Contemporary Mathematics

Nonimmersions of real projective spaces implied by tmf

Robert R. Bruner, Donald M. Davis, and Mark Mahowald

Abstract. Recently Hopkins and Mahowald constructed a new 2-primary ring spectrum tmf, satisfying H^{∗}(tmf) ≈ A//A_{2}. We use tmf to obtain new results regarding nonimmersions of real projective spaces in Euclidean space. The method is to say enough about the tmf-cohomology of a product of real projective spaces to obtain nonexistence of certain axial maps.

1. Main theorem and an example

In [14], a new 2-primary ring spectrum tmf was constructed, satisfying

H^{∗}(tmf) ≈ A//A_{2}.

All cohomology groups have coefficients in Z_{2} = Z/2, and A_{2} denotes the subalgebra of the mod 2 Steenrod algebra A generated by Sq^{1}, Sq^{2}, and Sq^{4}. We will say more about this spectrum tmf, which is related to topological modular forms and elliptic cohomology, in Section 5. The spectrum tmf has sometimes been called eo_{2}.

In this paper, we will use tmf to derive the following new result regarding nonimmersions of real projective space P^{n} in Euclidean space. Let ν(n) denote the exponent of 2 in n, α(n) the number of 1’s in the binary expansion of n, and p(n) the smallest 2-power greater than or equal to n.

Theorem 1.1. If α(M) = 4h + ϵ, then P^{8M+8h+δ} cannot be immersed (⊈) in R^{16M−8h+d} in the following cases:

1. M ≡ 0 mod \frac{1}{2}p(h + 1) and (ϵ, δ, d) = (0, −4, −1) or (−1, −6, −1).

2. M ≡ 0 mod p(h) and (ϵ, δ, d) = (−1, 2, 10) or (−2, 0, 12), or (−1, 0, 6) with h even.

3. M ≡ 2^{e_{0}} + 2^{e_{1}} mod 2^{e_{1}+1} with e_{0} < e_{1} and h ≤ 2^{e_{1}} − 2^{e_{0}}, or M a 2-power, and (ϵ, δ, d) = (4, 4, −12), (3, 2, −10), or (0, 0, 2).

These results are new provided α(M) ≥ 6 except in isolated cases for P^{n} with α(n) very small. They improve on previously known nonimmersions by arbitrarily many dimensions. The first new result is P^{n} ⊈ R^{2n−36} if n = 1536, the case α(M) = 6, M = 2^{7} + 2^{6} − 2 in part 2 of the theorem. The results should be

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

1991 Mathematics Subject Classification. 57R42, 55N20.

Key words and phrases. immersions, real projective spaces, elliptic cohomology.

compared with the result P^{2(m+α(m)−1)} ⊈ R^{4m−2α(m)} of [3]. That result was proved by applying the cohomology theory BP ⟨2⟩^{∗}(−) to detect axial maps, while ours here uses tmf^{∗}(−) in a similar way. Our results here improve on those results in a manner analogous to the way in which ko results improve on ku results, since, for E = ⟨tmf, BP ⟨2⟩, ko, ku⟩, we have H^{∗}(E) = A//B with B = ⟨A_{2}, E_{2}, A_{1}, E_{1}⟩, where A_{n} is the subalgebra generated by the first n + 1 Sq^{2^{i}}’s and E_{n} is the subalgebra generated by the first n + 1 Milnor primitives. For this and other reasons, tmf is sometimes called “higher real K-theory.” We will say more about the strength of the results in Section 4.

The method of proof is to prove nonexistence of axial maps. An axial map P^{m} × P^{n} → P^{k} is one whose restrictions to P^{m} × ∗ and ∗ × P^{n} are both homotopic to the inclusion maps. The following result is well known.

Proposition 1.2. ([22],[15]) If P^{n} can be immersed in R^{n+k}, then there exist axial maps P^{n} × P^{n} → P^{n+k} and P^{n} × P^{2^{L}−n−k−2} → P^{2^{L}−n−2}, where L > n/2.

In Section 2, we will discuss the computation of tmf^{∗}(P^{n}) and tmf^{∗}(P^{m}×P^{n}). We will show that there is an element X ∈ tmf^{8}(P^{n}) for which the corresponding map P^{n} → Σ^{8}tmf is nontrivial in H^{8}(−). If f is an axial map, then f^{∗}(X^{i}) = (X_{1} + X_{2})^{i}, where X_{i} corresponds to X in the ith factor. Theorem 1.1.1 is proved by showing that if the hypothesized immersion existed, then the first type of axial map in Proposition 1.2 would satisfy X^{2M−h} = 0 but (X_{1} + X_{2})^{2M−h} ≠ 0, an obvious contradiction. Parts 2 and 3 of Theorem 1.1 are deduced similarly using the second type of axial map in 1.2.

The hard part is showing that (X_{1} + X_{2})^{i} ≠ 0 ∈ tmf^{∗}(P^{m} × P^{n}). We will use the Adams spectral sequence (ASS) to give an almost complete determination of a quotient algebra of tmf^{∗}(P^{m} ∧ P^{n}) (with m and n even) generated by X_{1} and X_{2}. We illustrate with a typical group, and its use in obtaining a nonimmersion result.

Example 1.3. Let G = tmf^{8a+8b−72}(P^{8a−4} ∧ P^{8b−4}), and for 1 ≤ i ≤ 8, let y_{i} = X_{1}^{a−i}X_{2}^{b−9+i} ∈ G. Then G has a subgroup S (viz. the elements of filtration ≥ 12 in its ASS determination) such that G/S has presentation with generators y_{1}, …, y_{8} and relations

2^{4}y_{1}, 2^{8}y_{2}, 2^{8}y_{7}, 2^{4}y_{8}, (1.4)

2^{12}y_{i}, 3 ≤ i ≤ 6, (1.5)

2^{4}(y_{2} + y_{3} + y_{4} + y_{5} + y_{6} + y_{7}) − 2^{5}g, (1.6)

2^{8}(y_{3} + y_{5}) − 2^{9}g′, (1.7)

for some g, g′ ∈ G/S.

Diagram 1.8 illustrates the group of Example 1.3. Vertical and diagonal lines both indicate multiplication by 2, although sometimes this is only up to elements of higher filtration. Readers familiar with ASS charts should be aware that the entire group pictured below is (a portion of) a single homotopy (or generalized homology) group.

Diagram 1.8.

[figure: 1; 2; 3; 4; 5; 6; 7; 8]

The chart indicates that 2^{4}(y_{2} + ⋯ + y_{7}) is 0 in filtration 4, since each element in filtration 4 appears as a summand of 2^{4}y_{i} for 2 values of i, but, as (1.6) asserts, it can equal some element of larger filtration. The relations (1.4) are actually 0, not just of higher filtration, by Proposition 2.1. The chart implies (1.7) by showing that both 2^{8}y_{3} and 2^{8}y_{5} equal the first element in filtration 8, up to higher filtration.

We use this example to establish the first nonimmersion of Theorem 1.1.1 when h = 3. We will prove that if m is odd and α(m − 3) = 12, then there does not exist an axial map P^{8m−4} × P^{8m−4} → P^{16m−73}. Letting M = m − 3 yields the desired nonimmersion, thanks to 1.2. Since X^{2m−9} = 0 ∈ tmf^{∗}(P^{16m−73}) for dimensional reasons, it suffices to show

∑_{i=1}^{8} \binom{2m−9}{m−i}y_{i} ≠ 0 ∈ G/S, (1.9)

where G/S is as in Example 1.3 with a = b = m.

If ν\binom{2m−9}{m−1} < 4 or ν\binom{2m−9}{m−2} < 8, then (1.9) is nonzero due to its terms with i = 1 or 2, and we are done. We now assume to the contrary, and so the terms of (1.9) with i = 1, 2, 7, and 8 are 0 by (1.4). We have

ν\binom{2m−9}{m−3} = α(m − 3) + α(m − 6) − α(2m − 9)
= α(m − 3) + (α(m − 5) − 1 + ν(m − 5)) − (α(2(m − 5)) + 1)
= α(m − 3) − 2 + ν(m − 5)

and similarly

ν\binom{2m−9}{m−4} = α(m − 3) − 2 + ν(m − 3).

If α(m − 3) = 12 and m is odd, then one of these ν-values is 11 and the other is greater than 11, and so (1.9) is either 2^{11}(y_{3} + y_{6}) or 2^{11}(y_{4} + y_{5}). The only relations in filtration 11 are 2^{11}(y_{3} + y_{4} + y_{5} + y_{6}) and 2^{11}(y_{3} + y_{5}), from (1.6) and (1.7), and these do not imply that 2^{11}(y_{3} + y_{6}) or 2^{11}(y_{4} + y_{5}) is 0, establishing the nonimmersion.

There are three possible ways in which this project might be improved in the future. The most glaring is to eliminate the indeterminacy in relations such as (1.6) and (1.7), which should allow the removal of the congruence hypotheses in Theorem 1.1. The hope for doing this is to obtain an analogue of the [2]-series, which would give explicit relations in tmf^{∗}(P^{n}). The biggest obstruction to doing this is that v_{2}^{4} is not an element of tmf_{∗}. A second possible expansion of this work is to construct an orientation MO[8] → tmf, which would allow tmf to be incorporated more directly into obstruction theory. In particular, it should enable us to obtain new positive immersion results. The third extension is much more straightforward. It is just to consider tmf^{∗}(P^{m}∧P^{n}) with m and/or n odd. This will probably enable the deduction of some more nonimmersion results, at the expense of some complication of exposition.

We close this introductory section by observing that speculations about this approach to nonimmersions were already published in 1982 in [9, §4], although the details are not exactly as envisioned there. The main new ingredient is the existence of tmf, along with the computer programs of the first author. We would like to thank the Johns Hopkins University JAMI program, where all three authors spent at least one month in spring 2000, at which time this work was carried out.

2. The tmf-cohomology of P^{2m} ∧ P^{2n}

In this section, we determine, up to certain indeterminacy, a quotient algebra of tmf^{∗}(P^{m} ∧ P^{n}) (with m and n even) generated by X_{1} and X_{2}. The result, Theorem 2.7, is similar to Example 1.3.

