2−∞,Z2)
✻ ✻ ✻
Putting things together, we obtain that in dimensions less than −8, ExtA2(P−2−∞⊗
P−2−∞,Z2) consists of a chart described in Proposition 3.15 and partially illustrated
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
−26 −18 −10
✻✻✻✻✻✻ ✻ ✻✻✻
✻✻✻✻ ✻✻
The only possible differentials in the Adams spectral sequence of P−2−∞∧P−2−∞∧ 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]. Similarly to
(3.3), we have
tmf∗(P1 ∧ P1) ≈ tmf−∗−2(P−2−∞ ∧ P−2−∞),
and so we obtain a turned-around version of Diagram 3.18, of the same general sort
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 X i1X
2 described 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). Simi-
larly the classes L1X
2 and X
1L2 have Adams filtration 1, and so one would anticipate
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 spectral
sequence; however, the Atiyah-Hirzebruch spectral sequence shows this quite clearly.
The class X i1 is detected by H
8i(P1; π0(tmf)), while L is detected by H
1(P1; π1(tmf)).
NONIMMERSIONS IMPLIED BY TMF, REVISITED 15
Under the pairing, their product is detected in H8i+1(P1; π1(tmf)), clearly of Adams
filtration 1.
The last part of Theorem 3.7 deals with the action of c4 on the monomials X
Since tmf is a commutative ring spectrum, tmf∗(P1 ∧ P1) is a graded commutative
algebra over tmf∗. The action c4(X1X2) must be of the form
i≥0 γic
4(L1X
as these are the only elements in tmf8(P1∧P1), and the class must be invariant under
reversing factors. The divisibility of γ0 by 8 follows since c4 has Adams filtration 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 x11x
2 , p ∈ Z. Note that Sq2(x11x
2 ) = Sq
4 Sq6(x11x
2 ). Thus a basis
of A2/ Sq
2 acting on all x11x
2 spans C. The 24 elements in a basis of A/ Sq
acting on x11x
2 yield x
2 + x
2 + x
2 + x
2 , x
2 + x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
2 , x
and x41x
2 + x
2 . These classes with second components shifted by all multiples 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 more
elaborate. For the 32 elements θ in a basis of A2/ Sq
1, we list θ(x−11 x
2 ) and θ(x
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 elements
θ(x8i−11 x
2 ) and θ(x
2 ). We will show Q = 0. This will complete the proof of
Proposition 3.9, implying in particular that Sq1(x8i−11 x
2 ) and Sq
1(x8i−11 x
2 ) 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 x
i depends only on i mod 8. We illustrate with
the case in which degree ≡ 0 mod 8. The other 7 congruences are handled similarly,
although some are a bit more complicated.
A basis of A2/ Sq
1 in degree ≡ 2 mod 8 acting on x−11 x−12 yields the following
elements: x−11 x
2 + x
2 + x
2 , x
2 + x
2 + x
2 + x
2 + x
2 + x
16 DONALD M. DAVIS AND MARK MAHOWALD
and x41x
2 + x
2. A basis of A2/ Sq
1 in degree ≡ 6 mod 8 acting on x−11 x32 yields
the following elements: x21x
2 + x
2 + x
2 + x
2 + x
2 + x
2 + x
2 + x
2, and x
2 + x
2. Because we allow both components 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−11 x
2 + x
2 + x
2 become 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) are 0 in
Q. First, R8 implies that all classes X8i are congruent to one another. Since X0 is
0 in the quotient due to P/Z2, we conclude that all classes X8i are 0 in Q. Next,
R4 implies that all X8i+4 are congruent to one another. Since X4 +X8 ∈ C, and we
have just shown that X8 ≡ 0 in Q, we deduce that all X8i+4 are 0 in Q. Now we
use R2 + R7 to see that all X8i+2 + X8i+4 are congruent to one another, then that
X2 +X4 ∈ 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+1 are congruent to one
another, and since X1 ∈ 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 careful consid-
eration of the possible “other terms” in the axial class discussed in the Introduction.
