TABLE OF IMMERSIONS AND EMBEDDINGS OF REAL PROJECTIVE SPACES Don Davis, dmd1@lehigh.edu Please let me know of any questions, additions, or corrections. Most of the results are conveniently tabulated by listing d and e, where n = 2^i + d is the dimension of the real projective space, and m = 2^{i+1} + e is the dimension of the Euclidean space. The few exceptional cases, which occur just before a 2-power, are listed on the right. Credit is given to the first printed account of which I am aware. e e e e m m m m d i_min nonimm imm nonemb emb n nonimm imm nonemb emb 0 1 -2 Mi56 -1 W -1 P 0 H 1 2 -2 Mi56 -1 S 0 M62 1 H 3 3 dim 4 Mi56 4 H 5 H 2 3 -1 BB 0 S 0 M62 1 N 6 6 dim 7 Hi 8 M62 11 ES 3 3 -1 BB 0 S 0 M62 2 D97 7 7 dim 8 Hi 8 H 12 M64 4 4 1 G 2 N 1 As 5 M64 12 17 G 18 L 17 As 21 M64 5 4 5 AG 6 S 6 AGM 7 M64 13 21 G 22 S 21 G 23 M64 6 4 5 AG 6 S 6 AGM 7 Re 14 21 G 22 S 21 G 23 Re 7 4 5 AG 6 S 6 AGM 7 Re 15 21 J 22 S 22 AGM 24 Re 8 4 5 AG 7 R70 6 AGM 10 T 9 4 13 AG 14 S 13 AG 15 M64 10 4 13 AG 14 S 13 AG 16 T 11 4 13 AG 14 S 14 AGM 16 T 12 4 14 R85 15 D83 14 AGM 18 T 13 4 14 R85 18 R70 14 AGM 20 T 14 4 14 AD 19 D83 15 As 21 B79 15 5 14 AD 21 AGM 15 As 22 B73 31 52 J 53 AGM 52 J 54 B73 16 5 20 KW 23 R70 20 KW 26 T 17 5 29 AG 30 S 29 AG 31 M64 18 5 29 AG 30 S 29 AG 32 T 19 5 29 AG 30 S 30 AGM 32 T 20 5 30 R85 31 D83 30 AGM 34 T 21 5 30 R85 34 R70 30 AGM 36 T 22 5 30 AD 35 D83 31 As 37 B79 23 5 30 AD 37 AGM 31 As 38 B73 24 5 36 DM7 39 R70 36 DM7 42 T 25 5 36 DM7 42 R70 36 DM7 44 T 26 5 43 DZ 44 Si 43 DZ 46 T 27 5 43 DZ 44 Si 43 DZ 47 B79 28 6 43 DZ 46 D83 43 As 49 DZ 60 107 DZ 111 B79 107 As 113 Z 29 6 43 DZ 46 D83 43 As 52 T 61 107 DZ 114 R70 107 As 116 T 30 6 44 KW 46 D83 44 KW 53 B73 62 108 KW 115 DM 108 KW 117 B73 31 6 44 DM 48 DM 44 DM 53 B73 63 114 J 115 DM 114 J 117 B73 32 6 45 KW 51 D83 45 As 58 T 33 6 61 AG 62 S 61 AG 63 M64 34 6 61 AG 62 S 61 AG 64 T 35 6 61 AG 62 S 62 AGM 64 T 36 6 62 R85 63 D83 62 AGM 66 T 37 6 62 R85 66 R70 62 AGM 68 T 38 6 62 AD 67 D83 63 As 69 B79 39 6 62 AD 69 AGM 63 As 70 B73 40 6 68 DM7 71 R70 68 DM7 74 T 41 6 68 DM7 74 R70 68 DM7 76 T 42 6 75 DZ 76 Si 75 DZ 78 T 43 6 75 DZ 76 Si 75 DZ 79 B79 44 6 75 DZ 78 D83 75 As 81 DZ 45 6 75 DZ 78 D83 75 As 84 T 46 6 76 KW 78 D83 76 KW 85 B73 47 6 76 DM 80 DM 76 DM 85 B73 48 6 77 KW 83 D83 77 As 90 T 49 6 77 KW 90 R70 77 As 92 T 50 6 87 D83 93 AGM 87 D83 94 T 51 6 87 D83 93 AGM 87 D83 95 B79 52 6 90 AD 94 DM2 91 As 97 Z 53 6 90 AD 94 DM2 91 As 100 T 54 6 92 KW 94 DM2 92 KW 101 B73 55 6 92 DM 96 DM 92 DM 101 B73 56 6 93 KW 99 D83 93 As 106 T 57 6 93 KW 102 D83 93 As 108 T 58 6 98 DM7 103 D83 98 DM7 110 T 59 6 98 DM7 108 DM 98 DM7 111 B79 60 7 103 D83 108 D83 103 D83 113 Z 124 231 D83 238 B79 231 D83 241 Z 61 7 103 D83 108 BR 103 D83 116 B73 125 231 D83 238 B79 231 D83 244 B73 62 7 106 KW 109 BR 106 KW 116 B73 126 234 KW 238 B79 234 KW 244 B73 63 7 106 D78 112 DM 106 D78 116 B73 127 238 J 240 DM 238 J 244 B73 References AG. J.Adem and S.Gitler, "Nonimmersion theorems for real projective spaces," Bol Soc Mat Mex 9 (1964) 37-50. AGM. J.Adem, S.Gitler, and M.Mahowald, "Embedding and immersion of projective