We begin with a result which generalizes (1.4).

Proposition 2.1. There is a (nonunique) element X ∈ tmf^{8}(P^{2n}), compatible as n varies, for which the corresponding map P^{2n} → Σ^{8}tmf is nonzero in H^{∗}(−). Let c_{1} = c_{2} = 0, c_{3} = 1, and c_{4} = 3. For 1 ≤ d ≤ 4 and 1 ≤ i < m,

2^{4i−c_{d}}X^{m−i} = 0 ∈ tmf^{∗}(P^{8m−2d}). (2.2)

Proof. We use S-duality. It is convenient, although not necessary, to use stunted projective spaces with negative indices. See, for example, [18]. For i > 0, tmf^{i}(P^{n}) ≈ tmf_{−i−1}(P_{−n−1}). Note P_{m} always means P_{m}^{∞}. Since H^{∗}(tmf) ≈ A//A_{2}, by a well-known change-of-rings theorem the E_{2}-term of the ASS converging to tmf_{∗}(Y) is given by Ext_{A_{2}}(H^{∗}Y) for any space Y. Here and elsewhere, we omit Z/2 in the second component of Ext groups.

In [18], it was proved that

(2.3)  lim_{←} Ext_{A_{2}}(H^{∗}P_{−n}) ≈ ⨁_{i∈Z} Ext_{A_{1}}(Σ^{8i−1}Z_{2}).

Thus, for any integer i, lim_{←} Ext_{A_{2}}^{∗,∗+8i−1}(H^{∗}P_{−n}) is a free h_{0}-module on generators of filtration 4j for all j ≥ 0, and

lim_{←} Ext_{A_{2}}^{∗,∗+8i−2}(H^{∗}P_{−n}) = 0. (2.4)

Let g_{i} denote the nonzero element of lim_{←} Ext_{A_{2}}^{0,8i−1}(H^{∗}P_{−n}).

The groups Ext_{A_{2}}(H^{∗}P_{N}) were completely computed in [9, 2.3,2.4]. We have that for 1 ≤ d ≤ 4 and i > a,

Ext_{A_{2}}^{0,8i−1}(H^{∗}P_{8a+2d−1}) ≈ Z/2

with generator g_{i} satisfying h_{0}^{4i−4a−c_{d}−1}g_{i} ≠ 0, and

Ext_{A_{2}}^{s,s+8i−1}(H^{∗}P_{8a+2d−1}) = 0 if s ≥ 4i − 4a − c_{d}. (2.5)

Nothing in this tower can be killed by an Adams differential because it maps onto a tower of the same height in ko_{∗}(P_{8a+2d−1}) which is not killed by a differential. Moreover, g_{i} cannot support a differential by (2.4).

It follows from (2.3) that

lim_{←} tmf_{∗}(P_{−n}) ≈ ∏_{i∈Z} ko_{∗}(S^{8i−1})^{∧}.

Let G_{i} be an element of lim_{←} tmf_{8i−1}(P_{−n}) of Adams filtration 0. Then G_{i} is necessarily detected by g_{i}, but is not unique, even up to odd multiples, because it could be changed by elements in towers beginning in filtration ≥ 4. Restriction yields compatible elements G_{i} ∈ tmf_{8i−1}(P_{b}) for b ≤ 8i − 1. With i = −1, duality produces the compatible elements X ∈ tmf^{8}(P^{n}) for n ≥ 8. Since the dual of X^{m−i} lies in tmf_{−8m+8i−1}(P_{−8m+2d−1}), (2.2) follows from (2.5).

In the preceding proposition, we used that as A_{2}-modules H^{∗}(P_{b+8}^{t+8}) ≈ H^{∗}(Σ^{8}P_{b}^{t}), which implies an isomorphism of E_{2}-terms of the ASS after applying ∧tmf to the spectra. There is, in fact, an associated equivalence of spectra, which will be useful later in the paper.

Proposition 2.6. For any integers b and t, with t possibly infinite, there is an equivalence of spectra

P_{b+8}^{t+8} ∧ tmf ≃ Σ^{8}P_{b}^{t} ∧ tmf.

Proof. The elements X of Proposition 2.1 combine to give an element X ∈ tmf^{8}(P_{1}^{∞}). The exact tmf^{∗}(−) sequence induced by the cofibration P^{7} → P_{1}^{∞} → P_{8}^{∞} shows that X pulls back to an element X ∈ tmf^{8}(P_{8}), with the property that the associated map P_{8} → Σ^{8}tmf is nontrivial in H^{8}(−).

Now we observe that covering the diagonal map P → P × P is a map of Thom spectra T ((8 + n)ξ) → T (8ξ) ∧ T (nξ). We use the representation of these Thom spectra as stunted projective spaces, apply ∧tmf to the map, and follow by X : P_{8} → Σ^{8}tmf, obtaining

P_{8+n} ∧ tmf → P_{8} ∧ P_{n} ∧ tmf → Σ^{8}tmf ∧ P_{n} ∧ tmf → Σ^{8}P_{n} ∧ tmf,

where we have used that tmf is a ring spectrum at the last step. One readily verifies that this composite P_{8+n} ∧ tmf → Σ^{8}P_{n} ∧ tmf induces an isomorphism in cohomology (σ^{8}g_{8i−1} ↦ g_{8} ⊗ g_{8i−1} ↦ g_{8i+7}), and hence is an equivalence. Since the maps are compatible with respect to changing n, there are also equivalences of the finite projective spaces.

Now we state the main result of this section, generalizing Example 1.3, which is the case s_{1} = s_{2} = 2 and d = 9.

Theorem 2.7. Let s_{1} and s_{2} equal 1, 2, 3, or 4, and d ≥ 2. Let c_{s} be as in Proposition 2.1. Let G = tmf^{8a+8b−8d}(P^{8a−2s_{1}} ∧ P^{8b−2s_{2}}), and for 1 ≤ i < d, let y_{i} = X_{1}^{a−i}X_{2}^{b−d+i} ∈ G. Then G has a subgroup S such that G/S has presentation with generators y_{1}, …, y_{d−1} and relations

2^{4i−c_{s_{1}}} y_{i}, i ≥ 1, (2.8)

2^{4i−c_{s_{2}}} y_{d−i}, i ≥ 1, (2.9)

2^{4i+4} ∑_{j} \binom{j}{i}y_{j} − 2^{4i+5}x_{i}, i ≥ 0, some x_{i} ∈ G/S, (2.10)

2^{8k+ϵ}y_{2k+δ} if d = 6k + β, s_{1} ≥ t_{1}, and s_{2} ≥ t_{2}, (2.11)

where (ϵ, δ, β, t_{1}, t_{2}) is one of the following: (2, 1, 2, 1, 1), (−1, 0, −1, 1, 2), (−1, 0, −1, 2, 1), (−2, 0, −1, 1, 3), (−2, 0, −1, 3, 1), (3, 1, 3, 1, 4), (3, 2, 3, 4, 1), (−1, 0, 0, 2, 4), (−1, 1, 0, 4, 2), (−5, 0, −2, 4, 4), and

2^{e}y_{i} for all i, (2.12)

where e is the (d − 1)st largest element in the set with repetitions which consists of all exponents which occur in (2.8), (2.9), (2.10), and (2.11).

In (2.10), we initiate the convention, followed throughout the paper, that, unless indicated to the contrary, sums involving binomial coefficients are to be taken over all values of the summation variable for which the summand is nonzero.

The reader might find it worthwhile to compare Theorem 2.7 with the special case given in Example 1.3. In that case, the integer e of (2.12) is the 8th largest entry in {4, 8, 12, …, 4, 8, 12, …, 4, 8, 12, … }, which is 12.

We extract from Theorem 2.7 the information that will be used in proving nonimmersions. This result will be proved at the end of the section.

Corollary 2.13. The element ∑ k_{i}y_{i} is nonzero in the group G of 2.7 in the following cases:

1. d = 3h, and, for t = 1 or 2, s_{1} = s_{2} = t + 1, ν(k_{i}) ≥ 4h − t for h ≤ i ≤ 2h,

and ν(∑ \binom{h−1}{j}k_{h+j}) = 4h − t;

2. (d, t, s_{1}, s_{2}) = (3h, 1, 3, 2) or (3h, 2, 4, 3), or (3h + 1, 1, 4, 4) with h even,

ν(k_{i}) ≥ 4h − t for h + 1 ≤ i ≤ 2h, and

ν(∑ \binom{h−1}{j}k_{h+1+j}) = 4h − t. (2.14)

3. Either d = 3h + 3, (t, s_{1}, s_{2}) = (3, 2, 3) or (2, 3, 4), ν(k_{i}) ≥ 4h + t for h + 1 ≤ i ≤ 2h + 1, and ν(∑ \binom{h}{j}k_{h+1+j}) = 4h + t, or d = 3h + 1, s_{1} = 4, s_{2} = 2, ν(k_{i}) ≥ 4h − 1 for h + 1 ≤ i ≤ 2h + 1, and ν(∑ \binom{h−1}{j}k_{h+2+j}) = 4h − 1.

The derivation of Theorem 2.7 is roughly to run a computer program that computes Ext_{A_{2}}(H^{∗}(P_{2s_{1}−1} ∧ P_{2s_{2}−1})) through a range by minimal resolution. Of course, this is not a proof that an observed pattern persists throughout all dimensions. In order to do that, we first obtain a complete computation of the relevant part of Ext_{A_{2}}(H^{∗}(P_{3} ∧ R)), where R is the fiber of the so-called Kahn-Priddy map λ : P_{1} → S^{0}. This is done by embedding H^{∗}(P_{3} ∧ R) in a short exact sequence between modules whose Ext groups are completely known. Next we use cofibration sequences to deduce from this the relevant part of each Ext_{A_{2}}(H^{∗}(P_{2s_{1}−1}∧P_{2s_{2}−1})). Finally, we show that the elements corresponding under duality to multiples of X_{1}^{i}X_{2}^{j} are not killed by differentials in the ASS converging to tmf_{∗}(P_{2s_{1}−1}∧P_{2s_{2}−1}).