We show that, at least as far as the monomials cX i1X
2 in its powers are concerned,
NONIMMERSIONS IMPLIED BY TMF, REVISITED 17
the axial class equals u(X1+X2), where u is a unit in tmf
0(RP∞×RP∞). Thus the
ℓth power of the axial class is nonzero in tmf8ℓ(RP n×RPm) if and only if (X1+X2)ℓ
is nonzero there, and the latter is the condition which yielded the nonimmersions of
[6, 1.1]. Thus we have a complete proof of [6, 1.1].
If P n × Pm f−→Pm+k is an axial map, then there is a commutative diagram
P n × Pm f−−−→ Pm+k
P∞ × P∞ g−−−→ P∞,
where g is the standard multiplication of P∞, since P∞ = K(Z2, 1). Since X ∈
tmf8(Pm+k) has been chosen to extend over P∞, we obtain that f ∗(X) is the restric-
tion of g∗(X). By Theorem 3.7 and the symmetry of g, we must have
g∗(X) = X1 +X2 +
4(L1X
1 L2), (4.1)
for some integers κi. This is what we call the “axial class.” Then g
∗(Xℓ) equals the
ℓth power of (4.1). Using the formulas for L2i , LiXi, and c4(X1X2) in 3.1 and 3.7 and
the binomial theorem, this ℓth power can be written in terms of the basis described
in 3.7. If some κi’s are nonzero, the coefficients of X
2 in g
∗(Xℓ) will not equal
, 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 like 2
when multiplying by Xi, and if, for example, L1 is present without X1, then the terms
ci4L1X
2 cannot cancel our X
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 throughout
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 − 80X42 is the same as that in
(X1 +X2)
4, and L1X
2 is a separate basis element.
With this simplification, the axial class in (4.1) becomes
X1 +X2 + 2
2 ) (4.2)
18 DONALD M. DAVIS AND MARK MAHOWALD
for some integers κi. There was another term 2κ0(X1+X2), but it can be incorporated
into the leading (X1 +X2). The odd multiple that it can create is not important.
From Theorem 3.7, we have
c4(X1X2) = 16(X1 +X2) + 2
(4.3)
for some integers γk. The 16 comes from γ0 = 8 and Li = 2. Actually we don’t really
know that γ0 = 8, even just up to multiplication by a unit, but it is divisible by 8
and the possibility of equality must be allowed for. This gives
2 ) = 16(X
2 ) + 2
i+k+1
j+k+1
(4.4)
Here we use that in a graded tmf∗-algebra tmf
∗(X) with even-degree elements, c(xy) =
cx · y, for c ∈ tmf∗ and x, y ∈ tmf∗(X).
There is an iterative nature to the action of c4 in (4.4), but the leading coefficient
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 equals
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 Ai such that
24+iAi
Remark 4.6. This formula will be evaluated on (i.e. multiplied by) monomialsXk1X
One might worry that the negative powers of X1 or X2 in 4.5 will cause nonsensical
negative powers in c4X
2. This will, in fact, not occur because the monomials on
which we act always have total degree greater than the dimension of either factor.
Thus if, after multiplication by c4, a term with negative exponent of Xi appears, then
the accompanying X
3−i-term will be 0 for dimensional reasons.
Proof of Theorem 4.5. The defining equation (4.3) may be written as, with θ =
X1X2 and z =
X1/X2,
θ = 16(z + z−1) +
i(zi+1 + z−(i+1)). (4.7)
NONIMMERSIONS IMPLIED BY TMF, REVISITED 19
Let pi = z
i + z−i. We will show that
24+iAip2i+1 (4.8)
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
k = pΣiei + L,
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 γi which occur in (4.7). This is allowable
if we agree that when collecting terms, we only make crude estimates about their
2-divisibility. We have
θ = 16p1 + 2θp2 + 2θ
2p3 + 2θ
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 factor.
