Generalized axial maps and Euclidean immersions of lens spaces

Luis Astey, Donald M. Davis and Jesús González^{∗}

June 13, 2002

Abstract

Generalized axial maps are introduced and related to the immersion problem for 2-torsion lens spaces to produce a characterization of this problem in the metastable range. A direct analysis gives a complete solution of the immersion problem for the missing cases. Other variations of the problem, which extend classical constructions for projective spaces, are discussed.

Key words and phrases: Axial maps, immersions of lens spaces, obstruction theory, modified Postnikov towers. 2000 Mathematics Subject Classification: Primary 57R42. Secondary 55S45.

1 Introduction

The results in this paper are motivated by the well known relationship between the existence of Euclidean immersions of real projective spaces and the existence of axial maps of type (n, k), that is, maps of the form

P^{n} × P^{n} → P^{n+k}  (1)

which are homotopically nontrivial over each axis. B. J. Sanderson observed in [21] that an axial map as above can be obtained from a given immersion P^{n} ⊆ ℝ^{n+k} and that, at least in the metastable range (that is, when n < 2k), such an immersion can homotopically be recovered from (1). Sanderson’s work relies on results of Hirsch and Haefliger ([12, 14]) and takes advantage of the so-called twisted normal bundle associated to an immersion P^{n} ⊆ ℝ^{n+k}. The success of the technique depends on the fact that the canonical real line bundle over P^{n} has multiplicative order 2. However, this is precisely the main drawback in a first attempt to generalize the ideas for higher 2-torsion lens spaces. Indeed, on the one hand, Hirsch’s basic result on immersing manifolds [13] implies that the codimension in an optimal immersion for L^{2n+1}(2^{e}) —the (2n + 1)- dimensional 2^{e}-torsion lens space— agrees with the geometric dimension of −(n+1)ξ_{n,e}, where ξ_{n,e} is the realification of the canonical complex line bundle over L^{2n+1}(2^{e}); but on the other hand, ξ_{n,e} is not even a unit in KO(L^{2n+1}(2^{e})). We straighten out the situation by following a path, first suggested in [1] by Adem, Gitler and James, which naturally leads to the concept of generalized e-axial maps (to be formalized in Definition 2.2). Our main result is as follows.

∗The second author is supported by a grant from the Reidler Foundation. The third author is supported by CONACyT Grant No. 37296-E.

Theorem 1.1 If L^{2n+1}(2^{e}) immerses in Euclidean codimension k, then there is an eaxial map of the form a: S^{2n+1} ×_{e} P^{2n+1} → P^{2n+k+1}. The converse holds except perhaps for n = 2, 3 or 5.

Theorem 1.1 is valid for e = ∞ (as long as we interpret L^{2n+1}(2^{∞}) as ℂP^{n}) and this can help to understand the close relationship among Euclidean immersions of complex projective spaces and those for high 2-torsion lens spaces ([11, 22]).

In Section 2 we adapt the techniques in [1] to show that an e-axial map as in Theorem 1.1 must exist whenever L^{2n+1}(2^{e}) admits a codimension-k immersion. We also show that the converse holds, at least, in the metastable range. This last restriction misses only the cases with n < 8; they are studied in Section 3, where the validity of Theorem 1.1 is verified for n ∉ {2, 3, 5}^{1}. In any case, we settle the immersion problem for L^{2n+1}(2^{e}) when n < 9. The situation is summarized in (2), where we tabulate the optimal Euclidean dimensions in which L^{2n+1}(2^{e}) can be immersed —the last (well known) row, with e = ∞, is included for comparison purposes.

\begin{aligned}& n = 0 & n = 1 & n = 2 & n = 3 & n = 4 & n = 5 & n = 6 & n = 7 & n = 8 \\ e = 1 & 2 & 4 & 7 & 8 & 15 & 16 & 22 & 22 & 31 \\ e = 2 & 2 & 4 & 8 & 9 & 16 & 17 & 23 & 23 & 32 \\ 2 < e < ∞ & 2 & 4 & 8 & 10 & 16 & 18 & 23 & 23 & 32 \\ e = ∞ & ⋆ & 3 & 7 & 9 & 15 & 17 & 22 & 22 & 31 \end{aligned} (2)

The cases n = 2, 4, 8 are part of a general situation: with n = 2^{m} (m ≥ 1), it is well known that both P^{2n+1} and ℂP^{n} admit optimal immersions in ℝ^{4n−1}. In particular, any lens space L^{2n+1}(2^{e}) immerses in ℝ^{4n}, which is optimal for e ≥ 2 (see Section 3).

In Section 4 we give a more familiar characterization of a generalized axial map and relate the concept to the existence of 2^{e}-equivariant maps from an odd dimensional sphere into a Stiefel manifold, and to the sectioning problem of bundles over lens spaces (all of these being classical constructions for e = 1). In view of Lemma 2.7 and Remarks 2.6 and 4.2, it would be interesting to know to what extent a bilinear map technique can be used to improve a given immersion for P^{2n+1} to one for L^{2n+1}(2^{e}).

1We do not know whether n = 2, 3 or 5 are actual exceptional cases for Theorem 1.1.

2 Generalized axial maps

For a real vector bundle α over a space X, let S(α) and P (α) stand, respectively, for the sphere and projectivized bundles associated to α, and let h_{α} denote the canonical line bundle over P (α) splitting off π^{∗}(α), where π: P (α) → X is the projection.