We begin with the following key result.

Theorem 2.15. In filtration ≤ [(4d − 2)/3],

Ext_{A_{2}}^{∗,∗+8d−1}(H^{∗}(P_{3} ∧ R)) = 0

and Ext_{A_{2}}^{∗,∗+8d−2}(H^{∗}(P_{3} ∧ R)) as a module over Z_{2}[h_{0}] has generators g_{1}, …, g_{d} of filtration 0 and relations h_{0}^{4i}g_{i} (i ≥ 1), h_{0}^{4i+1}g_{d−i} (i ≥ 0), h_{0}^{4i+4} ∑ \binom{j}{i}g_{j} (i ≥ 0), and, if d = 3k + 2, then h_{0}^{4k+3}g_{k+1}.

There are other elements in Ext_{A_{2}}(H^{∗}(P_{3} ∧ R)) in bigradings not specified in Theorem 2.15. The entire Ext chart makes a very pleasing picture, with the towers described in the theorem having their tops along a line of slope 1/6, with h_{1}^{2} extending into the towers at regular spacing, and a few additional elements at the top of the chart moving along a line of slope 1/5. We illustrate with the complete chart in t − s ≤ 42. The action of h_{0}, h_{1}, and h_{2} is indicated.

Diagram 2.16.

[figure: 2; 6; 10; 14; 18; 22; 26; 30; 34; 38; 42]

The heights H_{d}(i) of the towers in Ext_{A_{2}}^{∗,∗+8d−2}(H^{∗}(P_{3} ∧R)) for d ≤ 8 are listed for illustration in Table 2.17. They satisfy H_{d+6}(i + 2) = H_{d}(i) + 8. From this information and the pattern of extensions from filtration 4i − 1 to 4i, the reader can extend the relevant portion of Diagram 2.16. The relation h_{0}^{4i+4} ∑ \binom{j}{i}g_{j} in Theorem 2.15 can be considered to be a consequence of similar extension patterns ([figure] ) from filtration 3 to 4, 7 to 8, …, 4i + 3 to 4i + 4.

Table 2.17.

d
H_{d}(i) 1 2 3 4 5 6 7 8
1 1 3 4 4 4 4 4 4
2  1 4 5 7 8 8 8
i 3  1 4 5 8 9 11
4  1 4 5 8 9
5  1 4 5 8
6  1 4 5
7  1 4
8  1

Note that H_{d}(i) is not, in general, the order of the element g_{i} because of the extensions into the tower just to the left. For example, the case d = 5 in Table 2.17 corresponds to the group in dimension 38 in Diagram 2.16. First note that the heights of the towers above the elements g_{i} in dimension 38 in 2.16 are 4, 7, 5, 4, and 1, resp., which is the information in 2.17. The relations in Theorem 2.15 when d = 5 are h_{0}^{4}g_{1}, h_{0}^{7}g_{2}, h_{0}^{5}g_{4}, h_{0}g_{5}, and h_{0}^{4}(g_{2} + g_{3} + g_{4}). One can easily verify that this is consistent with 2.16 in dimension 38.

Proof of Theorem 2.15. There is a short exact sequence of A_{2}-modules

0 → F → H^{∗}R⟶^{θ}Q → 0, (2.18)

where

F_{i} = \begin{cases}Z_{2} & if i = 1, 2, 4, or 8 \\ 0 & otherwise \end{cases}

and Sq^{4} Sq^{2} Sq^{1} ≠ 0 in F, and Q is the quotient of H^{∗}P_{−1} by the A_{2}-submodule generated by classes of degree less than −1. This is the 8-fold suspension of the module called Q in [9, §1,7]. We apply H^{∗}P_{3}⊗ to (2.18) and consider the long exact sequence in Ext_{A_{2}}(−). Since Q is built from subquotients Σ^{8i−1}A_{2}//A_{1} for i ≥ 0,

Ext_{A_{2}}(H^{∗}P_{3} ⊗ Q) ≈ ⨁_{i≥0} Ext_{A_{1}}(H^{∗}(Σ^{8i−1}P_{3})). (2.19)

We will need the following lemma, whose proof is postponed.

Lemma 2.20. Let {g_{1}, …, g_{d}} be the basis of Ext_{A_{2}}^{0,8d−2}(H^{∗}P_{3} ⊗ Q) such that g_{i} is dual to x_{1}^{8i−1}x_{2}^{8d−8i−1}. Then

v_{1}^{4}g_{i} = h_{0}^{4}(g_{i+1} + ∑_{j=1}^{i−1} g_{j}). (2.21)

The g_{i} on the LHS of (2.21) has d-value 1 less than that of the g_{j} on the RHS. We have

(2.22) Ext_{A_{2}}^{s,t}(H^{∗}P_{3} ⊗ F) ≈ ⨁_{i}(Ext_{A_{1}}^{s,t}(Σ^{8i+7}Z_{2}) ⊕ Ext_{A_{1}}^{s+1,t+1}(Σ^{8i−1}Z_{2}))

if t − s ≥ 6s + 6. One way to see this is to build it from Ext_{A_{2}}(H^{∗}(Σ^{i}P_{3})) for i = 1, 2, 4, and 8. We use the computation of Ext_{A_{2}}^{s,t}(H^{∗}P_{3}) in [9]; it is isomorphic to ⨁ Ext_{A_{1}}^{s,t}(Σ^{8i−1}Z_{2}) if t − s ≥ 6s + 4. The Ext_{A_{1}}(Σ^{8i+7}Z_{2}) in (2.22) comes from the 8-cell of F, while

Ext_{A_{1}}^{s,t}(Σ^{8i−1}F^{(4)}) ≈ Ext_{A_{1}}^{s+1,t+1}(Σ^{8i−1}Z_{2}),

where F^{(4)} denotes the 4-skeleton of F.

Another lemma with postponed proof follows.

Lemma 2.23. In the exact sequence

(2.24) → Ext_{A_{2}}^{s−1,t}(H^{∗}P_{3} ⊗ F)⟶^{∂} Ext_{A_{2}}^{s,t}(H^{∗}P_{3} ⊗ Q)⟶^{θ^{∗}} Ext_{A_{2}}^{s,t}(H^{∗}(P_{3} ∧ R)) →,

im(∂) in (s, t) = (1, 8d − 1) contains h_{0}g_{d}, and im(∂) in (s, t) = (4, 8d + 2) contains h_{0}^{4}(g_{1} + ⋯ + g_{d}).

We have given almost enough information to determine the requisite information about Ext_{A_{2}}(H^{∗}(P_{3} ∧ R)) from (2.24). The relation h_{0}^{4i}g_{i} is already present in Ext_{A_{2}}(H^{∗}P_{3} ⊗ Q). The relations h_{0}^{4i+1}g_{d−i} and h_{0}^{4i+4} ∑ \binom{j}{i}g_{j} follow inductively from the two lemmas. Indeed, the case i = 0 is immediate from im(∂) in Lemma 2.23. Applying v_{1}^{4} to the i = 0 relations in (s, t) = (1, 8d − 9) and (4, 8d − 6) yields, by Lemma 2.20, the relations

h_{0}^{5}(g_{d} + ∑_{j=1}^{d−2} g_{j}) and h_{0}^{8}(∑_{j=1}^{d−1}(d − j)g_{j} + g_{d} − g_{1}).

The first of these becomes h_{0}^{5}g_{d−1} after reduction using the relation h_{0}^{4}(g_{1}+⋯+g_{d}), while the second is reduced to h_{0}^{8} ∑ jg_{j} similarly, yielding the case i = 1. A similar argument works for larger values of i.

The final relation of Theorem 2.15 is not so easily seen. Here the minimal resolution computer program ([1],[2]) of the first author is very useful. It allows us to see all these Ext groups through a range of dimensions (t − s ≤ 78 has been used often). This allows us to see any unusual boundary morphisms. We can deduce that they are promulgated by v_{2}^{8}-periodicity.

The relations already considered imply that ∂ in (2.24) is surjective if

t − s = 8d − 2 and s ≥ \begin{cases}4k & if d = 3k − 1, 3k \\ 4k + 1 & if d = 3k + 1. \end{cases} (2.25)

The only elements in Ext_{A_{2}}^{s,s+8d−1}(H^{∗}P_{3} ⊗ F) not accounted for by (2.22) and with s less than the largest value for which (2.25) is not satisfied is a Z_{2} in

(t − s, s) = (15 + 24k, 2 + 4k). (2.26)

This can be seen by the method described after (2.22). The computer program mentioned above establishes the claim in t − s ≤ 78. All homomorphisms in the exact sequences that build Ext_{A_{2}}(H^{∗}P_{3} ⊗ F) from Ext_{A_{2}}(H^{∗}Σ^{i}P_{3}), i = 1, 2, 4, 8, commute with v_{2}^{8}, and, since by [9] Ext_{A_{2}}(H^{∗}P_{3}) has a faithful action of v_{2}^{8}, this implies that the behavior in t − s ≤ 78 continues indefinitely. These elements in (2.26) are mapped nontrivially by ∂. Again the safest way to see this is to use the computer program in low dimensions, and then v_{2}^{8}-periodicity. This accounts for the final relation in Theorem 2.15.

That Ext_{A_{2}}^{s,s+8d−1}(H^{∗}(P_{3} ∧ R)) = 0 for s ≤ [(4d − 2)/3] follows also from the exact sequence (2.24). Here we have Ext_{A_{2}}^{∗,∗+8d−1}(H^{∗}P_{3} ⊗ Q) = 0 by (2.19), and ∂ is injective on Ext_{A_{2}}^{s,s+8d−1}(H^{∗}P_{3} ⊗ F) for s ≤ [(4d − 2)/3].

We illustrate the above proof with a diagram of (2.24) for 12 ≤ t − s ≤ 19. Elements of Ext_{A_{2}}(H^{∗}P_{3} ⊗ Q) are indicated by •’s, elements of Ext_{A_{2}}(H^{∗}P_{3} ⊗ F) by ∘’s, and diagonal lines of negative slope are ∂.

