Abstract
We develop certain discriminant form techniques allowing one to “glue” two integral lattices into a new lattice. In particular, we introduce the notion of fusion of finite forms, which is closely connected with the more familiar notion of cobordism of such forms. We also consider the category of finite forms and their cobordisms. These techniques are used for describing the intersection form of a compact oriented 4k-manifold M obtained by gluing together two manifolds M1 and M2 along one or several boundary components (the intersection forms of M1 and M2 are assumed to be nondegenerate). Bibliography: 12 titles.
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Literature Cited
O. A. Ivanov, “A construction for integral symmetric bilinear forms,”Zap. Nauchn. Semin. LOMI,83, 63–66 (1979).
O. A. Ivanov and N. Yu. Netsvetaev, “On the behavior of the intersection form when gluing manifolds along a boundary,” Math. readings in Memory of M.Ya.Suslin, Saratov (1989).
O. A. Ivanov and N. Yu. Netsvetaev, “Gluing quadratic forms via gluing oriented manifolds,” Int. conf. on algebra in mem. Shirshov, Novosibirsk (1991), p. 14.
V. V. Nikulin, “Integral symmetric bilinear forms and some of their geometric applications,”Izv. Akad. Nauk SSSR, Ser. Math.,43, 111–177 (1979).
G. M. Brumfiel and J. M. Morgan, “Quadratic functions, the index modulo 8 and a ℤ/4-Hirzebruch formula,”Topology,12, 105–122 (1973).
J. Barge, J. Lannes, F. Latour, and P. Vogel, “Λ-spheres,”Ann. Scient. Ec. Norm. Sup., (4e serie), 4, 463–506 (1974).
J. Lannes, “Forms quandratiques d'enlacement sur l'anneau des entiers d'une corps de nombres,”Ann. Scient. Ec. Norm. Sup., (4e serie),8, 535–579 (1975).
W. Scharlau,Quadratic and Hermitian Forms, Springer-Verlag, Berlin-New York (1985).
A. Dimca, “On the homology and cohomology of complete intersections with isolated singularities,”Compositio Math.,58, 321–339 (1986).
J. H. Conway and N. J. A. Sloane,Sphere packings, Lattices and Groups, Springer-Verlag, Berlin-New York (1988).
O. Ya. Viro,Branched coverings of manifolds with boundary, Ph. D. Thesis, Leningrad (1974).
J. Levine, “Knot cobordism groups in codimension two,”Com. Math. Helv.,44, 229–244 (1969).
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Translated fromZapiski nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 115–132.
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Ivanov, O.A., Netsvetaev, N.Y. Cobordisms of finite quadratic forms and gluing oriented manifolds. J Math Sci 81, 2524–2534 (1996). https://doi.org/10.1007/BF02362422
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DOI: https://doi.org/10.1007/BF02362422