Abstract
We present the structure of symplectic cobordism ring MSp * in dimensions up to 51 and give a construction of an infinite series of elements Γ i , i = 1, 3,4, , . . . , of order four in this ring, where dim Γ i = 8i + 95. The key element of the series is Γ1 in dimension 103.
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Baas N.A.: On bordism theory of manifolds with singularities. Math. Scand. 33, 279–302 (1973)
Bakuradze, M.: Some relations in symplectic cobordisms. Proc. A. Razmadze Math. Inst. 104, 27–34 (1994, in Russian)
Bakuradze M.: Some calculations with transfer in symplectic cobordism. Soobshch. Akad. Nauk Gruzii 151(2), 208–211 (1995)
Bakuradze M.: The transfer and symplectic cobordism. Trans. Am. Math. Soc. 349(11), 4385–4399 (1997)
Bakuradze M.: On the Buchstaber subring in M Sp *. Georgian Math. J. 5(5), 401–414 (1998)
Bakuradze M., Jibladze M., Vershinin V.V.: Characteristic classes and transfer relations in cobordism. Proc. Am. Math. Soc. 131(6), 1935–1942 (2003)
Bakuradze, M., Nadiradze, R.: Cohomological realizations of two-valued formal groups and their applications. Soobshch. Akad. Nauk Gruzin. SSR 128(1), 21–24 (1987, in Russian)
Botvinnik B.I., Kochman S.O.: Singularities and higher torsion in symplectic cobordism. Can. J. Math. 46(3), 485–516 (1994)
Botvinnik B.I., Kochman S.O.: Adams spectral sequence and higher torsion in M Sp *. Publ. Mat. 40(1), 157–193 (1996)
Buchstaber, V.M.: Topological applications of the theory of two-valued formal groups. Izv. Akad. Nauk SSSR Ser. Mat. 42(1), 130–184, 215 (1978, in Russian)
Buchstaber, V.M.: Characteristic classes in cobordism and topological applications of the theory of single-valued and two-valued formal groups. Current Problems in Mathematics, vol. 10, pp. 5–178. VINITI, Moscow (1978, in Russian)
Ginzburg, V.L.: Cobordisms of symplectic and contact manifolds. Funktsional. Anal. i Prilozhen. 23(2), 27–31 (1989, in Russian); English translation in Funct. Anal. Appl. 23(2), 106–110 (1989)
Ivanovskiĭ, L.N.: Cohomology of algebras with simple systems of generators. Sibirsk. Mat. Ž. 14, 1231–1246 (1973, in Russian)
Kochman S.O.: The symplectic cobordism ring. I. Mem. Am. Math. Soc. 228, 206 (1980)
Kochman S.O.: The symplectic cobordism ring. II. Mem. Am. Math. Soc. 271, 170 (1982)
Kochman S.O.: The Hurewicz image of Ray’s elements in MSp *. Proc. Am. Math. Soc. 94(4), 715–717 (1985)
Kochman S.O.: Symplectic cobordism and the computation of stable stems. Mem. Am. Math. Soc. 104(496), x–88 (1993)
Landweber P.S.: Cobordism operations and Hopf algebras. Trans. Am. Math. Soc. 129, 94–110 (1967)
Liulevicius A.: Notes on homotopy of Thom spectra. Am. J. Math. 86, 1–16 (1964)
May J.P.: Matric Massey products. J. Algebra 12(4), 533–568 (1969)
Milnor J.W.: On the cobordism ring Ω* and a complex analogue. I. Am. J. Math. 82, 505–521 (1960)
Nadiradze R.: New examples of symplectic and self-conjugate manifolds (Russian, English, Georgian summary). Soobshch. Akad. Nauk Gruzin. SSR 95(2), 293–296 (1979)
Nadiradze, R.: Some remarks on Tors \({{}\;\Omega^ *_ {{Sp}}}\) . Soobshch. Akad. Nauk Gruzin. SSR 135(1), 33–36 (1989, in Russian)
Novikov, S.P.: Some problems in the topology of manifolds connected with the theory of Thom spaces. Dokl. Akad. Nauk SSSR 132, 1031–1034 (1960, in Russian); English translation in Sov. Math. Dokl. 1, 717–719 (1960)
Novikov, S.P.: Homotopy properties of Thom complexes. Mat. Sb. N. Ser. 57(99), 407–442 (1962, in Russian)
Novikov S.P.: The methods of algebraic topology from the viewpoint of cobordism theory. Math. USSR Izv. 1, 827–913 (1967)
Poincaré, H.: Anlysis Situs. Journal de l’École Polytechnique 1 (section 5, Homologies) (1895)
Pontriagin, L.S.: Classification of continuous maps of a complex into a sphere. I. Dokl. Akad. Nauk. SSSR 19(3), 147–149 (1938, in Russian)
Pontriagin, L.S.: Characteristic cycles on differentiable manifolds. Mat. Sbornik N. S. 21(63), 233–284 (1947, in Russian); English translation in AMS Translation, series 1, no. 32
Quillen D.: On the formal group laws of unoriented and complex cobordism theory. Bull. Am. Math. Soc. 75, 1293–1298 (1969)
Ray N.: Indecompozable in MSp *. Topology 10(4), 261–270 (1971)
Ray N.: Some results in generalized homology K-theory and bordism. Proc. Camb. Philos. Soc. 71, 283–300 (1972)
Ray N.: The symplectic bordism ring. Proc. Camb. Philos. Soc. 71, 271–282 (1972)
Rohlin, V.A.: 1951 A three-dimensional manifold is the boundary of a four-dimensional one. Dokl. Akad. Nauk. SSSR (N.S.) 81, 355–357 (1951, in Russian)
Segal D.M.: On the symplectic cobordism ring. Comment. Math. Helv. 45, 159–169 (1970)
Stong R.: Some remarks on symplectic cobordism. Ann. Math. 86(2), 425–433 (1967)
Stong, R.: Notes on Cobordism Theory. Mathematical Notes. Princeton University Press, Princeton; University of Tokyo Press, Tokyo, v+354+lvi pp (1968)
Sullivan, D.: Geometric periodicity and the invariants of manifolds. 1971 Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), pp. 44–75. Lecture Notes in Mathematics, vol. 197. Springer, Berlin (1970)
Thom R.: Quelques proprietes globales des varietes differentiables. Comment. Math. Helv. 28, 17–86 (1954)
Vershinin V.V.: The S.P. Novikov algebraic spectral sequence for the spectrum MSp. Sib. Math. J. 21, 19–31 (1980)
Vershinin V.V.: Computation of the symplectic cobordism ring in dimensions up to 32 and nontriviality of majority of triple products of Ray elements. Sib. Math J. 24, 41–51 (1983)
Vershinin V.V.: Symplectic cobordism with singularities. Math. USSR Izvestiya 22(2), 211–225 (1984)
Vershinin, V.V.: Cobordisms and Spectral Sequences. AMS (Translations of mathematical monographs; v. 130) (1993)
Vershinin V.V., Anisimov A.L.: A series of elements of order 4 in the symplectic cobordism ring. Can. Math. Bull. 38(3), 373–381 (1995)
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Communicated by Hvedri Inassaridze.
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Anisimov, A.L., Vershinin, V.V. Symplectic cobordism in small dimensions and a series of elements of order four. J. Homotopy Relat. Struct. 7, 31–152 (2012). https://doi.org/10.1007/s40062-012-0005-4
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DOI: https://doi.org/10.1007/s40062-012-0005-4