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 ncan be (differentiably) embedded was the subject of intense investigation du* *ring the 1960s and 1970s. The purpose of this paper is to survey the status of the q* *uestion, 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 the* *orem. Theorem 1.1. If n = 2i+ 3 11, then P ncan be embedded in R2n-4. As far as I can tell, this improves on previous embeddings by 1 dimension. In* *deed, Berrick's 1979 table ([4]) lists the best embedding for P nto be in R2n-3 when * *n = 2i+ 3, from [19], and I know of no embedding results for P nproved 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. 1 2 D. DAVIS ||_n_||6__ref_||____ref_ || | n |6 ref | ref | | 32 |63 [20] |64 [11] | |____|_______|________||||| || || || || || 2 |3|[20] |4|[11] || ||33 |64|[14] |65|[11] || || 3 |4|[14] |5|[11] || ||34 |64|[14] |65|[18] || || 4 |7|[20] |8|[11] || ||35 |64|[14] |66| new || || 5 |8|[14] |9|[11] || ||36 |65|[3] |69|[15] || || 6 |8|[14] |11[8]| || ||37 |70|[2] |71|[15] || || 7 |8|[14] |12[17]| || ||38 |70|[2] |71|[19] || || 8 |15[20]||16[11]| || ||39 |70|[2] |71|[19] || || 9 |16[14]||17[11]| || ||40 |70|[2] |74|[21] || ||10 |16[14]||17[18]||| ||41 |77|[1] |79|[15] || ||11 |16[14]||18|new || ||42 |77|[1] |80|[21] || ||12 |17[3]| |21[15]| || ||43 |78|[2] |80|[21] || ||13 |22[2]| |23[15]| || ||44 |78|[2] |82|[21] || ||14 |22[2]| |23[19]| || ||45 |78|[2] |84|[21] || ||15 |22[2]| |24[19]| || ||46 |79|[3] |85|[6] || ||16 |31[20]||32[11]| || ||47 |79|[3] |86|[5] || ||17 |32[14]||33[11]| || ||48 |83|[3] |90|[21] || ||18 |32[14]||33[18]||| ||49 |93|[1] |95|[15] || ||19 |32[14]||34|new || ||50 |93|[1] |96|[21] || ||20 |33[3]| |37[15]| || ||51 |94|[2] |96|[21] || ||21 |38[2]| |39[15]| || ||52 |94|[2] |98|[21] || ||22 |38[2]| |39[19]| || ||53 |94|[2] |100[21]| || ||23 |38[2]| |39[19]| || ||54 |95|[3] |101[6]| || ||24 |38[2]| |42[21]| || ||55 |95|[3] |102[5]| || ||25 |45[1]| |47[15]| || ||56 |99|[3] |106[21]| || ||26 |45[1]| |48[21]| || ||57 |99|[3] |108[21]| || ||27 |46[2]| |48[21]| || ||58 |103[7]| |110[21]| || ||28 |46[2]| |50[21]| || ||59 |103[7]| |111[6]| || ||29 |46[2]| |52[21]| || ||60 |107[3]| |114[21]| || ||30 |47[3]| |53[6]| || ||61 |107[3]| |116[21]| || |_31_|52[12]_|54[5]___| ||62 |107[3]| |117[5]| || |_63_|114[12]_|117[5]___| Note that some of the nonembedding results (those of [1], [12], and [7]) are ac* *tually obtained from nonimmersion results. 2.Proof of theorem In this section, we prove Theorem 1.1. Our method is that used by Mahowald in [15]. A main tool is the following result, which was proved in [15], following * *[8]. In EMBEDDINGS OF REAL PROJECTIVE SPACES 3 Section 3, we will add a few details to the argument given in [15]. Let q denot* *e the Hopf bundle over P q, and let " denote the trivial bundle. Theorem 2.1. Assume that P qembeds in Rp with normal bundle . o If q has n linearly independent sections and P n-1embeds in Sm-1 , then P q+nembeds in Rp+m . o ( q) (q + 1)" (p + 1)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 ([10]) * *that a topological embedding of an n-manifold in Rd can be approximated by a different* *iable embedding provided 2d 3(n + 1), in our application of this theorem. To prove Theorem 1.1, we apply Theorem 2.1 with q = 2i+ 2, p = 2i+1+ 1, n = 1, i+2 and m = 1. Throughout this section, we assume i 3. The embedding of P 2 i+1+1 in R2 was proved by Nussbaum in [18]. Theorem 1.1 will then follow from the following result (with = ) together with the embedding of P 0in S0. i+2 Proposition 2.2. If is an orientable (2i- 1)-plane bundle over P 2 such that (2i+ 3)" (2i+1+ 2); then has a nonzero section. Proof.We apply obstruction theory to the following diagram. i-2 S2 @R BSO(2i- 2) | | |? i+2 ___________- i P 2 BSO(2 - 1) The desired lifting of to BSO(2i- 2) is proved using modified Postnikov towe* *rs (MPTs), as introduced in [15] and refined in [9]. In the diagram below, Hn = K(* *Z; n) 4 D. DAVIS and Kn = K(Z2; n). We write Z=2 and Z2 interchangeably. Each vertical map is pa* *rt of a fiber