Definition 2.1 Let P^{k} (k ≤ ∞) be the k-dimensional real projective space. We say that a map a: P (α) → P^{N} is axial when the composite P (α) →^{a} P^{N} ↪ P^{∞} classifies h_{α}.

We are interested in the following particular situation. Let ℂP^{n} be the n-dimensional complex projective space and, for e ≥ 1, let L^{2n+1}(2^{e}) be the (2n + 1)-dimensional 2^{e}- torsion lens space, that is, the quotient of S^{2n+1} by the restriction to ℤ/2^{e} of the diagonal action of S^{1} in ℂ^{n+1}. Let η_{n} (also denoted by η_{n,∞}) be the canonical complex line bundle over ℂP^{n} and let η_{n,e} be its pull back under the projection L^{2n+1}(2^{e}) → ℂP^{n}. We set ξ_{n,e} = r(η_{n,e}), the realification of η_{n,e}.

Definition 2.2 An e-axial map is any axial map associated to a multiple of ξ_{n,e} (e ≤ ∞). Thus, a : P (mξ_{n,e}) → P^{N} is e-axial provided the composite P (mξ_{n,e}) →^{a} P^{N} ↪ P^{∞} classifies h_{mξ_{n,e}}.

The following considerations provide us with a more manageable description for P (mξ_{n,e}). It is well known and simple to see that the total space of the iterated m-fold Whitney sum of η_{n,e} is the quotient of S^{2n+1} × ℂ^{m} by the equivalence relation which identifies (wx, y) and (x, wy), for (x, y) ∈ S^{2n+1} × ℂ^{m} and w ∈ ℤ/2^{e}. We denote the resulting space and the corresponding associated sphere bundle by S^{2n+1} ×_{e} ℂ^{m} and S^{2n+1} ×_{e} S^{2m−1}, respectively. In these terms we see that the projectivization P (mξ_{n,e}) is the quotient of S^{2n+1} ×P^{2m−1} by the free action of ℤ/2^{e} given by w ·(x, y) = (wx, w^{−1}y). We also use the more suggestive notation S^{2n+1} ×_{e} P^{2m−1} for this space. It is convenient to extend these observations to the case e = ∞ by setting ℤ/2^{∞} = S^{1} (we agree to interpret L^{2n+1}(2^{∞}) as ℂP^{n}).

Remark 2.3 In the situation above, the (nonfree) action of ℤ/2^{e} on P^{2m−1} is induced through the quotient S^{2m−1} → P^{2m−1}; in any case, this yields a free action of ℤ/2^{e−1} on P^{2m−1}. For instance, when e = ∞, the free action is given by w · x = w^{1/2}y, where y ∈ S^{2m−1} is any representative for x ∈ P^{2m−1}, and w^{1/2} is any square root of w ∈ S^{1}.

On the same lines, we note that S^{2n+1} ×_{e} P^{2m−1} has a more symmetric expression as the quotient of P^{2n+1} × P^{2m−1} by the action of ℤ/2^{e−1} given by w(x, y) = (wx, w^{−1}y). Here ℤ/2^{e−1} is acting freely on each projective space as above. For e = 1 this will help us to relate the concepts of 1-axial maps and (regular) axial maps (see Remark 4.2).

Remark 2.4 Back to the general situation in Definition 2.1, for any map f: Y → X, the naturality of the construction for h_{α} shows that the composite P (f^{∗}(α)) → P (α) ⟶^{a} P^{N} (where the first map is induced by f) is axial provided a is.

More generally, for a sub-vector bundle β of f^{∗}α, the restriction to P (β) of the above composite yields an axial map P (β) → P^{N}. This gives an easy but important way of constructing axial maps: Assume α has a complement γ so that α ⊕ γ is a trivial ℓdimensional (say) bundle over X. Then the composite P (α) → P (α⊕γ) = X ×P^{ℓ−1} ⟶^{π_{2}} P^{ℓ−1} yields an axial map, where π_{2} stands for projection onto the second factor (note that, as in the context above, starting with the trivial ℓ-dimensional vector bundle over a point, we can interpret π_{2} as the map induced by X → ⋆).

The relevance of the construction comes from Hirsch’s Theorem on immersions of manifolds which claims that an m-dimensional manifold M admits an immersion in ℝ^{N} precisely when τ_{M} —the tangent bundle of M— admits an (N − m)-dimensional complement ν. In that case, the observations above yield axial maps P (τ_{M}) → P^{N−1} and P (ν) → P^{N−1}. The paper [1] studies the converse of these constructions for M a real projective space; we do a similar analysis for lens spaces.

Remark 2.5 One half of Theorem 1.1 is now self evident; namely, each Euclidean immersion of L^{2n+1}(2^{e}) in codimension k gives rise to an axial map S^{2n+1} ×_{e} P^{2n+1} → P^{2n+k+1}. Indeed, we only need to recall the bundle isomorphisms

τ_{L^{2n+1}(2^{e})} ⊕ 1 = (n + 1)ξ_{n,e}, for e < ∞, and τ_{ℂP^{n}} ⊕ 1_{ℂ} = (n + 1)η_{n,∞}.  (3)

Remark 2.6 In view of the final considerations in Remark 2.3, there are 2-fold covering projections

P^{2n+1} × P^{2m−1} → S^{2n+1} ×_{2} P^{2m−1} → ⋯ → S^{2n+1} ×_{e} P^{2m−1} → S^{2n+1} ×_{e+1} P^{2m−1}. (4)

Moreover, from Remark 2.4, we see that the composition of the last map in (4) with an (e + 1)-axial map yields a corresponding e-axial map. Similarly, the composition of the obvious projection S^{2n+1} ×_{e} P^{2m−1} → S^{2n+1} ×_{∞} P^{2m−1} with an ∞-axial map yields an e-axial map. In Section 4 we get a partial converse for these observations (Remark 4.2). In view of Theorem 1.1 and Remark 2.5, the considerations above should be compared with the fact that an immersion of ℂP^{n} in ℝ^{N} yields an immersion of L^{2n+1}(2^{e+1}) in ℝ^{N+1}; and the latter, in turn, induces a corresponding immersion for L^{2n+1}(2^{e}) (cf. [11]).