Diagram 2.27.

[figure: 14; 18]

Proof of Theorem 2.7. We begin by observing that the group G of Theorem 2.7 is isomorphic to tmf_{8d−1}(P_{2s_{1}−1} ∧ P_{2s_{2}−1}), using S-duality as in the proof of 2.1. Under this isomorphism and natural morphisms which we will consider, the classes y_{i} of 2.7 correspond to classes g_{i} of Theorem 2.15.

We begin by determining the relevant portion of Ext_{A_{2}}(H^{∗}(P_{3} ∧ P_{−1})). This is the E_{2}-term of the ASS converging to e_{∗}(P_{3} ∧P_{−1}). After shifting by 8 dimensions, using 2.6, this will correspond to the case s_{1} = 2, s_{2} = 4 of Theorem 2.7.

There is a cofibration R⟶^{i}P_{−1} → S^{0} which induces a short exact sequence in H^{∗}(−). We apply P_{3}∧ and obtain a long exact sequence

Ext_{A_{2}}^{s−1,t}(H^{∗}(P_{3} ∧ P_{−1})) ⟶^{λ_{∗}} Ext_{A_{2}}^{s−1,t}(H^{∗}P_{3})
(2.28)  ⟶^{∂} Ext_{A_{2}}^{s,t}(H^{∗}(P_{3} ∧ R))⟶^{i_{∗}} Ext_{A_{2}}^{s,t}(H^{∗}(P_{3} ∧ P_{−1})).

The upper edge of Ext_{A_{2}}(H^{∗}(P_{3}∧R)) has slope 1/5, while that of Ext_{A_{2}}(H^{∗}P_{3}) has slope 1/2, and so between slope 1/5 and 1/2, Ext_{A_{2}}(H^{∗}(P_{3} ∧P_{−1})) is isomorphic to Ext_{A_{2}}(H^{∗}P_{3}), which is given in [9]. However, the region in which we are interested lies below slope 1/5.

We study (2.28) with t − s = 8d − 2 and s ≤ [(4d − 2)/3]. In this range, Ext_{A_{2}}^{∗,∗+8d−1}(H^{∗}P_{3}) ≈ ⨁ Ext_{A_{1}}^{∗,∗+8d−1}(Σ^{8i−1}Z_{2}). We will show that these elements pull back under λ_{∗} to elements of tmf_{8d−1}(P_{3} ∧ P_{−1}). This implies that in this range (a) ∂ = 0 and hence i_{∗} is injective, and (b) elements in im(i_{∗}) are not hit by differentials in the ASS converging to tmf_{∗}(P_{3} ∧ P_{−1}). This yields tmf_{8d−2}(P_{3} ∧ P_{−1}) ≈ tmf_{8d−2}(P_{3} ∧ R) in filtration ≤ [(4d − 2)/3], and hence the result claimed for tmf^{8a+8b−8d}(P^{8a−4} ∧ P^{8b−8}) follows by duality. Note that the last relation of Theorem 2.15 becomes, in P_{3} ∧ P_{7}, h_{0}^{4k+3}g_{k+1} if d = 3k + 3, which corresponds to the relations parametrized by (3, 1, 3, 1, 4) and (−1, 0, 0, 2, 4) in 2.7.

To see this pulling back, note that, using 2.6,

tmf_{8d−1}(P_{3} ∧ P_{−1}) ≈ [P_{−∞}^{0}, Σ^{−8d}P_{3} ∧ tmf] ≈ [P_{−∞}^{0}, P_{3−8d} ∧ tmf].

There is a filtration-0 map P_{−∞}^{0} → P_{3−8d} ∧ tmf given by inclusion and collapse. Since this map is nonzero in H^{−1}(−), the corresponding element of tmf_{8d−1}(P_{3} ∧ P_{−1}) projects to the nonzero element of Ext_{A_{2}}^{0,8d−1}(H^{∗}P_{3}) in (2.28). The v_{1}^{4}-action on tmf_{∗}(−) implies that towers in Ext_{A_{2}}^{∗,∗+8d−1}(H^{∗}P_{3}) which begin in filtration 4i also pull back to elements of Ext_{A_{2}}(H^{∗}(P_{3} ∧ P_{−1})).

Other values of (s_{1}, s_{2}) in Theorem 2.7 are handled by cofibration sequences involving a change of 1 in one s_{i}. We illustrate with one case. The Ext charts in all cases can be viewed at [2].

The case (s_{1}, s_{2}) = (2, 1) is obtained using the exact sequence

(2.29) Ext_{A_{2}}^{s,t}(H^{∗}(P_{3} ∧ P_{−1}^{0}))⟶^{i_{∗}} Ext_{A_{2}}^{s,t}(H^{∗}(P_{3} ∧ P_{−1})) → Ext_{A_{2}}^{s,t}(H^{∗}(P_{3} ∧ P_{1}))

and the information just obtained about Ext_{A_{2}}(H^{∗}(P_{3} ∧ P_{−1})). The chart for P_{3} ∧ P_{−1}^{0} is obtained from charts of Σ^{−1}P_{3} and P_{3} by inserting boundary morphisms (d_{1}- differentials) corresponding to the action of h_{0}. Due to the period-8 shift between values s_{1} = 0 and 4, the relation in P_{3} ∧ P_{−1} corresponding to (2.9) is 2^{4i+1}g_{d−i}, which also appears in 2.15. The corresponding relation in P_{3} ∧P_{1} is supposed to be 2^{4i}g_{d−i}. The elements of Ext_{A_{2}}(H^{∗}(P_{3} ∧ P_{−1}^{0})) which hit the elements h_{0}^{4i+1}g_{d−i} in (2.29) are the beginnings of the “lightning flashes” obtained from the Ext_{A_{1}}(Z_{2})- like parts of Ext_{A_{2}}(H^{∗}(Σ^{j}P_{3})). Indeed, Ext_{A_{2}}(H^{∗}(P_{3} ∧ P_{−1}^{0})) has summands like Diagram 2.30 with initial element in (t−s, s) = (8d−2, 4i) throughout the relevant range, and these initial elements are mapped nontrivially by i_{∗} in (2.29).

Diagram 2.30.

[figure]

The extra relation in 2.7 parametrized by (2, 1, 2, 1, 1) says that if d = 6k + 2, then 2^{8k+2}g_{2k+1} is in the image of i_{∗} in (2.29). This is hit under i_{∗} by powers of v_{2}^{8} acting on an element of Ext_{A_{2}}^{2,16}(H^{∗}(P_{3} ∧ P_{−1}^{0})) corresponding to an element of Ext_{A_{2}}^{2,17}(H^{∗}P_{3}) which in [9, p.302] is the circled element in D′_{3} in t − s = 23.

We emphasize that we do not perform a direct verification that this element of Ext_{A_{2}}^{2,16}(H^{∗}(P_{3} ∧ P_{−1}^{0}))^{1} maps nontrivially under i_{∗}. Instead, we rely on the computer calculations of various Ext_{A_{2}}(H^{∗}(P_{2s_{1}−1} ∧ P_{2s_{2}−1})), and when we see new relations arising as s_{i} increases we can always interpret them in the relevant exact sequence. The action of v_{2}^{8} guarantees promulgation of nonzero boundary morphisms. Also, our knowledge of Ext_{A_{2}}(H^{∗}(P_{2s_{1}−1} ∧ P_{2s_{2}−1}^{2s_{2}})) (from [9]) guarantees that in the relevant range (below a line of slope 1/6), Ext_{A_{2}}(H^{∗}(P_{2s_{1}−1} ∧ P_{2s_{2}−1}^{2s_{2}})) has very few classes other than its regular family of lightning flashes, and these classes are all obtained from v_{2}^{8} acting on elements in the first 48 dimensions. This method accounts for all relations in (2.11).

The relations (2.8) and (2.9) of 2.7 are guaranteed by 2.1. It is important that these relations are strictly valid, not just up to higher filtration, as is the case with (2.10) (the 2^{4i+5}x_{i} being the higher filtration). The key relations (2.10) are implied by naturality from the corresponding relations in Theorem 2.15.

The final relation (2.12) specifies the filtration in which all h_{0}-towers from filtration 0 cease to exist. It could be derived from the other relations (2.8) to (2.11), although the indeterminacy in (2.10) makes such a derivation delicate. This relation (2.12) is listed primarily as a convenience for our later arguments.

1or similar elements in other cases

We illustrate in Diagram 2.31 with the chart of Ext_{A_{2}}(H^{∗}(P_{3}∧P_{1}) in t−s ≤ 34. Here we use a different basis in filtration 0 than was used for P_{3} ∧ R. The one used here is designed to simplify the appearance of the chart. Note that our analysis only involves the h_{0}-towers emanating from filtration 0 in dimensions 8d−2, a very small portion of the chart.

Diagram 2.31.

[figure: 4; 10; 18; 22; 26; 30; 34]

We close this section by providing the three omitted proofs of lemmas and a corollary.

Proof of Lemma 2.20. Let M denote the A_{2}-module H^{∗}P_{3} ⊗ Q. One can verify (by hand, even) the following relation in M_{8d−1} for 1 < j ≤ d. Here Sq(R) is Milnor basis notation.

(2.32)

Sq(1)(x_{1}^{8j−1}x_{2}^{8d−8j−1}) + Sq(5) ∑_{i=0}^{1} x_{1}^{8j−8i−1}x_{2}^{8d−8j+8i−5}

+ Sq(3, 2)(x_{1}^{8j−1}x_{2}^{8d−8j−9}) + Sq(6, 1) ∑_{i=j−1_{i≠j}}^{d−1} x_{1}^{8i−1}x_{2}^{8d−8i−9}

+ Sq(0, 2, 1)(x_{1}^{8j−9}x_{2}^{8d−8j−5}) + Sq(4, 2, 1) ∑_{i=j}^{d−2} x_{1}^{8i−1}x_{2}^{8d−8i−17}

The relation must be modified when j = 1 because then x_{1}^{8j−9} ∉ H^{∗}P_{3}. However, it turns out to be irrelevant because h_{0}^{4}g_{1} = 0.