sequence with the diagonal maps on either side of it in the diagram;* * e.g., K2ix K2i+1! E3 ! E2 ! K2i+1x K2i+2 is a fiber sequence. BSO(2i- 2) | | K2ix K2i+1 H Hj |? E3 HHj || K2i+2 K2i-1x K2i+1 H Hj |? E2 HHj || K2i+1x K2i+2 H2i-2H Hj |? E1 Hk1 | Hj K2ix K2i+2 | |? i+2________- i P 2 BSO(2 - 1) H Hj H2i-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 g* *ive rise to the k-invariants in the MPT. These are computed by the method initiated in [9] and used in many subsequent papers such as [7] and [13]. It is a matter* * of building a minimal resolution using Massey-Peterson algebras. The relations are* * as in the table below, with fi denoting the Bockstein. __________________________________ | fiw2i-2 | |_________________________________ | | k1i: (Sq 2+w2)(fiw2i-2) | | 2 | | k1i : (Sq 4+w4)(fiw2i-2) | |__2_+2___________________________ | | k2i : (Sq 2+w2)k1i | | 2 +1 2 | | k2i : Sq1k1i + (Sq 2Sq1+w3)k1i| |__2_+2_______2_+2_______________2_| | k3i : Sq1k2i + (Sq 2+w2)k2i | |__2_+2_______2_+2____________2_+1_ | EMBEDDINGS OF REAL PROJECTIVE SPACES 5 We illustrate how these relations are used, using k22i+1. Its relation means* * that, with the action map, k22i+1 (K2i-1x K2i+1) x E2 -! E2 ---! K2i+1 corresponds to the class 1 k22i+1+ Sq22i-1 1 + 2i-1 w2. It also means that in H*(E1), (2.3) Sq2 k12i+ w2k12i= 0: i-1 2i+2 2i+2 * 1 * * 2i Since H2 (P ; Z) = 0, lifts to a map `1 : P ! E1. Suppose `1(k2i) = * *ffl0x i+2 and `*1(K12i+2) = ffl2x2 . By (2.3), i 2i 2i+2 0 = ffl0Sq 2x2 + w2()ffl0x = 0 + ffl0x ; and so ffl0 = 0. If ffl2 = 1, then let `01denote the composite i+2x2i-2x`1 P 2 - ----! H2i-2x E1 -! E1: This will satisfy i-2 * 1 4 `01*(k12i+2)= (x2 x `1) (1 k2i+2+ Sq 2i-2 1 + 2i-2 w4) i+2 4 2i-2 2i-2 = x2 + Sq x + x . w4() i+2 2i+2 = x2 + x + 0 = 0: We don't have to worry about whether varying through H2i-2changes `*(k12i) beca* *use we have already shown that this is 0 for any lifting. Thus a lifting to E1 can * *be chosen i+2 which sends both level-1 k-invariants to 0, and hence lifts to a map `2 : P 2 * *! E2. To show that `*2(k22i+1) = 0, we need the following result, whose proof will * *appear at the end of this section. i+30 i Lemma 2.4. The map extends to a map P 2 -! BSO(2 - 1). * * i+3 Since our tower has no k-invariants in dimension 2i+3, 0lifts to a map `02: P* * 2 ! E2 by the same analysis used to lift . Note that the MPT was constructed through dimension 2i+2, i.e. the space at the top agrees with BSO(2i-2) through dimensi* *on 2i+ 2. Usually we would say that even though the space at the top may not agree with BSO(2i- 2) through dimension 2i+ 3, that is not of concern because in the * *end i+2 i we are just using the lifting of P 2 ; the (2 + 3)-cell is just used to detect* * relations 6 D. DAVIS i-2 at early stages of the lifting. However, in the case at hand, since ss2i+2(S2 * * ) = 0, the space at the top of the tower agrees with BSO(2i- 2) through dimension 2i+ * *3. i+1 0 * 2 2i+2 * * 3 If (`02)*(k22i+1) = ffl1x2 and (`2) (k2i+2) = ffl2x , then by the relati* *on for k2i+2, we have i+2 2 2i+1 2 2i+1 2i+3 0 = ffl2Sq 1x2 + ffl1Sq x + x . ffl1x = 0 + 0 + ffl1x ; and so ffl1 = 0. This is where we need Lemma 2.4. If ffl2 6= 0, then varying `2* * through K2i+1will yield a new lifting which sends k22i+2to 0, thanks to the term Sq1k12* *i+2in the relation k22i+2. i+2 Thus there is a lifting `002: P 2 ! E2 sending both k-invariants trivially, * *and hence i+2 a lifting `3 : P 2 ! E3. If `3 sends the lone k-invariant at height 3 nontri* *vially, then varying `3 through K2i+1gives a new lifting which sends the k-invariant to* * 0, and hence lifts to BSO(2i- 2), as desired. || Proof of Lemma 2.4.By [13, 3.1], (2i+1+ 2)2i+3 has 2i + 4 linearly independent i+3 i i+1 sections, and hence there is a map f0 : P 2 ! BSO(2 -2) which classifies (2 * *+2). Let f = iOf0, where i : BSO(2i-2) !fBSO(2i-1)iis the inclusion. By the hypothes* *is i+2 of Proposition 2.1, our map and ffifiP 2 differ by a map ffi which can be fac* *tored as i+2 i P 2 ! V2i-1! BSO(2 - 1); since