spaces," Bol Soc Mat Mex 10(1965) 84-88. As. L.Astey, "A cobordism obstruction to embedding manifolds," Ill Jour Math 31 (1987) 344-350. AD. L.Astey and D.Davis, "Nonimmersions of real projective spaces implied by BP," Bol Soc Mat Mex 24 (1979) 49-55. At. M.Atiyah, "Immersions and embeddings of manifolds," Topology 1 (1962) 125-132. BB. P.Baum and W.Browder, "The cohomology of quotients of classical groups," Topology 3 (1965) 305-336. BD. M.Bendersky and D.Davis, "Unstable BP-homology and desuspensions," Amer Jour Math 107 (1985) 833-852. B73. A.J.Berrick, Oxford Univ D.Phil thesis, 1973. B79. A.J.Berrick, The Smale invariant of an immersed projective space," Math Proc Camb Phil Soc 86 (1979) 401-412, BR. A.J.Berrick and A.D.Randall, "On equivariant maps and immersions of real projective spaces," Springer Verlag Lecture Notes in Math 1370 (1986) 57-62. BDM R.Bruner, D.Davis, and M.Mahowald, "Nonimmersions of real projective spaces implied by tmf," Contemp Math AMS 293 (2002) 45-68. D78. D.Davis, "Connective coverings of BO and immersions of projective spaces," Pac Jour Math 76 (1978) 33-42. D83. D.Davis, "Some new immersions and nonimmersions of real projective spaces," AMS Contemp Math 19 (1983) 51-64. D84. D.Davis, "A strong nonimmersion theorem for real projective spaces," Annals of Math 120 (1984) 517-528. D97. D.Davis, "Embeddings of real projective spaces," Bol Soc Mat Mex 4 (1998) 115-122. DDJM. D.Davis, G.Dula, J.Gonzalez, and M.Mahowald, "Immersions of RP^{2^e-1}," Alg Geom Topology 8 (2008) 997-1030. DM. D.Davis and M.Mahowald, "The immersion conjecture for RP^{8k+7} is false," Trans Amer Math Soc 236 (1978) 361-383. DM2. D.Davis and M.Mahowald, "A new spectrum related to 7-connected cobordism," Springer Verlag Lecture Notes in Math 1370 (1986) 126-134. DM7. D.Davis and M.Mahowald, "Nonimmersions of RP^n implied by tmf, revisited," Proc Complex Cobordism Conf, HHA (2008). DZ. D.Davis and V.Zelov, "Some new embeddings and nonimmersions of real projective spaces," Proc Amer Math Soc 128 (2000) 3731-3740. ES. D.Epstein and R.Schwarzenberger, "Imbeddings of real projective spaces," Annals of Math 76 (1962) 180-184. G. S.Gitler, "The projective Stiefel manifolds II-applications," Topology 7 (1968) 47-53. Hi. M.Hirsch, "Immersion of manifolds," Trans Amer Math Soc 93 (1959) 242-276. H. H.Hopf, "Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Raume," Vierteljschr Naturforsch Gesellschaft Zurich 85 (1940) 165-177. J. I.James, "On the immersion problem for real projective spaces," Bull Amer Math Soc 69 (1963) 231-238. KW. N.Kitchloo and W.S.Wilson, "The second real Johnson-Wilson theory and nonimmersions of RP^n, I and II" Proc Complex Cobordism Conf, HHA (2008). L. K.Lam, "Construction of nonsingular bilinear maps," Topology 6 (1967) 423-426. M62. M.Mahowald, "On the embeddability of the real projective spaces," Proc Amer Math Soc 13 (1962) 763-764. M64. M.Mahowald, "On obstruction theory in orientable fibre bundles," Trans Amer Math Soc 110 (1964) 315-349. Mil. R.J.Milgram, "Immersing projective spaces," Annals of Math 85 (1967) 473-482. Mi56. J.Milnor, "On the