EMBEDDINGS OF REAL PROJECTIVE SPACES

DONALD M. DAVIS

1. Statement of results

The question of finding the smallest Euclidean space in which real projective space P^{n} can be (differentiably) embedded was the subject of intense investigation during the 1960s and 1970s. The purpose of this paper is to survey the status of the question, and add a little bit to our knowledge by proving one new family of embeddings, using old methods of obstruction theory. Our new result is given in the following theorem.

Theorem 1.1. If n = 2^{i} + 3 ≥ 11, then P^{n} can be embedded in R^{2n−4}.

As far as I can tell, this improves on previous embeddings by 1 dimension. Indeed, Berrick’s 1979 table ([4]) lists the best embedding for P^{n} to be in R^{2n−3} when n = 2^{i} + 3, from [20], and I know of no embedding results for P^{n} proved subsequent to Berrick’s table. (There are, however, subsequent nonembedding results, notably those of [3].)

The following table lists the best nonembedding and embedding results for P^{n} of which I am aware, for n ≤ 63. Of course, most of these results fit into infinite families. Here we use the symbol ⊂ to refer to differentiable embeddings.

1991 Mathematics Subject Classification. 57R40.

Key words and phrases. embeddings, real projective spaces, obstruction theory.

n ⊄ ref ⊂ ref
2 3 [21] 4 [12]
3 4 [15] 5 [12]
4 7 [21] 8 [12]
5 8 [15] 9 [12]
6 8 [15] 11 [8]
7 8 [15] 12 [16]
8 15 [21] 16 [12]
9 16 [15] 17 [12]
10 16 [15] 17 [19]
11 16 [15] 18 new
12 17 [3] 21 [16]
13 21 [9] 23 [16]
14 21 [9] 23 [20]
15 22 [2] 24 [20]
16 31 [21] 32 [12]
17 32 [15] 33 [12]
18 32 [15] 33 [19]
19 32 [15] 34 new
20 33 [3] 37 [16]
21 38 [2] 39 [16]
22 38 [2] 39 [20]
23 38 [2] 39 [20]
24 38 [2] 42 [22]
25 45 [1] 47 [16]
26 45 [1] 48 [22]
27 46 [2] 48 [22]
28 46 [2] 50 [22]
29 46 [2] 52 [22]
30 47 [3] 53 [6]
31 52 [13] 54 [5]

n ⊄ ref ⊂ ref
32 63 [21] 64 [12]
33 64 [15] 65 [12]
34 64 [15] 65 [19]
35 64 [15] 66 new
36 65 [3] 69 [16]
37 70 [2] 71 [16]
38 70 [2] 71 [20]
39 70 [2] 71 [20]
40 70 [2] 74 [22]
41 77 [1] 79 [16]
42 77 [1] 80 [22]
43 78 [2] 80 [22]
44 78 [2] 82 [22]
45 78 [2] 84 [22]
46 79 [3] 85 [6]
47 79 [3] 86 [5]
48 83 [3] 90 [22]
49 93 [1] 95 [16]
50 93 [1] 96 [22]
51 94 [2] 96 [22]
52 94 [2] 98 [22]
53 94 [2] 100 [22]
54 95 [3] 101 [6]
55 95 [3] 102 [5]
56 99 [3] 106 [22]
57 99 [3] 108 [22]
58 103 [7] 110 [22]
59 103 [7] 111 [6]
60 107 [3] 114 [22]
61 107 [3] 116 [22]
62 107 [3] 117 [5]
63 114 [13] 117 [5]

Note that some of the nonembedding results (those of [1], [13], [9], and [7]) are actually obtained from nonimmersion results.

2. Proof of theorem

In this section, we prove Theorem 1.1. Our method is that used by Mahowald in [16]. A main tool is the following result, which was proved in [16], following [8]. In

EMBEDDINGS OF REAL PROJECTIVE SPACES 3

Section 3, we will add a few details to the argument given in [16]. Let ξ_{q} denote the Hopf bundle over P^{q}, and let ε denote the trivial bundle.

Theorem 2.1. Assume that P^{q} embeds in R^{p} with normal bundle ν.

• If ν ⊗ ξ_{q} has n linearly independent sections and P^{n−1} embeds in S^{m−1}, then P^{q+n} embeds in R^{p+m}.

• (ν ⊗ ξ_{q}) ⊕ (q + 1)ε ≈ (p + 1)ξ_{q}.