Corollary 2.9 below gives, in a certain range, the part of Theorem 1.1 not included in Remark 2.5. The proof requires the next auxiliary result whose verification, although elementary, is deferred until Section 4 (Proposition 4.3). In the rest of the section we take 1 ≤ e ≤ ∞.

Lemma 2.7 There is an e-axial map a: S^{2n+1} ×_{e} P^{2m−1} → P^{N} if and only if there is an e-skew map A: S^{2n+1} × S^{2m−1} → S^{N}, that is, a map satisfying A(x, −y) = −A(x, y) and A(w · x, y) = A(x, w · y) for all (x, y) ∈ S^{2n+1} × S^{2m−1} and w ∈ ℤ/2^{e}.

Corollary 2.8 Assume N − 2m ≥ n. If there is an e-axial map S^{2n+1} ×_{e} P^{2m−1} → P^{N}, then there exists an (N − 2m + 1)-dimensional vector bundle ν over L^{2n+1}(2^{e}) such that mξ_{n,e} ⊕ ν is trivial.

Proof: Let σ: S^{2n+1} ×_{e} ℂ^{m} → L^{2n+1}(2^{e}) × ℝ^{N+1} be obtained as the radial extension of a map S^{2n+1} ×_{e} S^{2m−1} → L^{2n+1}(2^{e}) × S^{N} whose first component is the bundle projection S(mξ_{n,e}) → L^{2n+1}(2^{e}) and whose second component is induced by an e-skew map as in Lemma 2.7. Then σ is a skew bundle map (in the usual sense) which, in view of [12, Theorem 1.2], can be deformed into a monomorphism of vector bundles. The result follows.

Corollary 2.9 If there is an e-axial map S^{2n+1} ×_{e} P^{2n+1} → P^{2n+k+1} when n < k, then L^{2n+1}(2^{e}) admits a Euclidean immersion in codimension k.

Proof: Take N = 2n + k + 1 and m = n + 1, then Corollary 2.8 shows that (n + 1)ξ_{n,e} admits a k-dimensional complement over L^{2n+1}(2^{e}) which, in view of (3) and Hirsch’s basic immersion theorem, corresponds to the normal bundle of a codimension-k Euclidean immersion of L^{2n+1}(2^{e}).

3 Low dimensional immersion problem

Remark 2.6 and the following result, which is a particular case of [1, Lemma 2.1], implies that the only cases of Theorem 1.1 not considered already in Section 2 are those with n ≤ 7.

Proposition 3.1 Suppose there is a 1-axial map P^{2n+1} × P^{2n+1} → P^{2n+k+1}, where n ≥ k. Then n ≤ 7.

We analyze each of these low dimensional cases. The following portion of table (2) consisting of known optimal immersions ([5, 6, 7]) will be helpful.

\begin{aligned}& n = 0 & n = 1 & n = 2 & n = 3 & n = 4 & n = 5 & n = 6 & n = 7 & n = 8 \\ ℝP^{2n+1} & 2 & 4 & 7 & 8 & 15 & 16 & 22 & 22 & 31 \\ ℂP^{n} & & 3 & 7 & 9 & 15 & 17 & 22 & 22 & 31 \end{aligned} (5)

In particular, as observed in Remark 2.6, this yields immersions of lens spaces L^{2n+1}(2^{e}). Part of the work below is to improve such an immersion (whenever it is possible) and analyze the optimality of the corresponding e-axial map. We only need to consider the case e > 1 in view of [1].

Case n = 2^{m}: For e ≥ 2, the immersion L^{2n+1}(2^{e}) ⊆ ℝ^{4n} in the introduction is optimal in view of [15, Corollary 1.1]. If moreover m ≥ 2, the nonimmersion L^{2n+1}(2^{e}) ⊈ ℝ^{4n−1} lies within the range of Corollary 2.9, verifying in this case the second part of Theorem 1.1.

Cases n = 6 and n = 7: Table (5) yields immersions L^{2n+1}(2^{e}) ⊆ ℝ^{23}. We show the corresponding e-axial maps are optimal. In view of the considerations on Remark 2.6, it suffices to show the non-existence of a 2-axial map S^{2n+1} ×_{2} P^{2n+1} → P^{22}. Moreover, we only need to consider the case n = 6 (take f to be the inclusion L^{13}(4) ↪ L^{15}(4), α = 8ξ_{7,2} and β = 7ξ_{6,2} in the considerations immediate to Remark 2.4) which, again, lies within the range of Corollary 2.9. Then the second part of Theorem 1.1, as well as the immersion problem in this case, follows from the nonimmersion L^{13}(4) ⊈ ℝ^{22} in the recent Ph. D. Thesis [22].