immersion of n-manifolds in (n+1)-space," Comm Math Helv 30 (1956) 275-284. Mi57. J.Milnor, "Lectures on characteristic classes," Princeton Univ, 1957, published in 1975 as Annals of Math Studies 76, with Stasheff. N. F.Nussbaum, "Obstruction theory of possibly nonorientable fibrations," Northwestern Univ PhD thesis (1970). P. F.Peterson, "Some nonembedding problems," Bol Soc Mat Mex 2 (1957) 9-15. R70. A.D.Randall, "Some immersion theorems for projective spaces," Trans Amer Math Soc 147 (1970) 135-151. R85. A.D.Randall, "On equivariant maps of Stiefel manifolds," AMS Contemp Math 36 (1985) 145-149. Re. E.Rees, "Embeddings of real projective spaces," Topology 10 (1971) 309- 312. S. B.Sanderson, "Immersions and embeddings of projective spaces," Proc London Math Soc 14 (1964) 137-153. Si. N.Singh, "On nonimmersion of real projective spaces," Topology and Applns 136 (2004) 233-238. St. B.Steer, "On the embedding of projective spaces in Euclidean space," Proc London Math Soc 21 (1970) 489-501. T. E.Thomas, "Embedding manifolds in Euclidean space," Osaka Jour Math 13 (1976) 163-186. W. H.Whitney, "The singularities of a smooth n-manifold in (2n-1)-space," Annals of Math 45 (1944) 247-293. Wi. W.S.Wilson, "Brown-Peterson metastability and the Bendersky-Davis conjecture," Publ RIMS Kyoto 20 (1984) 1037-1051. Z. V.Zelov, "Embeddings and immersions of real projective spaces," Lehigh Univ PhD thesis, 1997. In the following summary of results, a(n) is the number of 1s in binary expansion of n, and nu(n) is the exponent of 2 in n. All results with (non)(immersion/embedding) of real projective space in Euclidean space. 1940 Hopf * if n>1, then n not emb n+1 * n emb 2n * if n odd and n>1, then n emb 2n-1 1944 Whitney * n imm 2n-1 1956 Milnor * 3 imm 4 1957 Milnor * 2^r not imm 2^{r+1}-2 1957 Peterson * 2^r not emb 2^{r+1}-1 1959 Hirsch * 6 imm 7, and 7 imm 8 1962 Atiyah * n not imm n+sigma(n)-1 and n not emb n+sigma(n), where sigma(n) is smallest s such that s+nu((n+s) choose s) le 4a+(0,1,2,2,3,3,3,3) if n=8a+(0,1,2,3,4,5,6,7) 1962 Epst-Schw * if n even and a(n)>1, then n emb 2n-1 1962 Mahowald * 2^r + 1 not emb 2^{r+1} 1963 James * 2^r - 1 not imm 2^{r+1} - 2r - (3,2,2,4) if r=(0,1,2,3) mod 4 1964 Sanderson * if n=3 mod 4 and n>8, then n imm 2n-(6,8) if a(n-3) (=1, >1) * if n=1 mod 4 and n>8, then n imm 2n-3 1964 Mahowald * if n=3 mod 4 and n>3, then n emb 2n-2 * n=(0,1,2) mod 4, and (n,n-1,n-2) not = 2-pwr, then n emb 2n-3 1964 Adem-Gitler * n=1 mod 4 and a(n)=3 and n>16, then n not imm 2n-5 1965 Adem-Gitler-Mahowald * if n = 2^r + 5 and n>13, then n not emb 2n-4 * if n=3 mod 4 and a(n)=4, then n not emb 2n-8 * if n=3 mod 4 and a(n)=5, then n imm 2n-9 1965 Baum-Browder * if n = 2^r + 2 and n>6, then n not imm 2n-5 1967 Milgram * if n>7, n imm 2n-a(n)-(0,1,1,4) if n = (1,3,5,7) mod 8 1967 Lam * 12 imm 18 1968 Gitler * if n = 2^r + 4, then n not imm 2n-7 * 13 not imm 21 1970 Randall * if n=4 mod 8 and a(n)>2, then n imm 2n-7 * if n=0 mod 8 and a(n)>1, then n imm 2n-9 * if n=1 mod 4 and a(n)>3, then n imm 2n-8 1970 Nussbaum * if n = 2^r + 4, then n imm 2n-6 * if n = 2^r + 2 and