Now let T1 = 16p1 and, for i ≥ 2, let Ti = 2θi−1pi. Each term in the expansion of
θ 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 θi−1. For each of these Tj with j > 1,
choose j − 1 additional factors, and continue this procedure. This builds a tree, and
we don’t get an explicit product term until every branch ends with T1. Each selected
factor Tj with j > 1 contributes a factor 2pj. There will also be binomial coefficients
and the omitted γi’s occurring as additional factors.
For example, Diagram 4.9 illustrates the choices leading to one term in the expan-
sion of θ. This yields the term 2p2 · 2p4 · 16p1 · 2p2 · 16p1 · 2p3 · 16p1 · 2p2 · 16p1, which
equals 221(p17 +L), where L is a sum of pi with i < 17 and i odd. By induction, one
sees in general that the sum of the subscripts emanating from any node, including
the subscript of the node itself, is odd.
20 DONALD M. DAVIS AND MARK MAHOWALD
Diagram 4.9. A possible choice of terms
T2 T4 T2 T1
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 2
k+4(p2k+1+L).
Note that a term 2k+4p2i+1 with i < k obtained from L will be more 2-divisible than
the 2i+4p2i+1 term 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 then
T2 then T1. Incorporating γi’s may make terms even more divisible, but the claim of
(4.8) is only that p2i+1 occurs with coefficient divisible by 2
Now we incorporate 4.5 into (4.2) to obtain the following key result, which we prove
at the end of the section.
Theorem 4.10. The monomials ciX
2 in the nth power of the axial class in
tmf8n(RP∞ × RP∞) are equal to those in the nth power of
(X1 +X2)
24+iαi
, (4.11)
where u is an odd 2-adic integer and αi are 2-adic integers.
The factor which accompanies (X1+X2) in (4.11) is a unit in tmf
∗(RP∞×RP∞);
we referred to it earlier as u. Indeed, its inverse is a series of the same form, obtained
by solving a sequence of equations. This justifies the claim in the first paragraph of
this section regarding retrieval of the nonimmersions of [6, 1.1].
We must also observe that restriction to tmf8ℓ(RP n × RPm) of the non-X i1Xℓ−i2
parts of the basis of tmf8ℓ(RP∞ × RP∞) cannot cancel the X i1Xℓ−i2 terms essential
for the nonimmersion. This is proved by noting that these elements such as L1X
NONIMMERSIONS IMPLIED BY TMF, REVISITED 21
and ci4L1X
2 will restrict to a class of the same name in tmf
8ℓ(RP n × RPm), and
will be 0 there for dimensional reasons, since 8ℓ > 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
We let z =
X1/X2 and pj = z
j + z−j as in the proof of 4.5.
The summand with i = 2t becomes
2κi(X1 +X2)
X t1X
2jAjp2j+1
= 2κi(X1 +X2)(p2t + L)24i
k(p2k+i + L).
Here k is a sum of j-values taken from the various factors in the ith power. Also, in
pj + L, L denotes a combination of pt’s with t < j. Noting (p2t + L)(p2k+i + L) =
p2k+2i + L, this becomes
2(X1 +X2)2
k(p2k+2i + L). (4.12)
The argument when i = 2t + 1 is similar but slightly more complicated because
(X i+11 +X
2 ) is not divisible by (X1 +X2). We obtain
X i+11 +X
X1X2)2t+1
2jAjp2j+1
For one of the factors of the ith power, say the first, we treat p2j+1 as
X1+X2√
(p2j +L).
The expression then becomes
2(X1 +X2)pi+12
k(p2k+i−1 + L),
where k is obtained as in the previous case. We again obtain (4.12).
Thus when g∗(X) − (X1 +X2) is written as (X1 + X2)
βjp2j , the coefficient βj
satisfies ν(βj) ≥ (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 all along.