This completes the proof of Theorem 1.1 as well as the details in columns n = 4, 6, 7, 8 of (2) (the case n = 1 follows immediately from table (5) and Remark 2.6, whereas the case n = 0 is elementary). Table (5) and the next result complete the details in columns n = 2, 3, 5 of (2).

Proposition 3.2 (a) L^{11}(4) immerses in ℝ^{17}, while L^{11}(8) does not.

(b) L^{7}(4) immerses in ℝ^{9}, while L^{7}(8) does not.

(c) L^{5}(4) does not immerse in ℝ^{7}. L^{7}(4) does not immerse in ℝ^{8}. L^{11}(4) does not immerse in ℝ^{16}.

With respect to the missing cases in Theorem 1.1, except for the nonimmersion in (a), which lies in the range of Proposition 2.9, the authors do not know whether there exist axial maps corresponding to the nonimmersions in Proposition 3.2.

The last two nonimmersions in part (c) follow from [11, Theorem 1.1], whereas the proof of the first nonimmersion in part (c) is similar to, and simpler than (thus omitted), that of part (b).

Proof of Proposition 3.2 (b): The normal bundle ν of a possible immersion L^{7}(8) ⊆ ℝ^{9} would be two dimensional and orientable and, therefore, of the form ν = rη^{k} (0 ≤ k ≤ 7), where η = η_{3,3} and r stands for realification. In these conditions Hirsch’s theorem and (3) would yield an equation of the form

10 = r(η^{k}) ⊕ 4rη.  (6)

Since rη = r(η^{−1}), we only need to show (6) is impossible for 0 ≤ k ≤ 4. Recall ([18])

K̃O(L^{7}(8)) = ℤ/16 ⊕ ℤ/2  (7)

where the first summand is generated by rσ and the second by κ (σ = η − 1_{ℂ} and κ = ρ − 1_{ℝ}, with ρ the canonical real line bundle over L^{7}(8)). Note k must be positive in (6), otherwise we would have 8 = 4rη so that 0 = 4rσ, in contradiction to (7). Case k = 1 is also inconsistent with (6) as it yields 0 = 5rσ. To dispose of case k = 4 we recall η^{4} = cρ ([17, Proposition.3]), the complexification of ρ; thus, with k = 4, (6) yields 10 = rcρ ⊕ 4rη = 2ρ ⊕ 4rη and therefore 0 = 2κ + 4rσ, in contradiction to (7). Cases k = 2 and k = 3 are easier to consider after complexification (which, according to [18, Corollary 4.4], is monic under the present conditions). For instance, with k = 2, the complexification of (6) produces

0 = (η^{2} − 1 + η^{6} − 1) + 4(η − 1 + η^{7} − 1)
= (σ + 1)^{2} − 1 + (σ + 1)^{6} − 1 + 4(σ + (σ + 1)^{7} − 1)
= 160σ^{3} + 100σ^{2} + 40σ

(recall the well known relation σ^{4} = 0) which is incompatible with the fact ([18, Proposition 3.7]) that K̃(L^{7}(8) = ℤ/32 ⊕ ℤ/4 ⊕ ℤ/4 with respective generators σ, σ^{2} − 2σ and σ^{3} + 2σ^{2}. In an entirely similar way we rule out k = 3 proving, therefore, the nonimmersion L^{7}(8) ⊈ ℝ^{9}.

As for the second claim, according to [17], K̃O(L^{7}(4)) = ℤ/8 ⊕ ℤ/2, with respective generators rσ and κ + 2rσ; moreover there hold the relations 0 = (rσ)^{2} = −4rσ + 2κ, κ^{2} = −2κ and κ^{3} = 0. In particular, 0 = 2κ^{2} and 2κ = −2κ. Thus, using (3), the stable normal bundle of L^{7}(4) is −4rσ = 2κ which has geometric dimension at most 2. The required immersion follows from Hirsch’s Theorem.

The remainder of the section is devoted to the proof of Proposition 3.2 (a) which is, by far, the hardest part in Proposition 3.2. In view of the considerations in Remark 2.6, what we want is the critical 2-torsion required for the nonimmersions L^{11}(2^{e}) ⊈ ℝ^{17} (proved in [16] to hold for e ≥ 7). Consider the maps

L^{11}(4)⟶^{p_{1}}L^{11}(8)⟶^{p_{2}}ℂP^{5}⟶^{f}BSpin,  (8)

where p_{i} is the canonical projection map, while f classifies the stable normal bundle −6η of ℂP^{5}, with η = η_{5,∞}. By Hirsch’s theorem, it suffices to prove:

fp_{2}p_{1} factors through BSpin(6) → BSpin,  (9)

while

fp_{2} does not.  (10)

The method of proof is the obstruction-theoretic method of modified Postnikov towers (MPTs). We will use the following MPT, through dimension 11, where the left hand side of the bottom row is as in (8).

[figure: L(4)⟶^{p_{1}}L(8)⟶^{p_{2}}ℂP^{5}⟶^{f}; BSpin; ⟶ K(ℤ, 7) × K_{8}.; q_{1}; E_{1} ⟶^{k_{8}^{1},k_{10}^{1}} K_{8} × K_{10}; K(ℤ, 6) × K_{7}; q_{2}; E_{2} ⟶^{k_{10}^{2}} K_{10}; K_{7} × K_{9}; BSpin(6)]

(11)

Here and in what follows we abbreviate K(ℤ/2, m) by K_{m} and L^{11}(2^{e}) by L(2^{e}) (2 ≤ e ≤ 3). In this section all cohomology groups have coefficients in ℤ/2.