n>6, then n emb 2n-3 1970 Steer * if n odd, then n emb 2n-a(n)+1 (is best known if n=0,1 mod 8 and a(n)>7) 1971 Rees * 14 emb 23, and 15 emb 24 * if n = 2^r + 7 and n>15, then n emb 2n-7 1973 Berrick * if n>15, then n emb 2n-a(n)-(0,3) if n=(3,7) mod 8 1976 Thomas * if a(n) > (1,3,3,3,2,3) for n = (0,1,2,3,4,5) mod 8, then n emb 2n-6 1978 Davis-Mahowald * if n=7 mod 8 and a(n)=6, then n not imm 2n-18 * if n=7 mod 8 and n>63, then n imm 2n-(14,14,16,17,18) if a(n) = (6,7,8,9,>9) * if n=3 mod 8 and a(n)=(6,7,>7), then n imm 2n-(10,15,17) * if n=11 mod 16 and a(n)>8, then n imm 2n-18 * 63 imm 115 1978 Davis * if nu(n+1) >= a(n)-4 > 2, then n not imm 2n - 2^{a(n)-3} -4 * if a(n)=7 and nu(n+1) = 4 or 5, then n imm 2n-16 1979 Berrick * if n=3 mod 8 and a(n)>4 or n=6 mod 8 and a(n)>3, then n emb 2n-7 * if n=4 mod 8 and a(n)>3, then n imm 2n-9 * if n=6 mod 8 and a(n)>5, then n imm 2n-14 1979 Astey-Davis * a complicated subset of the result of D84, which says 2(m+a(m)-1) not imm 4m-2a(m). This subset contains the cases m = 2^r + (5,6,9,12,17,20,24,26,28), which comprise the entries in the above table 1983 Davis * if a(n)=4 and n=2 mod 8, then n not imm 2n-13 * if a(n)=5 and n=4 mod 8, then n not imm 2n-17 * if a(n)=3 and n=4 mod 8, then n imm 2n-9 * if a(n)=4 and n=6 mod 8, then n imm 2n-9 * if a(n)=5 and n=1 mod 8, then n imm 2n-12 * if a(n-1)=6, n>64, and n=0 mod 8, then n imm 2n-13 * if a(n)=5 and n=2 mod 8, then n imm 2n-13 * if a(n)=5, n>62, and n=14 mod 16, then n imm 2n-14 * if a(n)>4, n=12 mod 16, and nu(n+4)<7, then n imm 2n-12 1984 Davis * 2(m+a(m)-1) not imm 4m-2a(m). The first case included here but not in AD is m=54, which yields 114 not imm 208. But a stronger nonimmersion result for P^{114} was proved in D83. The cases a(m)=6 and 7 of D84 were proved in BD and Wi, respectively, subsequent to AD but prior to D84. The smallest values of n for which BD, Wi, and D84 give new nonimmersions of P^n are 500, 998, and 2024, respectively. 1985 Randall * if n=4 mod 8 and a(n)=3, then n not imm 2n-10 1986 Berrick-Randall * if n=(13,14) mod 16, a(n)>5, and nu(n+(3,2))<7, then n imm 2n-(14,15) 1986 Davis-Mahowald * if n=6 mod 16 and a(n)>4, then n imm 2n-14 1987 Astey * 2(m+a(m)-1) not emb 4m-2a(m)+1 1997 Davis * if n = 2^r + 3 > 7, then n emb 2n-4 1997 Davis-Zelov * if a(n)>3 and n=4 mod 8, then n emb 2n-7 * if a(n)=3 and n=8 mod 16, then n not imm 2n-13 * if a(n)=4 and n=10 mod 16, then n not imm 2n-9 2002 Bruner-Davis-Mahowald * if a(M)=4h+e, then 8M+8h+del not imm 16M-8h+d, for certain triples (e,del,d), provided M satisfies a certain congruence. First new result 1536 not imm 3036. 2004 Singh * if a(n)=4 and n=11 mod 16, then n imm 2n-10 2008 Kitchloo-Wilson * if (m,a(m)) mod 8 equals (2,7), (7,2), (3,6), (6,3), (3,5), or (0,0) then 2(m+a(m)-1) not imm 4m-2a(m)+2. * if (m,a(m)) mod 4 equals (0,3) or (1,2) then 2(m+a(m)) not imm 4m-2a(m)+2. * if (m mod 4, a(m) mod 8) equals (1,4) or (2,6) then 2(m+a(m)-1) not imm 4m-2a(m)+1. 2008 Davis-Mahowald * if a(M)=2, then 8M+8 not imm 16M+4 * if a(M)=3, then 8M+10 not imm 16M+2. 2008 Davis-Dula-Gonzalez-Mahowald * 2^e-1 imm 2^(e+1)-e-7.