Here the hypothesized embeddings need only be topological, and the embedding in the conclusion is only topological. We then use the result of Haefliger ([11]) that a topological embedding of an n-manifold in R^{d} can be approximated by a differentiable embedding provided 2d ≥ 3(n + 1), in our application of this theorem.

To prove Theorem 1.1, we apply Theorem 2.1 with q = 2^{i} + 2, p = 2^{i+1} + 1, n = 1, and m = 1. Throughout this section, we assume i ≥ 3. The embedding of P^{2^{i}+2} in R^{2^{i+1}+1} was proved by Nussbaum in [19]. Theorem 1.1 will then follow from the following result (with θ = ν ⊗ ξ) together with the embedding of P^{0} in S^{0}.

Proposition 2.2. If θ is an orientable (2^{i} − 1)-plane bundle over P^{2^{i}+2} such that

θ ⊕ (2^{i} + 3)ε ≈ (2^{i+1} + 2)ξ,

then θ has a nonzero section.

Proof. We apply obstruction theory to the following diagram.

[figure: P^{2^{i}+2}; BSO(2^{i} − 1); θ; BSO(2^{i} − 2); S^{2^{i}−2}]

The desired lifting of θ to BSO(2^{i} − 2) is proved using modified Postnikov towers (MPTs), as introduced in [16] and refined in [10]. In the diagram below, H_{n} = K(Z, n)

4 D. DAVIS

and K_{n} = K(Z/2, n). Each vertical map is part of a fiber sequence with the diagonal maps on either side of it in the diagram; e.g.,

K_{2^{i}} × K_{2^{i}+1} → E_{3} → E_{2} → K_{2^{i}+1} × K_{2^{i}+2}

is a fiber sequence.

[figure: P^{2^{i}+2}; θ; BSO(2^{i} − 1); H_{2^{i}−1}; E_{1}; K_{2^{i}} × K_{2^{i}+2}; K_{2^{i}+1} × K_{2^{i}+2}; K_{2^{i}+2}; E_{2}; E_{3}; BSO(2^{i} − 2); k^{1}; H_{2^{i}−2}; K_{2^{i}−1} × K_{2^{i}+1}; K_{2^{i}} × K_{2^{i}+1}]

The k-invariants correspond to elements in an Adams resolution of the stable sphere, where we kill the initial Z all at once. We need the relations which give rise to the k-invariants in the MPT. These are computed by the method initiated in [10] and used in many subsequent papers such as [7] and [14]. It is a matter of building a minimal resolution using Massey-Peterson algebras. The relations are as in the table below, with β denoting the Bockstein.

βw_{2^{i}−2}

k_{2^{i}}^{1} : (Sq^{2} +w_{2})(βw_{2^{i}−2})
k_{2^{i}+2}^{1} : (Sq^{4} +w_{4})(βw_{2^{i}−2})
k_{2^{i}+1}^{2} : (Sq^{2} +w_{2})k_{2^{i}}^{1}
k_{2^{i}+2}^{2} : Sq^{1} k_{2^{i}+2}^{1} + (Sq^{2} Sq^{1} +w_{3})k_{2^{i}}^{1}
k_{2^{i}+2}^{3} : Sq^{1} k_{2^{i}+2}^{2} + (Sq^{2} +w_{2})k_{2^{i}+1}^{2}

We illustrate how these relations are used, using k_{2^{i}+1}^{2}. Its relation means that, with μ the action map,

(K_{2^{i}−1} × K_{2^{i}+1}) × E_{2} ⟶^{μ} E_{2} ⟶^{k_{2^{i}+1}^{2}} K_{2^{i}+1}

corresponds to the class 1 ⊗ k_{2^{i}+1}^{2} + Sq^{2} ι_{2^{i}−1} ⊗ 1 + ι_{2^{i}−1} ⊗ w_{2}. It also means that in H^{∗}(E_{1}),

Sq^{2} k_{2^{i}}^{1} + w_{2}k_{2^{i}}^{1} = 0. (2.3)

Since H^{2^{i}−1}(P^{2^{i}+2}; Z) = 0, θ lifts to a map ℓ_{1} : P^{2^{i}+2} → E_{1}. Suppose ℓ_{1}^{∗}(k_{2^{i}}^{1}) = ϵ_{0}x^{2^{i}} and ℓ_{1}^{∗}(k_{2^{i}+2}^{1}) = ϵ_{2}x^{2^{i}+2}. By (2.3),

0 = ϵ_{0} Sq^{2} x^{2^{i}} + w_{2}(θ)ϵ_{0}x^{2^{i}} = 0 + ϵ_{0}x^{2^{i}+2},

and so ϵ_{0} = 0. If ϵ_{2} = 1, then let ℓ′_{1} denote the composite

P^{2^{i}+2} ⟶^{x^{2^{i}−2}×ℓ_{1}} H_{2^{i}−2} × E_{1} ⟶^{μ} E_{1}.