V2i-1is the fiber of BSO(2i- 1) ! BSO. By Lemma 2.5, any such ffi extends i+3 2i+3 over P 2 , and hence = f + ffi is the sum of two maps which extend over P * *. || Lemma 2.5. The restriction i+3 2i+2 [P 2 ; V2i-1] ! [P ; V2i-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 i+2c 2i+2 P 2 -! P2i-1,! V2i-1; with c the collapse map. This composite clearly extends to i+3 2i+3 P 2 ! P2i-1,! V2i-1; EMBEDDINGS OF REAL PROJECTIVE SPACES 7 which will establish the lemma. We begin by observing that ss*(V2i-1) sss*(P2i-1), the stable homotopy group* *s of a stunted projective space, through dimension at least 2i+ 3. These groups can * *be computed by the Adams spectral sequence, as done for example in [16, Table 8.16* *], to begin ssj(V2i-1) Z2 if j = 2i- 1, 2i+ 1, or 2i+ 2, with Adams filtrations 0, 1, and 2, respectively, and ss2i(V2i-1) = 0. By obstruction theory, this impli* *es that i+2 the order of [P 2 ; V2i-1] is no greater than 8. The claim is that these three* * classes extend cyclically. i+2 * *2i+3 One way to see this begins by noting that the desired group equals [P 2 ; P2* *i-1] i+3 2i and is in the stable range. There is a degree-1 map P22i-1! S . Let F be its f* *iber. We will show that in the exact sequence i+2 2i+2 2i+3 2i+2 2i (2.6) [P 2 ; F ] ! [P ; P2i-1] ! [P ; S ] the first group is 0, and the third group is Z=8. Since we have already observe* *d that the middle group has order no greater than 8, and has an element of Adams filtr* *ation 0, the claim made in the first line of this proof will be established. i The third group in (2.6) is isomorphic to ss-1(2 P-2i-3), using S-duality. T* *his is Z=8 by an elementary Adams spectral sequence calculation; it is the group in* * the column labeled 2 in [16, Table 8.14]. It is not difficult to check that H*(P-2i* *-3^F ) is, through dimension 0, a free module over the Steenrod algebra on classes of dime* *nsion -4, -2, and 0. Thus the Adams spectral sequence or elementary obstruction theory implies that ss-1(P-2i-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 t* *he proof given in [15]. The second part of 2.1 is elementary: ( ) (q + 1)" ( (q + 1)) ( o ") (p + 1)" (p + 1) 8 D. DAVIS The first part utilizes the following result. Throughout this section, embed* *dings () are not necessarily differentiable. Proposition 3.1. If nq Rp and P n-1 Sm-1 , then P q+n Rp+m . Proof of first part of Theorem 2.1.By hypothesis, n" . Tensoring with yields the first part of n Rp; with the second part clear since can be taken to be a tubular neighborhood of * *P q in Rp. The theorem now follows immediately from Proposition 3.1. || To prove Proposition 3.1, we will need the description of P n+qgiven in the f* *ollowing lemma. Lemma 3.2. There is a homeomorphism (3.3) P n-1[g D(nq) P n+q; where g : S(nq) ! P n-1 is defined by g([x; y]) = [x]. Here we use the model S(nq) = (Sn-1 x Sq)=(x; y) ~ (-x; -y). Proof.The space on the LHS of (3.3) is (3.4) Dn x Sq=((x; y) ~ (-x; -y); (x; y) ~ (x; y0) if|x| = 1): q_______ The desired homeomorphism sends [(x; y)] to [(x; 1 - |x|2y)] in Sn+q=z ~ -z. * *|| Proof of Proposition 3.1.Let f denote the embedding of D(nq) in Rp, and h the embedding of P n-1in Sm-1 . Using the descriptions of P n+qgiven by the LHS of (3.3) and (3.4), we define our embedding by OE([(x; y)])= (OE1(x; y); OE2(x; y)) OE1(x; y)= (1 - |x|)f(x; y) OE2(x; y)= |x|h([x=|x|]): Here x 2 Dn and y 2 Sq. We first show OE is well-defined. Because of the (1 - |* *x|)- factor, the second relation in (3.4) is not a problem, and because of the |x|-f* *actor, we have continuity at x = 0, even though x=|x| is not defined. Injectivity is prov* *ed as EMBEDDINGS OF REAL PROJECTIVE SPACES 9 follows. If OE2(x; y) = OE2(x0; y0), then x0= x. If also OE1(x; y) = OE1(x0; y0* *), then either |x| = 1 and hence [(x; y)] = [(x; y0)], or (x0; y0) = (x; y) so that [(x; y)] =* * [(x0; y0)]. || 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 s* *paces, Bol. Soc. Mat. Mex. 10 (1965) 84-88. 3. L. 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