On vanishing of all fourfold products of the Ray classes in symplectic cobordism

Bakuradze, Malkhaz

doi:10.1090/proc/15058

On vanishing of all fourfold products of the Ray classes in symplectic cobordism
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Proc. Amer. Math. Soc. 148 (2020), 4107-4115 Request permission

Abstract:

This paper provides certain computations with transfer associated with projective bundles of $\mathrm {Spin}$ vector bundles. One aspect is to revise the proof of the main result of [Trans. Amer. Math. Soc. 349 (1997), pp. 4385–4399] which says that all fourfold products of the Ray classes are zero in symplectic cobordism.

References

M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), no. suppl, suppl. 1, 3–38. MR 167985, DOI 10.1016/0040-9383(64)90003-5
Malkhaz Bakuradze, The transfer and symplectic cobordism, Trans. Amer. Math. Soc. 349 (1997), no. 11, 4385–4399. MR 1407478, DOI 10.1090/S0002-9947-97-01867-9
J. C. Becker and D. H. Gottlieb, The transfer map and fiber bundles, Topology 14 (1975), 1–12. MR 377873, DOI 10.1016/0040-9383(75)90029-4
V. M. Bukhshtaber and K. È. Fel′dman, The index of an equivariant vector field, and addition theorems for Pontryagin characteristic classes, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), no. 2, 3–28 (Russian, with Russian summary); English transl., Izv. Math. 64 (2000), no. 2, 223–247. MR 1770670, DOI 10.1070/im2000v064n02ABEH000282
Mark Feshbach, The transfer and compact Lie groups, Trans. Amer. Math. Soc. 251 (1979), 139–169. MR 531973, DOI 10.1090/S0002-9947-1979-0531973-8
V. G. Gorbunov, Symplectic cobordism of projective spaces, Mat. Sb. 181 (1990), no. 4, 506–520 (Russian); English transl., Math. USSR-Sb. 69 (1991), no. 2, 543–557. MR 1055526, DOI 10.1070/SM1991v069n02ABEH001376
Vassily Gorbunov and Nigel Ray, Orientations of Spin bundles and symplectic cobordism, Publ. Res. Inst. Math. Sci. 28 (1992), no. 1, 39–55. MR 1147850, DOI 10.2977/prims/1195168855
Nigel Ray, Indecomposables in $\textrm {Tors}\ \textrm {M}\textrm {Sp}_\ast$, Topology 10 (1971), 261–270. MR 300296, DOI 10.1016/0040-9383(71)90019-X
F. W. Roush, On some torsion classes in symplectic bordism, Preprint.
Jörg Schray and Corinne A. Manogue, Octonionic representations of Clifford algebras and triality, Found. Phys. 26 (1996), no. 1, 17–70. MR 1380124, DOI 10.1007/BF02058887
V. V. Vershinin, Calculation of the symplectic cobordism ring in dimensions up to $32$ and the nontriviality of the majority of ternary products of N. Ray’s elements, Sibirsk. Mat. Zh. 24 (1983), no. 1, 50–62, 191 (Russian). MR 688591