A bilinear form for Spin manifolds

Landweber, Peter S.; Stong, Robert E.

doi:10.1090/S0002-9947-1987-0876469-3

A bilinear form for Spin manifolds
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by Peter S. Landweber and Robert E. Stong

Trans. Amer. Math. Soc. 300 (1987), 625-640

DOI: https://doi.org/10.1090/S0002-9947-1987-0876469-3

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Abstract:

This paper studies the bilinear form on ${H^j}(M;{Z_2})$ defined by $\left [ {x, y} \right ] = x {\text {S}}{{\text {q}}^2}y[M]$ when $M$ is a closed Spin manifold of dimension $2j + 2$. In analogy with the work of Lusztig, Milnor, and Peterson for oriented manifolds, the rank of this form on integral classes gives rise to a cobordism invariant.

References

D. W. Anderson, E. H. Brown Jr., and F. P. Peterson, The structure of the Spin cobordism ring, Ann. of Math. (2) 86 (1967), 271–298. MR 219077, DOI 10.2307/1970690
William Browder, Remark on the Poincaré duality theorem, Proc. Amer. Math. Soc. 13 (1962), 927–930. MR 143205, DOI 10.1090/S0002-9939-1962-0143205-6
William Browder, Torsion in $H$-spaces, Ann. of Math. (2) 74 (1961), 24–51. MR 124891, DOI 10.2307/1970305
Edgar H. Brown Jr., Generalizations of the Kervaire invariant, Ann. of Math. (2) 95 (1972), 368–383. MR 293642, DOI 10.2307/1970804
E. E. Floyd, The number of cells in a non-bounding manifold, Ann. of Math. (2) 98 (1973), 210–225. MR 334256, DOI 10.2307/1970782
Steven M. Kahn, Involutions on odd-dimensional manifolds and the de Rham invariant, Topology 26 (1987), no. 3, 287–296. MR 899050, DOI 10.1016/0040-9383(87)90002-4
G. Lusztig, J. Milnor, and F. P. Peterson, Semi-characteristics and cobordism, Topology 8 (1969), 357–359. MR 246308, DOI 10.1016/0040-9383(69)90021-4
R. E. Stong, Appendix: calculation of $\Omega ^{\textrm {Spin}}_{11}(K(Z,4))$, Workshop on unified string theories (Santa Barbara, Calif., 1985) World Sci. Publishing, Singapore, 1986, pp. 430–437. MR 849112
W. Stephen Wilson, A new relation on the Stiefel-Whitney classes of spin manifolds, Illinois J. Math. 17 (1973), 115–127. MR 310883