This yields the claim of (4.11).
22 DONALD M. DAVIS AND MARK MAHOWALD
5. tmf-cohomology of CP∞ × CP∞
In [2], [4], and [8], it was noted, first by Astey, that the axial class using BP (or
BP 〈2〉) was u(X2 − X1), where u is a unit in BP ∗(P∞ ∧ P∞). 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 section
unnecessary. To do this, we calculate tmf∗(CP∞) and tmf∗(CP∞×CP∞) 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∞ and CP = CP∞
dR−−−→ RP × RP mR−−−→ RP
dC−−−→ CP × CP CP
1×(−1)
CP × CP mC−−−→ CP
The generator XR ∈ BP 2(RP ) satisfies XR = h∗(X). We also have that mC ◦ (1 ×
(−1))◦dC is null-homotopic. The key fact, which will fail for tmf, is BP ∗(CP×CP ) ≈
BP ∗[X1, X2].
The axial class is m∗R(XR). It equals (h× h)∗(1× (−1))∗m∗C(X). But
(1× (−1))∗m∗C(X) ∈ ker(d∗C).
By the above “key fact,” d∗C is the projection BP
∗[X1, X2]→ BP ∗[X ] in which each
Xi 7→ X . The kernel of this projection is the ideal (X2 −X1). To see this, just note
that in grading 2n a kernel element must be
2 with
ci = 0, and hence is
2 −Xn1 ) =
1(X2 −X1)
n−i−1−j
Thus (1 × (−1))∗m∗C(X) = (X2 − X1)u for some u ∈ BP ∗(CP × CP ). This u is
a unit by consideration of its reduction to H∗(−;Z), as in [2]. Since h∗(u) will then
be a unit in BP ∗(RP ×RP ) and h∗(Xi) = XRi, we obtain the claim about the axial
class being a unit times XR2 −XR1.
NONIMMERSIONS IMPLIED BY TMF, REVISITED 23
In order to see if there is any chance of adapting this to tmf, we compute tmf∗(CP∞)
and tmf∗(CP∞ × CP∞) in positive gradings. We begin with the relevant Ext calcu-
lations.
Let bo = Ext
(Z2,Z2). Recall that a chart for this is given as in Diagram 5.1,
extended with period (t− s, s) = (8, 4).
Diagram 5.1. Ext
(Z2,Z2)
0 4 8
· · ·
Let M10 denote the A2-module 〈1, Sq4, Sq2 Sq4, Sq4 Sq2 Sq4〉.
Lemma 5.2. There is an additive isomorphism
(M10,Z2) ≈ bo[v2],
where v2 ∈ Ext1,7(−).
Thus the chart for Ext
(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→ Σ7M10 → A2//A1 → M10 → 0.
This yields a spectral sequence which builds Ext
(M10,Z2) from
∗−i,∗−7i
(A2//A1,Z2).
Since Ext
(A2//A1,Z2) ≈ bo, one easily checks that there are no possible differen-
tials in this spectral sequence.
Let Cmn = H
∗(CPmn ;Z2).
24 DONALD M. DAVIS AND MARK MAHOWALD
Theorem 5.3. There is an additive isomorphism
(C∞−∞,Z2) ≈
Σ8p−2bo[v2].
Of course Σ applied to a module or an Ext group just means to increase the t-grading
by 1.
Proof. There is a filtration of C∞−∞ with Fp/Fp−1 ≈ Σ8p−2M10 for p ∈ Z. We have
Sq2 ι8p−2 = Sq
4 Sq2 Sq4 ι8p−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-isomorphism
claimed in the theorem.
Corollary 5.4. In gradings (t− s) less than −1,
(C−2−∞,Z2) ≈
Σ8p−2bo[v2].
Proof. There is an exact sequence
→ Exts−1,tA2 (C
−1,Z2)→ Ext
(C−2−∞,Z2)→ Ext
(C∞−∞,Z2)
q∗−→ Exts,tA2(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(C
−1,Z2).