Let g_{i}, as in the statement of the lemma, be dual to x_{1}^{8i−1}x_{2}^{8d−8i−1}, and let g′_{i} be dual to x_{1}^{8i−1}x_{2}^{8d−8i−9}. We claim

v_{1}^{4}g′_{i} = h_{0}^{4} ∑_{j∈S_{i}} g_{j}, (2.33)

where j ∈ S_{i} iff Sq(6, 1)(x_{1}^{8i−1}x_{2}^{8d−8i−9}) appears in the relation (2.32). By (2.32), this will yield the claim of the lemma.

The portion of the minimal A_{2}-resolution of Z_{2} needed to obtain v_{1}^{4} is (using t − s as subscripts):

H_{0} : Sq(1)G;
H_{1} : Sq(2)G;
H_{3} : Sq(4)G;
J_{0} : Sq(1)H_{0};
J_{2} : Sq(2)H_{1} + Sq(3)H_{0};
J_{3} : Sq(1)H_{3} + Sq(0, 1)H_{1} + Sq(4)H_{0};
K_{0} : Sq(1)J_{0};
K_{3} : Sq(1)J_{3} + Sq(2)J_{2} + Sq(4)J_{0};
L_{8} : Sq(3, 1)K_{3} + Sq(6, 1)K_{0}.

Let D and C be minimal resolutions of M and Z_{2}, respectively, and let ϕ : D → C be a morphism such that ϕ_{0} is dual to y′_{i} := x_{1}^{8i−1}x_{2}^{8d−8i−9}.

D_{0} ⟵^{∂_{1}} D_{1} ⟵^{∂_{2}} D_{2} ⟵^{∂_{3}} D_{3} ⟵^{∂_{4}} D_{4}
ϕ_{0}↓  ^{ϕ_{1}}↓  ^{ϕ_{2}}↓  ^{ϕ_{3}}↓  ^{ϕ_{4}}↓
C_{0} ⟵^{δ_{1}} C_{1} ⟵^{δ_{2}} C_{2} ⟵^{δ_{3}} C_{3} ⟵^{δ_{4}} C_{4}

In this proof, OT means “other terms.” If h is a generator of D_{1}, then ϕ_{1}(h) = Sq(2, 1)H_{3} + OT iff Sq(6, 1)y′_{i} is part of ∂_{1}(h), which corresponds to one of the relations (2.32). This is seen by noting that the only ways in which Sq(6, 1)G occurs in im(δ_{1}) are as δ_{1}(Sq(2, 1)H_{3}) = Sq(6, 1)G and δ_{1}(Sq(4, 1)H_{1}) = (Sq(6, 1)+ Sq(0, 3))G, but since Sq(0, 3) does not appear in the relations (2.32), it cannot appear in im(ϕ_{0}∂_{1}).

For each generator h of D_{1}, there is a corresponding generator j of D_{2} which maps to Sq(1)h + OT. Then ϕ_{1}(h) = Sq(2, 1)H_{3} + OT iff ϕ_{2}(j) = Sq(2, 1)J_{3} + OT. Some care is required to see this. The sketch of argument is ϕ_{1}(h) = Sq(2, 1)H_{3}+OT iff ϕ_{1}∂_{2}(j) = Sq(3, 1)H_{3} +OT iff δ_{2}ϕ_{2}(j) = Sq(3, 1)H_{3} +OT iff ϕ_{2}(j) = Sq(2, 1)J_{3} + OT. A main worry is that δ_{2}(Sq(5)J_{3}) contains the term Sq(3, 1)H_{3}, and so Sq(5)J_{3} could conceivably appear in ϕ_{2}(j) in place of Sq(2, 1)J_{3}. However, δ_{2}(Sq(5)J_{3}) also contains (Sq(5, 1) + Sq(1, 0, 1))H_{1}, which could be in im(ϕ_{1}∂_{2}) only if ϕ_{1}(h) contained (Sq(4, 1) + Sq(0, 0, 1))H_{1}, and this would require that the relation for h contain Sq(0, 3), which we have already observed does not happen.

A similar argument shows that ϕ_{2}(j) = Sq(2, 1)J_{3}+OT iff ϕ_{3}(k) = Sq(2, 1)K_{3}+ OT, where ∂_{3}(k) = Sq(1)(j) + OT. Here again, the most delicate part is ruling out the possibility that ϕ_{3}(k) might contain Sq(5)K_{3} rather than Sq(2, 1)K_{3}. This time we observe that δ_{3}(Sq(5)K_{3}) contains (Sq(7) + Sq(1, 2))J_{2}, which cannot appear as a term in im(ϕ_{2}∂_{3}).

The fourth step is the easiest: ϕ_{3}(k) = Sq(2, 1)K_{3} + OT iff ϕ_{4}(ℓ) = L_{8}, where ∂(ℓ) = Sq(1)(k) + OT. Now, if ϵ : C_{4} → Z_{2} is dual to L_{8}, then ϵϕ_{4} : D_{4} → Z_{2} represents v_{1}^{4}g′_{i}. By the analysis just completed of the map of resolutions, ϵϕ_{4} sends nontrivially generators which correspond to h_{0}^{4} times the duals of basis elements x_{1}^{8j−1}x_{2}^{8d−8j−1} of M for which the relation Sq(1)x_{1}^{8j−1}x_{2}^{8d−8j−1} + OT contains the term Sq(6, 1)x_{1}^{8i−1}x_{2}^{8d−8i−9}, establishing (2.33).

Proof of Lemma 2.23. We prove that h_{0}θ^{∗}(g_{d}) = 0 and h_{0}^{4}θ^{∗}(g_{1} + ⋯ + g_{d−1}) = 0.

For the first, we first observe that h_{0}g_{d} ≠ 0 in Ext_{A_{2}}(H^{∗}P_{3} ⊗ Q) because

Sq^{1}(x_{1}^{8d−1}x_{2}^{−1}) = Sq^{2}(x_{1}^{8d−2}x_{2}^{−1}) in H^{∗}P_{3} ⊗ Q,

and so there is a relation involving Sq^{1}(x_{1}^{8d−1}x_{2}^{−1}). However, in H^{∗}(P_{3} ∧ R), Sq^{2}(x_{1}^{8d−2}x_{2}^{−1}) contains another term, x_{1}^{8d−2}x_{2}^{1}. There is no relation in H^{∗}(P_{3} ∧ R) involving Sq^{1}(x_{1}^{8d−1}x_{2}^{−1}) because for all monomials m in degree 8d − 1 − t, t = 1, 2, or 4, Sq^{t}(m) is the sum of an even number of monomials, with the single exception of Sq^{1}(x_{1}^{8d−1}x_{2}^{−1}). Thus h_{0}g_{d} = 0 in Ext_{A_{2}}(H^{∗}(P_{3} ∧ R)).

For the second, we observe that there is an A_{2}-module homomorphism

H^{∗}(P_{3} ∧ R) → H^{∗}(Σ^{8d−9}P_{1})

through dimension 8d − 1 sending

x_{1}^{3}x_{2}^{8d−11} ↦ σ^{8d−9}x^{1},
x_{1}^{8i+3}x_{2}^{8d−8i−9} ↦ σ^{8d−9}x^{3}, 0 ≤ i < d,
x_{1}^{8i−1}x_{2}^{8d−8i−1} ↦ σ^{8d−9}x^{7}, 1 ≤ i < d.

To do this, we need merely note that, under this morphism, the relations which give rise to the elements of Ext_{A_{2}}^{1,t}(H^{∗}(P_{3} ∧ R)) in t = 8d − 6, 8d − 5, 8d − 3, and 8d − 1 map to relations in H^{∗}(Σ^{8d−9}P_{1}). Ignoring terms acting on classes of degree less than 8d − 8 (which certainly map to 0), the relations in H^{∗}(P_{3} ∧ R) are

Sq^{2}(x_{1}^{3}x_{2}^{8d−11}),
Sq^{1}(x_{1}^{8i+3}x_{2}^{8d−8i−9}) + Sq(0, 1)(x_{1}^{3}x_{2}^{8d−11}), 1 ≤ i < d,
(Sq^{3} + Sq(0, 1))(x_{1}^{3}x_{2}^{8d−9}),
Sq^{1}(x_{1}^{7}x_{2}^{8d−9}) + Sq^{5}(x_{1}^{11}x_{2}^{8d−17}) + Sq(2, 1)(x_{1}^{3}x_{2}^{8d−9}),
Sq^{1}(x_{1}^{8i−1}x_{2}^{8d−8i−1}) + Sq^{5}(x_{1}^{8i−5}x_{2}^{8d−8i−1} + x_{1}^{8i+3}x_{2}^{8d−8i−9})
+(Sq^{7} + Sq(4, 1) + Sq(0, 0, 1))(x_{1}^{3}x_{2}^{8d−11}), 2 ≤ i < d.

For example, these relations correspond, when d = 5, to the classes in filtration 1 in Diagram 2.16 with 33 ≤ t − s ≤ 38.

One easily checks that these relations map to 0 in H^{∗}(Σ^{8d−9}P_{1}). For example, the image of the fourth relation is

σ^{8d−9}(Sq^{1} x^{7} + Sq^{5} x^{3} + Sq(2, 1)x^{3}) = σ^{8d−9}(x^{8} + 0 + x^{8}) = 0.

Thus there is a morphism

Ext_{A_{2}}^{s,t}(H^{∗}Σ^{8d−9}P_{1}) → Ext_{A_{2}}^{s,t}(H^{∗}(P_{3} ∧ R))

for t − s ≤ 8d − 2 sending the generator G in (s, t) = (0, 8d − 2) to g_{1} + ⋯ + g_{d−1}. Since h_{0}^{4}G = 0, we must have h_{0}^{4}(g_{1} + ⋯ + g_{d−1}) = 0.