MPTs were introduced in [19] and [10]. Each vertical map in the above diagram is part of a fiber sequence preceded by the map from the fiber represented by a diagonal arrow, and followed by the classifying map represented by a horizontal arrow. The information of the diagram can be obtained from the Adams spectral sequence (ASS) of the stunted real projective space P_{6}—the quotient of P^{∞} by pinching P^{5} to a point— which is, through dimension 12, the fiber of BSpin(6) → BSpin. This ASS can be found in Table 8.7 on page 59 of [20]. We re-create it in the next chart.

(12)

[figure: π_{∗}(P_{6}), ∗ ≤ 10; 6; 8; 10]

The generalized Eilenberg-MacLane spaces on the right side of (11) give rise to the k-invariants for the MPT; these classes have dimensions 1 greater than those of the corresponding elements of π_{∗}(P_{6}). We will prove that

f lifts to a map ℂP^{5}⟶^{ℓ_{2}}E_{2}.  (13)

Note that the nonimmersion ℂP^{5} ⊈ ℝ^{16} ([23]) implies that ℓ_{2}^{∗}(k_{10}^{2}) ≠ 0. We will analyze the indeterminacy of this lifting and obtain the following result, which implies (9) and (10).

Proposition 3.3 There is a lifting of fp_{2}p_{1} to E_{2} which sends k_{10}^{2} trivially, but every lifting of fp_{2} to E_{2} sends k_{10}^{2} nontrivially.

As a first step toward proving (13), it is useful to factor f as

ℂP^{5}⟶^{h}ℍP^{2}⟶^{f′}BSpin,

where h is the canonical map to the quaternionic projective space, and f′ classifies the stable bundle −3H, with H the quaternionic Hopf bundle. Since \binom{−3}{2} = 6 is even, f′ lifts to a map ℓ_{1} : ℍP^{2} → E_{1}. Since H^{10}(ℍP^{2}) = 0, ℓ_{1}h sends k_{10}^{1} to 0 ∈ H^{10}(ℂP^{5}).

In order to determine the indeterminacy for lifting ℂP^{5} and L(2^{e}) in this MPT, we must know the relations which give rise to the k-invariants. These are computed by the method initiated in [10] and utilized in many subsequent papers by the second-named author (e.g. [8]) and also in papers of K. Y. Lam and/or D. Randall. It is a matter of building a minimal resolution using Massey-Peterson algebras. The relations for the MPT in (11) are given in the table below, where β is the integral Bockstein whose mod-2 reduction equals Sq^{1}.

βw_{6}
w_{8}

k_{8}^{1} : Sq^{2}βw_{6} + Sq^{1}w_{8}
k_{10}^{1} : (Sq^{4}+w_{4})βw_{6} + Sq^{3}w_{8}

k_{10}^{2} : Sq^{2}Sq^{1}k_{8}^{1} + Sq^{1}k_{10}^{1}

If ℓ_{1}^{∗}(k_{8}^{1}) ≠ 0, we vary the map ℓ_{1}h : ℂP^{5} → E_{1} through the first factor of the fiber K(ℤ, 6) × K_{7}. The primary indeterminacy is computed using the above relations. The relation for k_{10}^{1} means that the action map μ : (K(ℤ, 6) × K_{7}) × E_{1} → E_{1} sends k_{10}^{1} to

1 ⊗ k_{10}^{1} + Sq^{4}ι_{6} ⊗ 1 + ι_{6} ⊗ w_{4} + Sq^{3}ι_{7} ⊗ 1.

With y denoting the generator of H^{2}(ℂP^{5}), the new lifting ℓ′_{1}

ℂP^{5} ⟶^{y^{3}×ℓ_{1}h} K(ℤ, 6) × K_{7} × E_{1} ⟶^{μ} E_{1}

sends k_{10}^{1} to

h^{∗}ℓ_{1}^{∗}(k_{10}^{1}) + Sq^{4}(y^{3}) + y^{3} · w_{4}(−6η) = 0 + y^{5} + y^{5} = 0.

Thus h^{∗}ℓ_{1}^{∗}(k_{10}^{1}) is unchanged. A similar computation shows

ℓ′_{1}^{∗}(k_{8}^{1}) = h^{∗}ℓ_{1}^{∗}(k_{1}^{8}) + Sq^{2}(y^{3}) = y^{4} + y^{4} = 0.

Since ℓ′_{1}^{∗} sends both k-invariants to 0, it lifts to a map ℓ_{2} : ℂP^{5} → E_{2}, yielding (13).

Let ℓ_{8}^{2} = ℓ_{2}p_{2}, ℓ_{4}^{2} = ℓ_{2}p_{2}p_{1} and ℓ_{2^{e}}^{1} = q_{2}ℓ_{2^{e}}^{2}. Since Sq^{1} = 0 in H^{∗}(L(2^{e})), the primary indeterminacy for ℓ_{4}^{2} and ℓ_{8}^{2} is 0. This means that varying through either factor of K_{7} × K_{9} will not change the image of k_{10}^{2} in H^{10}(L(2^{e})).

Secondary indeterminacy requires checking whether varying ℓ_{2^{e}}^{1} through K(ℤ, 6)×K_{7} and then lifting the altered map might change the image of k_{10}^{2}. First note that if we vary through K(ℤ, 6), the image of k_{8}^{1} is changed, and so the altered map no longer lifts to E_{2}.