The A-modules C∞1 and Σ
2C−2−∞ are dual. Thus, by [9, Prop 4],
(Z2,C
1 ) ≈ Ext
(Σ2C−2−∞,Z2).
There is a ring structure on Ext
(Z2,C
1 ). We deduce the following result, which
is pictured in Diagram 5.12.
Corollary 5.5. In (t− s) gradings ≤ 0, there is a ring isomorphism
(Z2,C
1 ) ≈ bo[v2][X ],
where X ∈ 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,−8 equal the class
in Ext0,−8i for each i > 0.
NONIMMERSIONS IMPLIED BY TMF, REVISITED 25
The Ext groups computed here are the E2-term of the ASS converging to tmf
−∗(CP∞).
We will consider the differentials in this spectral sequence after performing the Ext
calculation relevant for tmf∗(CP∞ × CP∞).
Now we consider C−2−∞⊗C−2−∞. 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 nontrivial
A2-morphism C
ρ−→ΣjZ2.
Lemma 5.6. Let
K = ker(C−2−∞ ⊗C−2−∞
ρ−→C−2−∞ ⊗ Σ−10Z2).
Let S denote the set of all classes x8i−21 x
2 with i ≤ −1 and j ≤ −2, together with
the classes x8i−21 x
2 with i ≤ −1 and j ≤ −1. Then K is the direct sum of a free
A2//E2-module on S with a single relation Sq
4 Sq2 Sq4(x−101 x
2 ) = 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
2 yields the additional
elements x−21 x
2 , x
2 , x
2 , x
x−21 x
2 , and x
2. The action of A2//E2 on x
2 yields the
additional elements x01x
2 + x
2, and x
2 + x
2. Each exponent can be decreased by any multiple of 8.
One can easily check that in each grading all classes in C−2−∞ ⊗C−2−∞ are obtained
exactly once from the described elements in K together with C−2−∞ ⊗ Σ−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 that it
generalizes to all gradings ≡ 4 mod 8. Letting Xi denote xi1x−28−i2 , we have:
(1) From generators in −28, we obtain just X−10 in K. The class
X−18 is in C
−∞ ⊗ Σ−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.
26 DONALD M. DAVIS AND MARK MAHOWALD
Note in (4) that X0 and X−28 do not appear because each component must be ≤ −4
and the components sum to −28.
One easily checks that the 11 classes listed above, including X−18, form a basis for
the space spanned by X−4, . . . , X−24, in an orderly fashion that clearly generalizes 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,C
1 )⊗ ExtA2(Z2,C∞1 )→ ExtA2(Z2,C∞1 ⊗C∞1 ).
Let Xi denote the class in grading −8 coming from the ith factor. Then we obtain
Theorem 5.7. The algebra Ext
(Z2,C
1 ⊗ C∞1 ) in gradings ≤ −8 is isomorphic
to Z2[X1, X2]〈X1X2, y−12〉 with y2−12 = X21X2 + X1X22 . The monomials of the form
X i1X
2y−12 are acted on freely by Z2[v0, v1, v2]. Let Sn denote the Z2-vector space with
basis the monomials X i1X
2 , and define a homomorphism ǫ : Sn → Z2 by sending
each monomial to 1. Then Z2[v0, v1, v2] acts freely on ker(ǫ), while bo[v2] acts freely
on Sn/ ker(ǫ). Thus in dimensions t − s ≤ −8 Ext∗,∗A2 (Z2,C
1 ⊗ C∞1 ) has, for each
i > 0, i copies of Σ−8i−4Z2[v0, v1, v2] and i copies of Σ
−8i−16Z2[v0, v1, v2], and also one
copy of Σ−8i−8bo[v2].