This will satisfy

ℓ′_{1}^{∗}(k_{2^{i}+2}^{1}) = (x^{2^{i}−2} × ℓ_{1})^{∗}(1 ⊗ k_{2^{i}+2}^{1} + Sq^{4} ι_{2^{i}−2} ⊗ 1 + ι_{2^{i}−2} ⊗ w_{4})
= x^{2^{i}+2} + Sq^{4} x^{2^{i}−2} + x^{2^{i}−2} · w_{4}(θ)
= x^{2^{i}+2} + x^{2^{i}+2} + 0 = 0.

We don’t have to worry about whether varying through H_{2^{i}−2} changes ℓ^{∗}(k_{2^{i}}^{1}) because we have already shown that this is 0 for any lifting. Thus a lifting to E_{1} can be chosen which sends both level-1 k-invariants to 0, and hence lifts to a map ℓ_{2} : P^{2^{i}+2} → E_{2}.

To show that ℓ_{2}^{∗}(k_{2^{i}+1}^{2}) = 0, we need the following result, whose proof will appear at the end of this section.

Lemma 2.4. The map θ extends to a map P^{2^{i}+3} ⟶^{θ′} BSO(2^{i} − 1).

Since our tower has no k-invariants in dimension 2^{i}+3, θ′ lifts to a map ℓ′_{2} : P^{2^{i}+3} → E_{2} by the same analysis used to lift θ. Note that the MPT was constructed through dimension 2^{i} +2, i.e. the space at the top agrees with BSO(2^{i} −2) through dimension 2^{i} + 2. Usually we would say that even though the space at the top may not agree with BSO(2^{i} − 2) through dimension 2^{i} + 3, that is not of concern because in the end we are just using the lifting of P^{2^{i}+2}; the (2^{i} + 3)-cell is just used to detect relations at early stages of the lifting. However, in the case at hand, since π_{2^{i}+2}(S^{2^{i}−2}) = 0, the space at the top of the tower agrees with BSO(2^{i} − 2) through dimension 2^{i} + 3.

If (ℓ′_{2})^{∗}(k_{2^{i}+1}^{2}) = ϵ_{1}x^{2^{i}+1} and (ℓ′_{2})^{∗}(k_{2^{i}+2}^{2}) = ϵ_{2}x^{2^{i}+2}, then by the relation for k_{2^{i}+2}^{3}, we have

0 = ϵ_{2} Sq^{1} x^{2^{i}+2} + ϵ_{1} Sq^{2} x^{2^{i}+1} + x^{2} · ϵ_{1}x^{2^{i}+1} = 0 + 0 + ϵ_{1}x^{2^{i}+3},

and so ϵ_{1} = 0. This is where we need Lemma 2.4. If ϵ_{2} ≠ 0, then varying ℓ_{2} through K_{2^{i}+1} will yield a new lifting which sends k_{2^{i}+2}^{2} to 0, thanks to the term Sq^{1} k_{2^{i}+2}^{1} in the relation k_{2^{i}+2}^{2}.

Thus there is a lifting ℓ′′_{2} : P^{2^{i}+2} → E_{2} sending both k-invariants trivially, and hence a lifting ℓ_{3} : P^{2^{i}+2} → E_{3}. If ℓ_{3} sends the lone k-invariant at height 3 nontrivially, then varying ℓ_{3} through K_{2^{i}+1} gives a new lifting which sends the k-invariant to 0, and hence lifts to BSO(2^{i} − 2), as desired.

Proof of Lemma 2.4. By [14, 3.1], (2^{i+1} + 2)ξ_{2^{i}+3} has 2^{i} + 4 linearly independent sections, and hence there is a map f′ : P^{2^{i}+3} → BSO(2^{i}−2) which classifies (2^{i+1}+2)ξ. Let f = i∘f′, where i : BSO(2^{i}−2) → BSO(2^{i}−1) is the inclusion. By the hypothesis of Proposition 2.1, our map θ and f|P^{2^{i}+2} differ by a map δ which can be factored as

P^{2^{i}+2} → V_{2^{i}−1} → BSO(2^{i} − 1),

since V_{2^{i}−1} is the fiber of BSO(2^{i} − 1) → BSO. By Lemma 2.5, any such δ extends over P^{2^{i}+3}, and hence θ = f + δ is the sum of two maps which extend over P^{2^{i}+3}.