Proof of Corollary 2.13. This proof is quite similar to the discussion of Example 1.3 and to the proof of [3, 1.3] given on [3, p.524]. We begin with case 1.

We first observe that if, for some i < h,

ν(k_{i}) < 4i − c_{s_{1}} or ν(k_{d−i}) < 4i − c_{s_{2}},

then our element ∑ k_{i}y_{i} must be nonzero, as it cannot be obtained as a consequence of relations (2.8)–(2.12). Proving this requires a bit of argument, which occupies the next two paragraphs.

The relations (2.8)–(2.12) can be manipulated into a form similar to that of [3, 1.4], except that where [3, 1.4] has a coefficient 2^{e}, we have 2^{4e−c_{s}}, and the binomial coefficients occurring as coefficients are only valid mod 2. For example, if h = 4 and t = 1, the relations can be arranged, after some manipulation, as the rows of the matrix

(\begin{matrix}2^{4} \\ & 2^{4} & 2^{4} & 2^{4} & 2^{4} & 2^{4} & 2^{4} & 2^{4} & 2^{4} & 2^{4} \\ & & 2^{8} & 2^{8} & 2^{8} & 2^{8} & 2^{8} & 2^{8} & 2^{8} \\ & & & 2^{8} & 2 · 2^{8} & 2^{8} & 2 · 2^{8} & 2^{8} & 2 · 2^{8} \\ & & & & 2^{12} & 2 · 2^{12} & 2^{12} & 2 · 2^{12} \\ & & & & & 2^{12} & 2^{12} & 2 · 2^{12} \\ & & & & & & 2^{16} \\ & & & & & & & 2^{16} \\ & & & & & & & & 2^{12} \\ & & & & & & & & & 2^{8} \\ & & & & & & & & & & 2^{4} \end{matrix}) (2.34)

with the understanding that each 2^{4i} entry is really an unspecified odd integer times 2^{4i}, while each 2 · 2^{4i} entry is really an unspecified even integer times 2^{4i}.

If, say, ν(k_{1}) ≥ 4, ν(k_{2}) ≥ 8, but ν(k_{3}) = 11, then use (2.8) to eliminate k_{1}y_{1} + k_{2}y_{2}. Now the third row of the above matrix of relations can be used to eliminate k_{3}y_{3} at the expense of changing our element to ∑_{i≥4} u_{i}2^{11}y_{i} with u_{i} odd. Use the fourth row of (2.34) to change this to

w_{5}2^{11}y_{5} + A_{6}2^{11}y_{6} + w_{7}2^{11}y_{7} + A_{8}2^{11}y_{8} + ⋯

with w_{i} odd and A_{i} even. No subsequent row can eliminate the lead term of this, and so the original class was nonzero. This procedure works in general, and if it is ν(k_{d−i}) which is too small, the matrix of relations can be considered from right to left, and the same argument invoked.

Now we can assume that ν(k_{i}) ≥ 4i−c_{s_{1}} and ν(k_{d−i}) ≥ 4i−c_{s_{2}} for i < h, and so the class in question reduces (using (2.8) and (2.9)) to ∑_{i=h}^{2h} k_{i}y_{i} with ν(k_{i}) ≥ 4h−t for all i. We will rewrite k_{h+j} as 2^{4h−t}k′_{j}. By (2.12), 2^{4h−t+1}y_{i} = 0 for all i, and so the analysis becomes effectively a mod 2 analysis, and the indeterminacy in (2.10) becomes inconsequential.

For example, in the case, h = 4, t = 1, considered above, restricting attention to columns 4 through 7,^{2} we wish to know whether a given vector

v = (k′_{0}2^{15}, k′_{1}2^{15}, k′_{2}2^{15}, k′_{3}2^{15})

is in the span of the matrix

M = (\begin{matrix}2^{8} & A_{1}2^{8} & u_{1}2^{8} & A_{2}2^{8} \\ 0 & 2^{12} & A_{3}2^{12} & u_{2}2^{12} \\ 0 & 0 & 2^{12} & u_{3}2^{12} \end{matrix}) (2.35)

considered mod 2^{16}, with u_{i} odd and A_{i} even. The only way to obtain the first component, k′_{0}2^{15}, of our vector is by 2^{7}k′_{0} times the first row of M. Since we are working mod 2^{16}, the precise values of the coefficients A_{1}, u_{1}, and A_{2} do not matter, only their parity. Working similarly with the other entries and rows, we find that v is in the span of M iff the mod 2 vector (k′_{0}, k′_{1}, k′_{2}, k′_{3}) is in the span of

2column 8 is not required

(\begin{matrix}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{matrix}) ,

which is obtained from M by just considering parity of coefficients. Now case 1 of the corollary follows from the case (r = 3, h = 4) of Lemma 2.38.

The proof of case 2 of the corollary is almost identical to that of case 1. In each of the three subcases, the analysis of the relations reduces, similarly to case 1, to determining whether a vector

v := (k′_{0}2^{4h−t}, k′_{1}2^{4h−t}, …, k′_{h−1}2^{4h−t})

is in the span of a mod 2^{4h−t+1} matrix whose rows are multiples by various 2powers of the corresponding rows of the matrix M_{h} of Lemma 2.38 with r = 2. The condition (2.14) is what Lemma 2.38 says is necessary for our vector to not be in the span of the relations.

The reason that case 1 of 2.13 uses r = 3 in Lemma 2.38 while case 2 uses r = 2 is that the columns in the matrix (such as (2.35)) of case 1 are h through 2h − 1, while those in case 2 are h + 1 through 2h, and this causes a corresponding change in exactly which binomial coefficients occur in the various rows. Indeed the relevant matrix M_{h} is obtained by first manipulating the relations into a form illustrated by the prototype matrix (2.34), and then considering the (h − 1) × h submatrix whose upper left hand element is in position (h, h) or (h + 1, h + 1) (depending on which case we are in) of the larger matrix. Since, for relevant values of i, the (i, j)th entry of the big matrix is 2^{pwr}\binom{[(i−2)/2]+j−i}{j−i}, the effect on the entries of M_{h} is obtained.

But why the difference in which submatrix we consider? The difference is that in case 1, we use (2.8) for i ≤ h − 1, while in case 2, we use it for i ≤ h. This difference is due to the fact that, with e as in (2.12), we have 4h − c_{s_{1}} = e in case 1, but 4h − c_{s_{1}} < e in case 2.

We require h to be even in the third subcase of case 2, for if h is odd, the last subcase of (2.11) implies 2^{4h−1}y_{h+1} = 0 (when d = 3h + 1), and then (2.12) implies that 2^{4h−1} annihilates all y_{i} in this dimension, and so an element ∑ k_{i}y_{i} satisfying the conditions of the third subcase of case 2 of 2.13, except that h is not even, would be 0, contrary to our conclusion.

The first two subcases of Case 3 are almost identical to Case 1. One apparent difference is the use of d = 3h + 3 here rather than d = 3h as in Case 1. The reason for using 3h + 3 here is to make the congruence hypothesis in Theorem 1.1.3 look nicer and be the same in all subcases.

If h = 3 in the first subcase of Case 3, the matrix of relations would be just like (2.34) except that the last three entries would become 2^{11}, 2^{7}, and 2^{3}. As already discussed, these tail entries do not play a role in the results that we derive. If h = 3 in the second subcase of Case 3, the exponents of 2 which are divisors of the successive rows of the matrix of relations when it is put in a form similar to (2.34) are 3,4,7,8,11,12,15,13,9,5,1. The fact that the largest of these is 15, rather than 16 as it was in subcase 1, is what drives our hypothesis about wanting various ν-values to be 14. The way in which we have selected which values of (s_{1}, s_{2}) to use is to use the largest values which have a given maximum order in the group corresponding to a fixed value of d.

The third subcase of Case 3 contains one slightly different feature. We illustrate when h = 4. The reader can easily adapt the argument to the general case.

The relations are 2^{1}y_{1}, 2^{5}y_{2}, 2^{9}y_{3}, 2^{13}y_{4}, 2^{12}y_{10}, 2^{8}y_{11}, 2^{4}y_{12}, 2^{16}y_{i}, 5 ≤ i ≤ 9, 2^{4} ∑ y_{j} mod 2^{5}, 2^{8} ∑ jy_{j} mod 2^{9}, and 2^{12} ∑ \binom{j}{2}y_{j} mod 2^{13}. These can be manipulated in matrix form to

(\begin{matrix}2^{1} \\ & 2^{4} & 2^{4} & 2^{4} & 2^{4} & 2^{4} & 2^{4} & 2^{4} & 2^{4} & 2^{4} & 2^{4} \\ & & 2^{5} & 2^{5} & 2^{5} & 2^{5} & 2^{5} & 2^{5} & 2^{5} & 2^{5} & 2^{5} \\ & & & 2^{8} & 2 · 2^{8} & 2^{8} & 2 · 2^{8} & 2^{8} & 2 · 2^{8} & 2^{8} \\ & & & & 2^{9} & 2 · 2^{9} & 2^{9} & 2 · 2^{9} & 2^{9} & 2 · 2^{9} \\ & & & & & 2^{12} & 2^{12} & 2 · 2^{12} & 2 · 2^{12} \\ & & & & & & 2^{13} & 2^{13} & 2 · 2^{13} \\ & & & & & & & 2^{16} \\ & & & & & & & & 2^{16} \\ & & & & & & & & & 2^{12} \\ & & & & & & & & & & 2^{8} \\ & & & & & & & & & & & 2^{4} \end{matrix}) (2.36)

where, as with (2.34), each 2^{i} (resp. 2 · 2^{i}) means an odd (resp. even) integer times 2^{i}.

We are given a row (k_{1}, …, k_{12}) with ν(k_{i}) ≥ 15 for 5 ≤ i ≤ 9. Similarly to Case 1, one can show that if ν(k_{i}) < 4i − 3 for some i ≤ 4 or ν(k_{i}) < 4(13 − i) for some i ≥ 10, then the row is not in the span of (2.36).