Here Z2[X1, X2]〈X1X2, y−12〉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
(Z2,C
1 ⊗C∞1 ) ≈ Ext
∗,∗−4
(C−2−∞ ⊗C−2−∞,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 Σ−10 in the cokernel part of Lemma 5.6, and
that Theorem 5.5 was only valid in dimension ≤ 0. In the range under consideration,
the relation on the top class in Lemma 5.6 does not affect Ext.
The ring structure in filtration 0 comes from HomA2(Z2,C
1 ⊗C∞1 ) being isomorphic
to elements of C∞1 ⊗C∞1 annihilated by Sq2 and Sq4, which has as basis all elements
x4i1 ⊗ x
2 and (x
1 ⊗ x
2 )(x
1 ⊗ x22 + x21 + x42).
NONIMMERSIONS IMPLIED BY TMF, REVISITED 27
Now we show that Ext
1,−8n+2
(Z2,C
1 ⊗ C∞1 ) = Z2, and h1 times each mono-
mial in Ext
0,−8n
(Z2,C
1 ⊗ C∞1 ) equals the nonzero element here. An element in
1,−8n+2
(Z2,C
1 ⊗C∞1 ) = Z2 is an equivalence class of morphisms
Σ2A2 ⊕ Σ4A2
h−→C∞1 ⊗C∞1
which increase grading by 8n− 2, and yield a trivial composite when preceded by
Σ4A2 ⊕ Σ8A2
Sq2 Sq6
0 Sq4
−−−−−−−−−→ Σ2A2 ⊕ Σ4A2.
Morphisms h which can be factored as
Σ2A2 ⊕ Σ4A2
Sq2,Sq4−−−−→ A2
k−→C∞1 ⊗C∞1 (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 Σ2A2 or Σ
4A2 to any one of the following classes:
2 , x
2 , x
(5.9)
The classes are listed in this order because any two adjacent monomials are equivalent
using as k in (5.8) the morphism sending the generator to the indicated classes in
succession:
2 , x
For example, (Sq2, Sq4)(x11x
2 ) = (x
2 , x
2 ). Thus all classes in (5.9) are equiva-
lent to one another.
That h1 times any monomialX
2 equals this nonzero element of Ext
1,8n+2
(Z2,C
C∞1 ) follows from usual Yoneda product consideration. If 0 ← Z2 ← C0 ← C1 ←
is the beginning of a minimal A2-resolution, with C1 = Σ
1A2 ⊕ Σ2A2 ⊕ Σ4A2,
then h1X
2 is represented by the composite C1 → C0 → C∞1 ⊗ C∞1 sending
ι2 7→ ι 7→ X i1Xn−i2 , and this is equivalent to the element described in the previous
paragraph.
Here is a schematic way of picturing Theorem 5.7. We first list the generators 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
28 DONALD M. DAVIS AND MARK MAHOWALD
left half of Diagram 5.11 arises from one tower in dimensions −24 and −16, while the
Z2[v0, v1, v2]-structure in the right half of diagram 5.11 arises from the other towers
in Diagram 5.10.
Diagram 5.10. Generators of ExtA2(Z2,C
1 ⊗C∞1 )
−28 −24 −20 −16 −12
✻✻✻ ✻✻ ✻✻ ✻ ✻
Diagram 5.11. Structure on two types of generators
0 010 10
✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻✻ ✻ ✻
Now we consider the differentials in the ASS converging to tmf∗(CP∞) and then for
tmf∗(CP∞ ∧ CP∞). The gradings are negated when considered as tmf-cohomology
groups. Corollary 5.5 gives the E2-term converging to [Σ
∗CP∞1 , tmf] ≈ tmf
−∗(CP∞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 v41 and v2 acting on generators in
lower grading. The elements indicated by •’s are involved in differentials, as explained
later.
NONIMMERSIONS IMPLIED BY TMF, REVISITED 29
Diagram 5.12. A portion of E2 for [Σ
∗CP∞, tmf]
−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 [Σ∗CP∞, tmf],
∗ < 1, are given by
−2k+1) = hǫ+11 v
for ǫ = 0, 1, i, j ≥ 0, k ≥ 1.