Lemma 2.5. The restriction

[P^{2^{i}+3}, V_{2^{i}−1}] → [P^{2^{i}+2}, V_{2^{i}−1}]

of sets of homotopy classes of maps is surjective.

Proof. We will show that the target group is Z/8, generated by the class of the composite

P^{2^{i}+2} ⟶^{c} P_{2^{i}−1}^{2^{i}+2} ↪ V_{2^{i}−1},

with c the collapse map. This composite clearly extends to

P^{2^{i}+3} → P_{2^{i}−1}^{2^{i}+3} ↪ V_{2^{i}−1},

EMBEDDINGS OF REAL PROJECTIVE SPACES 7

which will establish the lemma.

We begin by observing that π_{∗}(V_{2^{i}−1}) ≈ π_{∗}^{s}(P_{2^{i}−1}), the stable homotopy groups of a stunted projective space, through dimension at least 2^{i} + 3. These groups can be computed by the Adams spectral sequence, as done for example in [17, Table 8.16], to begin π_{j}(V_{2^{i}−1}) ≈ Z/2 if j = 2^{i} − 1, 2^{i} + 1, or 2^{i} + 2, with Adams filtrations 0, 1, and 2, respectively, and π_{2^{i}}(V_{2^{i}−1}) = 0. By obstruction theory, this implies that the order of [P^{2^{i}+2}, V_{2^{i}−1}] is no greater than 8. The claim is that these three classes extend cyclically.

One way to see this begins by noting that the desired group equals [P^{2^{i}+2}, P_{2^{i}−1}^{2^{i}+3}] and is in the stable range. There is a degree-1 map P_{2^{i}−1}^{2^{i}+3} → S^{2^{i}}. Let F be its fiber. We will show that in the exact sequence

[P^{2^{i}+2}, F] → [P^{2^{i}+2}, P_{2^{i}−1}^{2^{i}+3}] → [P^{2^{i}+2}, S^{2^{i}}] (2.6)

the first group is 0, and the third group is Z/8. Since we have already observed that the middle group has order no greater than 8, and has an element of Adams filtration 0, the claim made in the first line of this proof will be established.

The third group in (2.6) is isomorphic to π_{−1}(Σ^{2^{i}}P_{−2^{i}−3}), using S-duality. This is Z/8 by an elementary Adams spectral sequence calculation; it is the group in the column labeled 2 in [17, Table 8.14]. It is not difficult to check that H^{∗}(P_{−2^{i}−3} ∧F) is, through dimension 0, a free module over the Steenrod algebra on classes of dimension −4, −2, and 0. Thus the Adams spectral sequence or elementary obstruction theory implies that π_{−1}(P_{−2^{i}−3} ∧ F), which equals the first group in (2.6), is 0.

3. Review of Mahowald proof

In this section, we prove Theorem 2.1, merely fleshing out a few details in the proof given in [16]. The second part of 2.1 is elementary:

(ν ⊗ ξ) ⊕ (q + 1)ε ≈ (ν ⊕ (q + 1)ξ) ⊗ ξ
≈ (ν ⊕ τ ⊕ ε) ⊗ ξ
≈ (p + 1)ε ⊗ ξ
≈ (p + 1)ξ

8 D. DAVIS

The first part utilizes the following result. Throughout this section, embeddings (⊂) are not necessarily differentiable.

Proposition 3.1. If nξ_{q} ⊂ R^{p} and P^{n−1} ⊂ S^{m−1}, then P^{q+n} ⊂ R^{p+m}.

Proof of first part of Theorem 2.1. By hypothesis, nε ⊂ ν⊗ξ. Tensoring with ξ yields the first part of

nξ ⊂ ν ⊂ R^{p},

with the second part clear since ν can be taken to be a tubular neighborhood of P^{q} in R^{p}. The theorem now follows immediately from Proposition 3.1.

To prove Proposition 3.1, we will need the description of P^{n+q} given in the following lemma.

Lemma 3.2. There is a homeomorphism

P^{n−1} ∪_{g} D(nξ_{q}) ≈ P^{n+q}, (3.3)

where g : S(nξ_{q}) → P^{n−1} is defined by g([x, y]) = [x]. Here we use the model S(nξ_{q}) = (S^{n−1} × S^{q})/(x, y) ∼ (−x, −y).