Now assume ν(k_{i}) ≥ 4i − 3 for i ≤ 4 and ν(k_{i}) ≥ 4(13 − i) for i ≥ 10. The relations 2^{1}y_{1}, …, 2^{4}y_{12} may be used to eliminate k_{1}, k_{2}, k_{3}, k_{4}, k_{10}, k_{11}, and k_{12}. We use rows 5, 6, and 7 of (2.36) to try to eliminate the other k_{i}.

For 0 ≤ j ≤ 4, let k′_{j} = k_{5+j}/2^{15} mod 2. We find that ∑ k_{i}y_{i} = 0 iff (k′_{0}, …, k′_{4}) is in the span of the mod 2 matrix

(\begin{matrix}1 & 0 & 1 & 0 & 1 \\ & 1 & 1 & 0 & 0 \\ & & 1 & 1 & 0 \end{matrix}) . (2.37)

The only combination of rows which could possibly equal (k′_{0}, …, k′_{4}) is

k′_{0}R_{1} + k′_{1}R_{2} + (k′_{0} + k′_{1} + k′_{2})R_{3},

where R_{1}, R_{2}, R_{3} are the rows of (2.37). Consideration of columns 4 and 5 shows that our vector is 0 iff k′_{0} + k′_{1} + k′_{2} + k′_{3} ≡ 0 mod 2 and k′_{1} + k′_{2} + k′_{3} + k′_{4} ≡ 0 mod 2.

The first of these necessary conditions, if applied to an axial map by the method employed in Case 3 of Section 3, would yield nonimmersions already proved in [3], and so we do not pursue it. The second of the necessary conditions is the one quoted in the third subcase of 2.13.3.

The novelty in this final subcase was our obtaining two necessary conditions, and just choosing the second.

We close this section with a lemma used in the above proof.

Lemma 2.38. Let r = 2 or 3, and let M_{h} denote the (h − 1) × h matrix over Z/2 with (i, j)th entry

\binom{[(h + i − r)/2] + j − i}{j − i}

for 1 ≤ i ≤ h − 1 and 1 ≤ j ≤ h. A vector v = (k_{0}, k_{1}, …, k_{h−1}) is in the span of M_{h} iff

∑_{j=0}^{h−1} \binom{h−1}{j}k_{j} = 0 ∈ Z/2.

Proof. For 1 ≤ i ≤ h−1, let R_{i} denote the ith row of M_{h}. Using combinatorial identities, one can show that the combination of rows which agrees with v in the first h − 1 entries is

∑_{i=1}^{h−1} R_{i} ∑_{j=0}^{i−1} \binom{[(i + h − r + 1)/2]}{i − j − 1}k_{j}.

Its entry in the last column is ∑_{j=0}^{h−2} \binom{h−1}{j}k_{j}, which must equal k_{h−1}, yielding the lemma.

3. Deduction of nonimmersions

In this section we deduce Theorem 1.1 from Corollary 2.13.

Proof of Theorem 1.1.1. We use the first type of axial map in 1.2. Thus it will suffice to prove that, if t = 1 or 2,

(X_{1} + X_{2})^{2M−h} ≠ 0 ∈ tmf^{∗}(P^{8M+8h−2t−2} ∧ P^{8M+8h−2t−2}) (3.1)

if M ≡ 0 mod \frac{1}{2}p(h + 1) and α(M) = 4h + 1 − t. By Corollary 2.13.1 with a = b = M + h, (3.1) will hold provided ν\binom{2M−h}{M+h−i} ≥ 4h − t for h ≤ i ≤ 2h and

ν(∑_{j=0}^{h−1} \binom{h−1}{j}\binom{2M−h}{M−j}) = 4h − t. (3.2)

The sum in (3.2) equals \binom{2M−1}{M}, and since, ν\binom{2M−1}{M} = α(M) − 1, our result follows from the following lemma.

Lemma 3.3. If 2^{e} ≤ h < 2^{e+1} and M ≡ 0 mod 2^{e}, then ν\binom{2M−h}{M−j} ≥ α(M) − 1 for 0 ≤ j ≤ h.

Proof. We must show that if δ_{1} ≥ δ_{2} ≥ 0 and δ_{1} + δ_{2} = h, then

α(M − δ_{1}) + α(M − δ_{2}) ≥ α(M) − 1 + α(2M − h). (3.4)

Case 1: δ_{1} < 2^{e}.

Let M = 2^{e}m. Then (3.4) reduces to

α(m − 1) + α(2^{e} − δ_{1}) + α(m − 1) + α(2^{e} − δ_{2}) ≥ α(m) − 1 + α(m − 1) + α(2^{e+1} − h),

which is true since α(m − 1) ≥ α(m) − 1 and

α(2^{e} − δ_{1}) + α(2^{e} − δ_{2}) − α(2^{e+1} − h) = ν\binom{2^{e+1} − h}{2^{e} − δ_{1}} ≥ 0.

Case 2: δ_{1} ≥ 2^{e}, M = 2^{e+1}m.

Same argument as Case 1, with e replaced by e + 1. Case 3: δ_{1} = 2^{e} + Δ ≥ 2^{e}, M = 2^{e} + 2^{e+1}n.

Here (3.4) reduces to

α(n − 1) + α(2^{e+1} − Δ) + α(n) + α(2^{e} − δ_{2}) ≥ α(n) + α(n) + α(2^{e} − Δ − δ_{2}),

which is true since

α(2^{e+1} − Δ) + α(2^{e} − δ_{2}) − α(2^{e} − Δ − δ_{2}) = ν\binom{2^{e+1} + 2^{e} − Δ − δ_{2}}{2^{e+1} − Δ} + 1.

Proof of Theorem 1.1.2. Using the second type of axial map in 1.2, we have for 1 ≤ s_{i} ≤ 4 that if

(X_{1} + X_{2})^{2^{L}−m+1} ≠ 0 ∈ tmf^{∗}(P^{8m−2s_{1}} ∧ P^{8(2^{L−3}−2m+1+d)−2s_{2}}),

then

P^{8m−2s_{1}} ⊈ R^{16m−8d−10+2s_{2}}. (3.5)

With m = M + h + 1, the three subcases in 2.13.2 turn (3.5) into the three nonimmersions of 1.1.2. We will see that the hypotheses of 1.1.2 (namely α(M) = 4h − t and M ≡ 0 mod p(h)) imply that the conditions of 2.13.2 are satisfied when

k_{i} = \binom{2^{L}−m+1}{m−i} = (−1)^{m−i}\binom{2m−i−2}{m−2}, (3.6)

which will imply 1.1.2. The sum condition of (2.14) becomes

ν(∑_{j=0}^{h−1} \binom{h−1}{j}\binom{2^{L}−M−h}{M−j}) = 4h − t, (3.7)

which is satisfied since the LHS equals ν\binom{2^{L}−M−1}{M} = α(M). The other condition of 2.13.2 is the content of the following lemma.

Lemma 3.8. If 2^{e} < h ≤ 2^{e+1} and M ≡ 0 mod 2^{e+1}, then

ν\binom{2M+h−j−1}{M−j} ≥ α(M) for 0 ≤ j ≤ h − 1. (3.9)

Proof. Letting H = h−1 and M = 2^{e+1}n, the desired inequality is equivalent to

α(n − 1) + α(2^{e+1} − j) + α(n) + α(H) ≥ α(n) + α(H − j) + α(n)

for 0 ≤ j ≤ H < 2^{e+1}, and this reduces to

ν(n) + e + 1 ≥ ν\binom{H}{j} + ν(j), (3.10)

which is implied by the following result.

Proposition 3.11. If 0 ≤ j ≤ H < 2^{e+1}, then ν\binom{H}{j} + ν(j) ≤ e.

Proof. The proof is by induction on e. We write H = 2H′+ϵ_{0} and j = 2j′+ϵ_{1} with ϵ_{i} = 0 or 1. The hardest case is when ϵ_{0} = 0 and ϵ_{1} = 1. In this case we have

ν\binom{2H′}{2j′+1} = α(2j′ + 1) + α(2H′ − 2j − 1) − α(2H′)
= 1 + α(j′) + α(H′ − j′) + ν(H′ − j′) − α(H′)
= 1 + ν\binom{H′}{H′−j′} + ν(H′ − j′) ≤ e.

Proof of Theorem 1.1.3. With m = M +h+1, the three subcases in 2.13.3 turn (3.5) into the three nonimmersions of 1.1.3. With k_{i} again given by (3.6), the sum condition of 2.13.3 is satisfied because

ν(∑ \binom{h}{j}\binom{2^{L}−M−h}{M−j}) = ν\binom{2^{L}−M}{M} = α(M) − 1 = 4h + t

in the first two subcases, and

ν(∑ \binom{h−1}{j}\binom{2^{L}−M−h}{M−j−1}) = ν\binom{2^{L}−M−1}{M−1} = α(M) − 1 = 4h − 1

in the third. The other condition of 2.13.3 is the content of the following proposition.

Proposition 3.12. If M ≡ 2^{e_{0}} + 2^{e_{1}} mod 2^{e_{1}+1} with e_{0} < e_{1} and 1 ≤ h ≤ 2^{e_{1}} − 2^{e_{0}}, or M is a 2-power, then

ν\binom{2M+h−j−1}{M+h−1} ≥ α(M) − 1

for 0 ≤ j ≤ h.

Proof. The proposition is vacuously true if M is a 2-power. If M = 2^{e_{1}+1}a + 2^{e_{1}} + 2^{e_{0}} and 1 ≤ h ≤ 2^{e_{1}} − 2^{e_{0}}, then

ν\binom{2M + h − j − 1}{M + h − 1} = ν\binom{2^{e_{1}+2}a + D_{1}}{2^{e_{1}+1}a + D_{0}}

with 2^{e_{1}+1} < D_{1} < 2^{e_{1}+2} and D_{1} − 2^{e_{1}+1} < D_{0} < 2^{e_{1}+1}. Thus \binom{D_{1}}{D_{0}} is even, and

ν\binom{2M+h−j−1}{M+h−1} = α(a) + ν\binom{D_{1}}{D_{0}} ≥ α(M) − 1.