Let F_{0,2} denote the fiber of E_{2} → BSpin. There is a fibration

K_{7} × K_{9} → F_{0,2} → K(ℤ, 6) × K_{7}

and compatible actions

\xymatrix{F_{0,2} × E_{2} \ar[d] \ar[r] & E_{2} \ar[d] \\ K(ℤ, 6) × K_{7} × E_{1} \ar[r] & E_{1}.}

Because of the above remark about K(ℤ, 6), we consider just K_{7}, and define X_{1} by the pullback diagram

\xymatrix{X_{1} \ar[d] \ar[r] & F_{0,2} \ar[d] \\ K_{7} \ar[r]^{i_{2}} & K(ℤ, 6) × K_{7}.}

There is a fiber sequence

X_{1} → K_{7} ⟶^{Sq^{1},Sq^{3}} K_{8} × K_{10}  (14)

and a commutative diagram of actions

\xymatrix{X_{1} × E_{2} \ar[d] \ar[r]^{μ′} & E_{2} \ar[d] \\ K_{7} × E_{1} \ar[r] & E_{1}.}

Let x_{7}: L(2^{e}) → K_{7} be the nontrivial map. Since Sq^{1} = 0 in H^{∗}(L(2^{e})), x_{7} lifts to a map x′: L(2^{e}) → X_{1}, and the new lifting ℓ′_{2}: L(2^{e}) → E_{2} is given by μ′(x′ × ℓ_{2^{e}}^{2}). It satisfies

(ℓ′_{2})^{∗}(k_{10}^{2}) = (ℓ_{2^{e}}^{2})^{∗}(k_{10}^{2}) + (x′)^{∗}j^{∗}(k_{10}^{2}),

where j : X_{1} → E_{2} is the inclusion.

The fibration (14) and class j^{∗}(k_{10}^{2}) ∈ H^{10}(X_{1}) comprise a secondary cohomology operation, associated to the Adem relation(s) (Sq^{2}Sq^{1})Sq^{1} + Sq^{1} · Sq^{3}. Note that each term is an Adem relation by itself, and, since the mod-4 Bockstein β_{2} (that is, with respect to the nontrivial extension 0 → ℤ/2 → ℤ/8 → ℤ/4 → 0, the boundary operation defined on ℤ/2-classes with a ℤ/4-representative) is the secondary operation associated with the Adem relation Sq^{1} · Sq^{1}, our secondary operation is Sq^{2}β_{2} + β_{2}Sq^{2}. Thus, since (ℓ_{2^{e}}^{2})^{∗}(k_{10}^{2}) is the nonzero class x_{10},

(ℓ′_{2})^{∗}(k_{10}^{2}) = x_{10} + Sq^{2}(β_{2}x_{7}) + β_{2}(Sq^{2}x_{7}).

Since Sq^{2}(H^{8}(L(2^{e}))) = 0 and Sq^{2}(H^{7}(L(2^{e}))) ≠ 0, and β_{2} = 0 in H^{∗}(L(8)) but β_{2}(x_{2i−1}) ≠ 0 ∈ H^{∗}(L(4)), we obtain

(ℓ′_{2})^{∗}(k_{10}^{2}) \{ \begin{matrix}= 0 & for L(4), \\ ≠ 0 & for L(8), \end{matrix}

which implies Proposition 3.3.

4 Characterizations for axial maps

As in the previous section, we only consider cohomology groups with coefficients in ℤ/2. Let x ∈ H^{1}(L^{2n+1}(2^{e})) (when e < ∞) and y ∈ H^{2}(L^{2n+1}(2^{e})) denote the nontrivial elements (so that y = x^{2} for e = 1, but x^{2} = 0 for 1 < e < ∞). Let z ∈ H^{1}(S^{2n+1} ×_{e} P^{2m−1}) be the Stiefel-Whitney class of the canonical real line bundle over S^{2n+1}×_{e}P^{2m−1} (recall S^{2n+1} ×_{e} P^{2m−1} is the projectivization of mξ_{n,e}). By the Leray-Hirsch Theorem, the projection S^{2n+1} ×_{e} P^{2m−1} → L^{2n+1}(2^{e}) injects H^{∗}(L^{2n+1}(2^{e})) into H^{∗}(S^{2n+1} ×_{e} P^{2m−1}) and, as an algebra over the former ring, the latter is generated by z subject to the single relation

0 = ∑_{i=0}^{m} \binom{m}{i} z^{2m−2i}y^{i} = (z^{2} + y)^{m}.  (15)

Consider the maps ι: P^{2m−1} → S^{2n+1} ×_{e} P^{2m−1} and Δ: L^{2d+1}(2^{e}) → S^{2n+1} ×_{e} P^{2m−1}, where d = min{n, m − 1}, defined as follows: (a) ι is the composite P^{2m−1}↪S^{2n+1} × P^{2m−1} → S^{2n+1} ×_{e} P^{2m−1}, where the first map is inclusion into the second factor; (b) Δ is induced by the composite S^{2d+1}→S^{2n+1} × S^{2m−1} → S^{2n+1} ×_{e} P^{2m−1}, where the first map takes x into the pair (x, \overline{x}). Here \overline{x} means conjugating all complex coordinates in x ∈ S^{2d+1} ⊂ ℂ^{d+1}.

Proposition 4.1 Assume N ≥ 2m. A map a: S^{2n+1} ×_{e} P^{2m−1} → P^{N} is e-axial if and only if aΔ is null-homotopic but aι is essential.