Proof. The space on the LHS of (3.3) is

D^{n} × S^{q}/((x, y) ∼ (−x, −y), (x, y) ∼ (x, y′) if |x| = 1). (3.4)

The desired homeomorphism sends [(x, y)] to [(x, \sqrt{1 − |x|^{2}}y)] in S^{n+q}/z ∼ −z.

Proof of Proposition 3.1. Let f denote the embedding of D(nξ_{q}) in R^{p}, and h the embedding of P^{n−1} in S^{m−1}. Using the descriptions of P^{n+q} given by the LHS of (3.3) and (3.4), we define our embedding by

ϕ([(x, y)]) = (ϕ_{1}(x, y), ϕ_{2}(x, y))
ϕ_{1}(x, y) = (1 − |x|)f(x, y)
ϕ_{2}(x, y) = |x|h([x/|x|]).

Here x ∈ D^{n} and y ∈ S^{q}. We first show ϕ is well-defined. Because of the (1 − |x|)- factor, the second relation in (3.4) is not a problem, and because of the |x|-factor, we have continuity at x = 0, even though x/|x| is not defined. Injectivity is proved as follows. If ϕ_{2}(x, y) = ϕ_{2}(x′, y′), then x′ = ±x. If also ϕ_{1}(x, y) = ϕ_{1}(x′, y′), then either |x| = 1 and hence [(x, y)] = [(±x, y′)], or (x′, y′) = (±x, ±y) so that [(x, y)] = [(x′, y′)].

References

1. J. Adem and S. Gitler, Nonimmersion theorems for real projective spaces, Bol. Soc. Mat. Mex. 9 (1964) 37-50.

2. J. Adem, S. Gitler, and M. Mahowald, Embedding and immersion of projective spaces, Bol. Soc. Mat. Mex. 10 (1965) 84-88.

3. L. Astey, A cobordism obstruction to embedding manifolds, Ill. Jour. Math. 31 (1987) 344-350.

4. A. J. Berrick, Projective space immersions, bilinear maps, and stable homotopy groups of spheres, SV Lecture Notes in Mathematics 788 (1979) 1-22.

5. , Oxford Univ. D. Phil. thesis (1973).

6. , The Smale invariants of an immersed projective space, Math. Proc. Camb. Phil. Soc. 86 (1979) 401-412.

7. D. M. Davis, Some new immersions and nonimmersions of real projective spaces, AMS Contemporary Math. 19 (1983) 51-64.

8. D. B. A. Epstein and R. L. E. Schwarzenberger, Imbeddings of real projective spaces, Annals of Math. 76 (1962) 180-184.

9. S. Gitler, The projective Stiefel manifolds, II-applications, Topology 7 (1968) 47-53.

10. S. Gitler and M. E. Mahowald, The geometric dimension of real stable vector bundles, Bol. Soc. Mat. Mex. 11 (1966) 85-107.

11. A. Haefliger, Plongements différentiable dans le domaine stable, Comm. Math. Helv. 37 (1962) 155-176.

12. H. Hopf, Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Räume, Vierteljschr. Naturforsch. Gesellschaft Zurich 85 (1940) 165-177.

13. I. M. James, On the immersion problem for real projective spaces, Bull. Amer. Math. Soc. 69 (1963) 231-238.

14. K. Y. Lam, Sectioning vector bundles over real projective spaces, Quar. Jour. Math. Oxford 23 (1972) 97-106.

15. M. E. Mahowald, On the embeddability of the real projective spaces, Proc. Amer. Math. Soc. 13 (1962) 763-764.

16. , On obstruction theory in orientable fibre bundles, Trans. Amer. Math. Soc. 110 (1964) 315-349.

17. , The metastable homotopy of S^{n}, Mem. Amer. Math. Soc. 72 (1967).

18. M. E. Mahowald and R. J. Milgram, Embedding real projective spaces, Annals of Math. 87 (1968) 411-422.

19. F. Nussbaum, Obstruction theory of possibly nonorientable fibrations, Northwestern Univ. PhD thesis (1970).

20. E. Rees, Embeddings of real projective spaces, Topology 10 (1971) 309-312.

21. R. Thom, Espaces fibrés en spheres et carrés de Steenrod, Ann. Sci. Ecole Norm. Sup. 69 (1952) 109-182.

22. E. Thomas, Embedding manifolds in Euclidean space, Osaka Jour. Math. 13 (1976) 163-186.

Lehigh University, Bethlehem, PA 18015 E-mail address: dmd1@lehigh.edu