Here h1, v
1, and v2 have the usual Ext
s,t gradings (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. The
elements indicated by •’s are involved in the differentials. The resulting picture is
nicer if the filtrations of all classes built on X−2k+1 are increased by 1. There is
a nontrivial extension (multiplication by 2) in dimension −6 due to the preceding
differential. This is equivalent to the way that bu∗ is formed from bo∗ and Σ
2bo∗.
We obtain Diagram 5.14 from Diagram 5.12 after the differentials, extensions, and
filtration shift are taken into account.
30 DONALD M. DAVIS AND MARK MAHOWALD
Diagram 5.14. Diagram 5.12 after differentials and filtration shift
−16 −8 0
The regular sequence of towers in the chart beginning in filtration 1 in dimension −10
is interpreted as vi1v2, i ≥ 0.
After negating dimensions to switch to cohomology indexing, we obtain the follow-
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∗(CP∞1 ) ≈ Z(2)[Z16](bo∗ ⊕ v2Z(2)[v1, v2]).
Here Z16 ∈ tmf16(CP∞1 ), and |v1| = −2 and |v2| = −6.
Recall that bo∗ = bo−∗ with bo∗ as suggested in 5.1. Much of the ring structure
of tmf∗(CP∞1 ) 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∞) 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 [Σ−16CP, 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 θ, this bundle
satisfies w2(θ) and p1(θ) = 0. Here we use from [18] that p1 generates the infinite
cyclic summand in H4(BSO;Z) and satisfies r∗(p1) = c
1 − 2c2 under BU
r−→BSO,
and ρ∗(p1) = 2e1 under BSpin
ρ−→ BSO, where H4(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(θ)) = (c1(9H −H3))2 − 2c2(9H −H3) = (6x)2 − 2 · 18x2 = 0.
Thus e1(θ) = 0, hence CP
∞ θ−→BSpin → K(Z, 4) is trivial, and so θ lifts to a map
CP∞ → BO[8]. Hence its Thom spectrum induces a degree-1 map T (θ) → MO[8].
Since ψ3(H) = H3 −H , by [19] θ is J(2)-equivalent to 8(H − 1), and hence its Thom
spectrum is T (8(H − 1)) = Σ−16CP∞8 . Using the Ando-Hopkins-Rezk orientation
([1]) MO[8]→ tmf, we obtain our desired class as the composite
Σ−16CP∞1
col−→ Σ−16CP∞8
T (θ)−−→MO[8]→ tmf . (5.16)
We will deduce our differentials from the d3-differential E
3 → E
3 in the ASS
converging to π∗(tmf). This can be seen in [13, p.537] or [11, Thm 2.2]. See Remark
5.17 for additional explanation. It is not difficult to show that, with M10 as in 5.2,
the morphism
(Z2,Z2)→ Exts,tA2(M10,Z2)
induced by the nontrivial A2-map M10 → Z2 sends the Z2 in Ext7,23A2 (Z2,Z2) which is
not part of the infinite tower to h21v
We prefer to think about the ASS for tmf∗(Σ
2CP−2−∞), which, as we have noted,
is isomorphic to that of [Σ∗CP∞1 , tmf]. The E2-term was described in 5.4. Let
S−16 → Σ2CP−2−∞ ∧ tmf correspond to the map in (5.16). Since E2(CP−2−∞ ∧ tmf)
in negative dimensions is built from copies of ExtA2(M10,Z2), we deduce from the
previous paragraph that h21v
1v2g−16 in the ASS for tmf∗(Σ
2CP−2−∞) must be hit by 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 h1v
1g−8, as indicated by the dotted line in Diagram 5.12. Naturality
32 DONALD M. DAVIS AND MARK MAHOWALD
of differentials with respect to h1 and v
1 implies the differentials of 5.13 for ǫ = 0, 1,
all i, j = 0, and k = 1. Using the diagonal map of CP∞1 and the multiplication
of tmf, powers of (5.16) give similar nontrivial elements in [Σ−16kCP∞1 , tmf] for 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 is a map
S−10 → Σ2CP−2−∞ ∧ tmf hitting v2g−16, and the argument of the previous paragraph
implies that d2(h1v
1v2g−8) = h
2g−16 and then other related differentials. This
now establishes the differentials of 5.13 when j = 1, and sets in motion an inductive
argument to establish these differentials for all j ≥ 1.