Proof: The inclusion P^{N} ↪ P^{∞} has S^{N} as homotopy fiber and, therefore, induces a one-to-one map [X, P^{N}] → [X, P^{∞}] for any CW complex X of dimension less than N. Thus, in this situation, the (non)triviality of a given homotopy class X → P^{N} can be decided by its effect on H^{1}( ). Let ω ∈ H^{1}(P^{N}) be the generator and, for a: S^{2n+1} ×_{e} P^{2m−1} → P^{N} set a^{∗}(ω) = rx + sz, where r, s ∈ ℤ/2 and x and z are as established at the beginning of the section. By definition a is e-axial precisely when r = 0 and s = 1. Hence, by the dimensional restriction, it suffices to verify the relations ι^{∗}(x) = 0, Δ^{∗}(x) = x, ι^{∗}(z) ≠ 0 and Δ^{∗}(z) = 0 (only the last two when e = ∞). As x lies in the image of π^{∗}, the first one follows at once from the fibration

P^{2m−1} →^{ι} S^{2n+1} ×_{e} P^{2m−1} →^{π} L^{2n+1}(2^{e}).

The second one holds since πΔ is the inclusion L^{2d+1}(2^{e}) ↪ L^{2n+1}(2^{e}). The third one is a consequence of the general fact that, when restricted to the fiber of π, the canonical line bundle over S^{2n+1} ×_{e} P^{2m−1} agrees with the universal bundle over P^{2m−1}. The last relation follows from dimensional reasons for e = ∞, and by naturality (Remark 2.4) of the following diagram when e < ∞.

[figure: L^{2d+1}(2^{e}); Δ; S^{2n+1} ×_{e} P^{2m−1}; P (mξ_{n,e}); ℂP^{d}; Δ; S^{2n+1} ×_{∞} P^{2m−1}; P (mξ_{n,∞}).]

Remark 4.2 In view of Remark 2.3, the last result implies that a 1-axial map is just an axial map in the usual sense. Moreover, a 1-axial map P^{2n+1} × P^{2m−1} → P^{N} factoring through the maps in (4) induces a map S^{2n+1} ×_{e+1} P^{2m−1} → P^{N} which is (e + 1)-axial if and only if the first condition in Proposition 4.1 holds (note that, for e = ∞, the first condition is always automatic). At any rate, since the projection L^{2d+1}(2^{e}) → L^{2d+1}(2^{e+1}) induces the trivial morphism in mod 2 cohomology of dimension 1, we see that the composite S^{2n+1} ×_{e} P^{2m−1} → S^{2n+1} ×_{e+1} P^{2m−1}→P^{N} would indeed be e-axial.

Proposition 4.3 The following conditions are equivalent:

(a) There is an e-axial map S^{2n+1} ×_{e} P^{2m−1} → P^{N}.

(b) There is an e-skew map S^{2n+1} × S^{2m−1} → S^{N}.

(c) The N + 1 iterated Whitney sum of the canonical line bundle over S^{2n+1} ×_{e} P^{2m−1} admits an everywhere nontrivial section.

Proof: Note that the principal ℤ/2-bundle associated to the canonical line bundle over S^{2n+1} ×_{e} P^{2m−1} is given by the obvious projection S^{2n+1} ×_{e} S^{2m−1} → S^{2n+1} ×_{e} P^{2m−1}.

In these terms, any e-axial map a: S^{2n+1} ×_{e} P^{2m−1} → P^{N} can be covered by a ℤ/2equivariant map α: S^{2n+1} ×_{e} S^{2m−1} → S^{N}. Conversely, any such α covers, by definition, some e-axial map. This proves the equivalence between (a) and (b). The equivalence between (a) and (c) follows from [9, Proposition 1.3].

We close the section by discussing the relationship of both the immersion problem for 2-torsion lens spaces and axial maps to the existence of equivariant maps into Stiefel type manifolds (in the case e = 1, this was first considered in [4] and studied in more detail in [3]).

Let Mono(ℝ^{2m}, ℝ^{N+1}) be the Stiefel manifold of orthogonal monomorphisms, and let Skew(S^{2m−1}, S^{N}) be the space of (regular) skew maps, that is, continuous maps q: S^{2m−1} → S^{N} satisfying q(−x) = −q(x). In [12] (see also [14]) it is proved that the inclusion

Mono(ℝ^{2m}, ℝ^{N+1}) ↪ Skew(S^{2m−1}, S^{N})  (16)

is a (2N − 4m + 1)-homotopy equivalence. Note that (16) is a ℤ/2^{e}-equivariant map, where w ∈ ℤ/2^{e} acts by precomposing with x ↦ wx. Now, while the adjoint of a Gauss map mξ_{n,e} → ℝ^{N+1} (as the one given in Corollary 2.8) can be interpreted as a twisted ℤ/2-equivariant map g: S^{2n+1} → Mono(ℝ^{2m}, ℝ^{N+1}) —that is, g(wx) = w^{−1}g(x), for x ∈ S^{2n+1} and w ∈ ℤ/2^{e}—, the adjoint of a e-skew map as in (b) of Proposition 4.3 gives a corresponding twisted ℤ/2-equivariant map g: S^{2n+1} → Skew(S^{2m−1}, S^{N}). In particular, this yields an indirect approach for Theorem 1.1 as well as for the fact that, in the metastable range, any twisted equivariant map from S^{2n+1} into Skew(S^{2m−1}, S^{N}) can equivariantly be deformed into Mono(ℝ^{2m}, ℝ^{N+1}). For the sake of completeness, we provide a direct proof for this last fact.