4. Comparison with known results

In [3], it was proved that

P^{2(m+α(m)−1)} ⊈ R^{4m−2α(m)}. (4.1)

Until now, (4.1) implied all known nonimmersions of P^{n} for n = 64A + d with 0 ≤ d ≤ 62 and α(A) ≥ 5.^{3} Our Theorem 1.1 improves upon (4.1) for many such values of n. We illustrate in Table 4.2, which also illustrates that we are far from being able to prove that our nonimmersions are best possible. We tabulate here for n = 128A + d with d even, α(A) = 6, and 16 ≤ d ≤ 76, the number e such that P^{n} ⊈ R^{256A+e} was proved in (4.1), denoted Previous, and then the best known value of e now, taking Theorem 1.1 into account. Improvement is obtained in nearly half of the cases, by as much as 17 dimensions. The value of e for which P^{n} is known to immerse in P^{256A+e} is also listed. The smallest gap between known immersion and nonimmersion dimensions in this range is 21 dimensions. All the immersions in this range are from [10]. We feel that the final answer will be close to the current nonimmersion dimension.

3For other values of n, (4.1) was often, but not always, the best known result.

Table 4.2.

d Previous Now Immerse
16  −6  −4  21
18  −6  −2  21
20  2  2  28
22  2  2  28
24  4  4  36
26  8  15  36
28  18  18  44
30  18  18  44
32  20  22  53
34  24  26  53
36  24  28  60
38  32  32  60
40  32  32  68
42  34  47  68
44  50  50  76
46  50  50  76
48  52  54  85
50  56  58  85
52  56  60  92
54  64  64  92
56  64  66  100
58  66  66  100
60  66  79  108
62  80  80  108
64  80  82  117
66  82  82  117
68  86  86  124
70  86  86  124
72  94  94  132
74  94  111  132
76  114  114  140

Note that it is always true in this table that either Previous(n) ≥ Now(n) − 2 or Previous(n + 2) ≥ Now(n). It was stated in [3] that, in this sense, (4.1) was within 2 dimensions of all known nonimmersions, namely the 2 dimensions can come from the Euclidean space or the projective space, or 1 from each. Our result, 1.1, does not change this: if P^{n} ⊈ R^{m} is proved in 1.1, then either P^{n} ⊈ R^{m−2} or P^{n+2} ⊈ R^{m} is implied by (4.1). However, in the interim, it was proved in [11] that if i > j > 3, then P^{2^{i}+2^{j}+10} ⊈ R^{2^{i+1}+2^{j+1}+11}, but all that is implied by (4.1) is P^{2^{i}+2^{j}+8} ⊈ R^{2^{i+1}+2^{j+1}+2} and P^{2^{i}+2^{j}+12} ⊈ R^{2^{i+1}+2^{j+1}+10}. So all that can be claimed is that (4.1) “comes within 3” of all known nonimmersions.

It is not true that every case of Theorem 1.1 is new. If α(M) < 6, then the results of 1.1 were proved in [5] or [6]. In those papers (and [10]), obstruction theory using MO[8] was employed to obtain nonimmersions. Since there is an A-module splitting H^{∗}(MO[8]) ≈ A//A_{2} ⊕ W for some A-module W, obstruction theory using MO[8] and using tmf will be similar, and identical in a range of dimensions. The advantage of tmf is our complete knowledge of possible obstructions to our obstructions. Because of its plethora of additional classes, we could only use MO[8] through a rather restricted range of dimensions.

The other cases in which 1.1 is not new is when the projective space which it shows not to immerse has dimension 2^{i}, 2^{i} + 2, 2^{i} + 4, 2^{i} + 2^{j} + 2, 2^{i} + 2^{j} + 4, or 2^{i} + 2^{j} + 2^{k} + 4. Strong (usually best possible) results are known for these spaces by (4.1) and earlier results. For example, 1.1.2b with h = 2^{e}, M = (2^{i} − 1)2^{e} with i = 2^{e+2} − 2 says that, with j = i + e + 3, P^{2^{j}} ⊈ R^{2^{j+1}−24·2^{e}+12}, whereas it is classical that P^{2^{j}} ⊈ R^{2^{j+1}−2}.

Aside from this, our results are new, as delineated in the following result. Here when we say that one nonimmersion result improves on another by d dimensions, we mean that it obtains a nonimmersion in a Euclidean space d dimensions larger than the other.

Proposition 4.3. With the exceptions described in the preceding paragraphs, Theorems 1.1.2 and 1.1.3c improve on previously known results by 2 dimensions. If ν(M) = 0, …, 10, Theorems 1.1.3a and 1.1.3b improve on known results by

d = 4, 6, 12, 16, 18, 28, 30, 36, 40, 42, 54 (4.4)

dimensions, respectively, while Theorems 1.1.1a and 1.1.1b improve on previous results by d + 3 and d + 1 dimensions, respectively, where d is as in (4.4).

For ν(M) > 10, the improvement will continue to increase in a pattern suggested by (4.4).

Proof. In parts 2a, 2b, 2c, and 3c of Theorem 1.1, the best previously known results were from (4.1) with m = 4M + 2, 4M + 2, 4M + 1, and 4M, respectively. Note that we choose m as large as possible so that 2(m+α(m)−1) ≤ the dimension of the projective space being studied. In part 3a, the best known results were from (4.1) with m = 4M − δ, where δ = 2, 2, 3, 4, 4, 6, 6, 7, 8, 8, 10 as ν(M) goes from 0 to 10. The relevance of ν(M) is from the equation

α(M − 1) = α(M) − 1 + ν(M),

which we have been using frequently without comment. Parts 1a and 1b use these same values of m = 4M − δ.

5. The spectrum tmf

In 1991, Hopkins and Miller defined a new collection of spectra, one of which is a 2-primary spectrum denoted EO_{2}. A thorough account of one approach to these spectra is given in [20]. See also [13] and [12], which emphasize the relationship of EO_{2} with elliptic curves and elliptic cohomology, and [19], which applies the Hopkins-Miller spectra to obtain new results in homotopy theory.

One approach starts with a formal group of height 2 over F_{4}, obtained from an elliptic curve. Using work of Lubin-Tate and the Landweber Exact Functor Theorem, one can associate to this a spectrum E_{2}. Work of Goerss, Hopkins, and Miller shows that E_{2} has the structure of E_{∞}-ring spectrum with the Morava stabilizer group S_{2} acting as a group of automorphisms. The profinite group S_{2} has a maximal finite subgroup G_{2} of order 24. The homotopy fixed point spectrum of E_{2} under the action of G_{2} is defined to be the Hopkins-Miller spectrum EO_{2}.

One key property of EO_{2} is

EO_{2} ∧ DA_{1} ≃ E_{2}. (5.1)

Here DA_{1}, called “the double of A_{1},” is a finite spectrum with

H_{∗}(DA_{1}) = Z_{2}[ξ_{1}^{2}, ξ_{2}^{2}]/(ξ_{1}^{8}, ξ_{2}^{4}), (|ξ_{i}| = 2^{i} − 1).

In 1992-3, Hopkins and Mahowald defined a connective version of EO_{2}, originally called eo_{2}, and now named tmf. It sits above the connected cover EO_{2}[0, ∞] in a certain pullback square. From (5.1), one deduces

tmf ∧ DA_{1} ≃ BP ⟨2⟩,

and from this the key property H^{∗}(tmf) = A//A_{2}. The construction and properties of tmf are described in [14] and [21].

The existence of such a spectrum caused some consternation, for Davis and Mahowald had published in [8] that such a spectrum could not exist. In that paper, they presented a valid deduction that if there is a certain self-map

v_{2}^{8} : Σ^{48}M(2, v_{1}^{4}) → M(2, v_{1}^{4}), (5.2)

then A//A_{2} cannot be realized as the cohomology of a spectrum. In [7], they had presented an argument for the existence of the map (5.2).

This argument in [7] for the existence of (5.2) involved detailed computations using the stable homotopy groups of spheres. In particular, they needed to know that the class \{\overline{h_{0}^{3}h_{3}h_{5}}\} ∈ π_{39}(M) satisfies v_{1}^{4}\{\overline{h_{0}^{3}h_{3}h_{5}}\} = 0. The mistake in their argument was their claim ([7, pp 653-4]) that v_{1}^{4}\{\overline{h_{0}^{3}h_{3}h_{5}}\} could not equal \{e_{0}r\}, a class of Adams filtration 10 in the 47-stem.

The argument presented noted that η{e_{0}r} ≠ 0 and claimed to show

ηv_{1}^{4}\{\overline{h_{0}^{3}h_{3}h_{5}}\} = 0.

The latter involved a delicate argument involving 5-fold Toda brackets. The precise mistake involved evaluation of a certain bracket ⟨2α, 2ι, η⟩.

The authors of [7] had another, unpublished, argument for v_{1}^{4}\{\overline{h_{0}^{3}h_{3}h_{5}}\} ≠ \{e_{0}r\}. This was that, if ϵ denotes the element in the 8-stem which is not in the image of J, then ϵv_{1}^{4}\{\overline{h_{0}^{3}h_{3}h_{5}}\} = 0, while ϵ\{e_{0}r\} ≠ 0. The latter equals \{P e_{0}r\}, which was thought to survive the ASS. Kochman’s work, ([16]), not ASS-based, also seemed to show {P e_{0}r} nonzero in homotopy in the 55-stem. The existence of tmf caused Kochman and Mahowald to take another look, resulting in their paper [17], which produced a previously unknown differential in the ASS, showing {P e_{0}r} = 0.

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[16] S. O. Kochman, Stable homotopy groups of spheres, a computer-assisted approach, Lecture Notes in Math, Springer-Verlag 1423 (1990).

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[18] 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.

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Wayne State University, Detroit, MI 48202

E-mail address: rrb@@math.wayne.edu

Lehigh University, Bethlehem, PA 18015

E-mail address: dmd1@@lehigh.edu

Northwestern University, Evanston, IL 60201

E-mail address: mark@@math.nwu.edu