No further differentials in the spectral sequence are possible, by dimensional 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 merely
refers to “a homotopy argument.” The current authors cited Giambalvo’s result in
[11] without additional argument. We provide some more detail here regarding this
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,
∗(MO[8]),Z2) ≈ Ext∗,∗A2(Z2 ⊕ Σ
Z2,Z2).
One way of obtaining the differentials from 15 to 14, as in [13], is to note that 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 product
formula B = 〈A, h0, h1〉, where A and B are as in Diagram 5.18. This can be seen
as v41 times 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 η.
NONIMMERSIONS IMPLIED BY TMF, REVISITED 33
But only 0 can be divisible by η 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
14 16 18
The differentials in the ASS converging to tmf∗(CP
−∞∧CP−2−∞) are implied by the
same considerations that worked for CP−2−∞. The Z2[v0, v1, v2]-parts in Theorem 5.7
cannot support differentials by dimensionality and h1-naturality. For the bo-like part,
we prefer thinking about it as [Σ∗+4CP∞1 ∧ CP∞1 , tmf] ≈ tmf−∗−4(CP∞1 ∧ CP∞1 ),
where the product structure is more apparent.
Let Zn denote the nonzero element of Ext
0,−8n
(Z2,C
1 ⊗C∞1 )/ ker(h1). By Theorem
5.7, Zn can be represented by X
2 for 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 infinite
cycles. Hence by the proof of Theorem 5.13,
2Z2k−1) = h
2 Z2k
for ǫ = 0, 1, i, j ≥ 0, and k ≥ 2.
Finally, X1X2 is an infinite cycle since there is nothing that it can hit. Also,
h1v2X1X2 and h
1v2X1X2 are not hit by differentials since Ext
(Z2,C
1 ⊗C∞1 ) = 0
by Theorem 5.7. We obtain the following.
Theorem 5.19. In grading ≥ 10, there is an isomorphism of graded abelian groups
tmf∗(CP∞1 ∧CP∞1 ) ≈ yZ(2)[v1, v2, X1, X2]⊕
In·Z(2)[v1, v2]⊕Z(2)[Z](bo∗⊕v2Z(2)[v1, v2]),
34 DONALD M. DAVIS AND MARK MAHOWALD
where |y| = 12, |Xi| = 8, |Z| = 16, |v1| = −2, and |v2| = −6. Here In = ker(Fn
ǫ−→Z),
where Fn is a free abelian group with basis {X i1Xn−i2 : 1 ≤ i < n}, and ǫ(X i1Xn−i2 ) = 1.
Thus In consists of all polynomials of grading n with sum of coefficients equal to 0.
We could have extended the description in 5.19 down to grading 8, but the description
would have been slightly more complicated, since it would include h1v2Z and h
1v2Z.
The motivation for this section was to see if perhaps
ker(tmf∗(CP∞ × CP∞) 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∞ ∧ CP∞). To obtain tmf∗(CP∞ × CP∞), we
add on two copies of tmf∗(CP∞), which was described in 5.15. Denote by Z1 and
Z2 the generators in tmf
16(CP∞ × CP∞). Monomials Z i1Zn−i2 should equal Zn of
5.19 plus perhaps elements of I2n of 5.19. The class y of 5.19 plus perhaps a sum 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∞ × RP∞)
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.
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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