Proposition 4.4 Under the conditions above, and if N − 2m ≥ n, any twisted ℤ/2equivariant map S^{2n+1} → Skew(S^{2m−1}, S^{N}) can equivariantly be deformed into a twisted ℤ/2-equivariant map S^{2n+1} → Mono(ℝ^{2m}, ℝ^{N+1}).

Proof: The map in the hypothesis can be interpreted as a section s for the bundle on the right hand side in the diagram below.

[figure: Mono(ℝ^{2m}, ℝ^{N+1}); Skew(S^{2m−1}, S^{N}); S^{2n+1} ×_{ℤ/2^{e}} Mono(ℝ^{2m}, ℝ^{N+1}); π_{1}; S^{2n+1} ×_{ℤ/2^{e}} Skew(S^{2m−1}, S^{N}); π_{2}; L^{2n+1}(2^{e}); L^{2n+1}(2^{e}); s]

Now, (16) and the long exact homotopy sequences associated to these fibrations show that the middle horizontal map in the diagram is a (2N −4m+1)-homotopy equivalence.

Therefore, under the present hypothesis, s can be deformed into a map

σ: L^{2n+1}(2^{e}) → S^{2n+1} ×_{ℤ/2^{e}} Mono(ℝ^{2m}, ℝ^{N+1})

such that π_{1} ∘σ is homotopic to the identity. However, as π_{1} is a fibration, the homotopy lifting property assures that σ can further be deformed into an actual section for π_{1}.

As with the technique of bilinear maps, it would be interesting to explore the extent to which the above ideas can push known immersions of real projective spaces to immersions of 2-torsion lens spaces and complex projective spaces.

References

[1] J. Adem, S. Gitler, and I. M. James. On axial maps of a certain type. Bol. Soc. Mat. Mexicana (2), 17:59–62, 1972.

[2] M. F. Atiyah. K-theory. W. A. Benjamin, Inc., 1967.

[3] A. J. Berrick and A. D. Randall. On equivariant maps and immersions of real projective spaces. Lecture Notes in Math. 1370, Springer-Verlag, 57–62, 1989.

[4] P. E. Conner and E. E. Floyd. Fixed point free involutions and equivariant maps. Bull. Amer. Math. Soc. 66:416–441, 1960.

[5] M. C. Crabb . Immersing projective spaces in Euclidean space. Proc. Roy. Soc. Edinburgh Sect. A 117(1-2):155–170, 1991.

[6] D. M. Davis. Table of immersions and embeddings of real projective spaces. Avail-

able at the web site http://www.lehigh.edu/~dmd1/immtable.

[7] D. M. Davis and M. E. Mahowald. Immersions of complex projective spaces and the generalized vector field problem. Proc. London Math. Soc. (3), 35(2):333–344, 1977.

[8] D. M. Davis and M. Mahowald. The geometric dimension of some vector bundles over projective spaces. Trans. Amer. Math. Soc., 205:295–315, 1975.

[9] S. Gitler and D. Handel. The projective Stiefel manifolds—I. Topology, 7:39–46, 1968.

[10] S. Gitler and M. E. Mahowald. The geometric dimension of real stable vector bundles. Bol. Soc. Mat. Mex., 11:85–107, 1966.

[11] J. González. Connective K-theoretic Euler classes and non-immersions of 2^{k}-lens spaces. J. London Math. Soc. (2) 63(1):247–256, 2001.

[12] A. Haefliger and M. W. Hirsch. Immersions in the stable range. Ann. of Math. (2), 75:231–241, 1962.

[13] M. W. Hirsch. Immersions of manifolds. Trans. Amer. Math. Soc., 93:242–276, 1959.

[14] I. M. James. The space of bundle maps. Topology 2:45–59, 1963.

[15] B. Junod. A non-immersion result for lens spaces L^{n}(2^{m}). Math. J. Okayama Univ. 37:137–151, 1995.

[16] B. Junod. Quelques résultats de non-immersion des espaces lenticulaires L^{n}(2^{m}). C. R. Acad. Sci. Paris, 306:279–282, 1988.

[17] T. Kobayashi and M. Sugawara. K_{Λ}-rings of lens spaces L^{n}(4). Hiroshima Math. J. 1:253–271, 1971.

[18] T. Kobayashi and M. Sugawara. On stable homotopy types of stunted lens spaces, II. Hiroshima Math. J. 7(3):689–705, 1977.

[19] M. E. Mahowald. On obstruction theory in orientable fibre bundles. Trans. Amer. Math. Soc., 110:315–349, 1964.

[20] M. E. Mahowald. The metastable homotopy of S^{n}. Mem. Amer. Math. Soc. 72, 1967.

[21] B. J. Sanderson. A non-immersion theorem for real projective space. Topology, 2:209–211, 1963.

[22] T. Shimkus. Immersions of 2-torsion lens spaces. Ph.D. Thesis. Department of Mathematics, Lehigh University, Bethlehem, PA. 2002.

[23] F. Sigrist and U. Suter. On immersions CP^{n} ⊆ R^{4n−2α(n)}. In Algebraic topology (Proc. Conf., Univ. British Columbia, Vancouver, B.C., 1977), pages 106–115. Springer, Berlin, 1978.

Departamento de Matemáticas, CINVESTAV–IPN, México

lastey@math.cinvestav.mx

Lehigh University, Bethlehem, PA 18015

dmd1@lehigh.edu

Departamento de Matemáticas, CINVESTAV–IPN, México

jesus@math.cinvestav.mx