On the topological structure of even-dimensional complete intersections

Libgober, A. S.; Wood, J. W.

doi:10.1090/S0002-9947-1981-0626495-9

On the topological structure of even-dimensional complete intersections
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by A. S. Libgober and J. W. Wood PDF

Trans. Amer. Math. Soc. 267 (1981), 637-660 Request permission

Abstract:

A topological connected sum decomposition into indecomposable pieces is given for complete intersections, and these pieces are described by plumbing constructions. The principal technical results are structure theorems for the intersection form on the middle dimensional homology and the submodule of spherical classes.

References

N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
William Browder, Surgery on simply-connected manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65, Springer-Verlag, New York-Heidelberg, 1972. MR 0358813
William Browder, Complete intersections and the Kervaire invariant, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 88–108. MR 561215
Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 0354657

History of the theory of numbers

André Haefliger, Differentiable imbeddings, Bull. Amer. Math. Soc. 67 (1961), 109–112. MR 123337, DOI 10.1090/S0002-9904-1961-10522-3
André Haefliger, Differentiable links, Topology 1 (1962), 241–244. MR 148076, DOI 10.1016/0040-9383(62)90106-4
F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. MR 0202713
F. Hirzebruch, W. D. Neumann, and S. S. Koh, Differentiable manifolds and quadratic forms, Lecture Notes in Pure and Applied Mathematics, Vol. 4, Marcel Dekker, Inc., New York, 1971. Appendix II by W. Scharlau. MR 0341499
Ravindra S. Kulkarni and John W. Wood, Topology of nonsingular complex hypersurfaces, Adv. in Math. 35 (1980), no. 3, 239–263. MR 563926, DOI 10.1016/0001-8708(80)90051-1
A. S. Libgober, A geometrical procedure for killing the middle dimensional homology groups of algebraic hypersurfaces, Proc. Amer. Math. Soc. 63 (1977), no. 2, 198–202. MR 440576, DOI 10.1090/S0002-9939-1977-0440576-9
A. Libgober, On the topology of some even-dimensional algebraic hypersurfaces, Topology 18 (1979), no. 3, 217–222. MR 546791, DOI 10.1016/0040-9383(79)90004-1
A. Libgober, Some properties of the signature of complete intersections, Proc. Amer. Math. Soc. 79 (1980), no. 3, 373–375. MR 567975, DOI 10.1090/S0002-9939-1980-0567975-9
Richard Mandelbaum and Boris Moishezon, On the topological structure of simply-connected algebraic surfaces, Bull. Amer. Math. Soc. 82 (1976), no. 5, 731–733. MR 413135, DOI 10.1090/S0002-9904-1976-14136-5
Yu. I. Manin, Cubic forms, 2nd ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. Algebra, geometry, arithmetic; Translated from the Russian by M. Hazewinkel. MR 833513
John Milnor, On simply connected $4$-manifolds, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 122–128. MR 0103472

Morse theory

Lectures on the

cobordism theorem

John Milnor and Dale Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. MR 0506372
Shigeyuki Morita, The Kervaire invariant of hypersurfaces in complex projective spaces, Comment. Math. Helv. 50 (1975), no. 3, 403–419. MR 385897, DOI 10.1007/BF02565759
Boris Moishezon, Complex surfaces and connected sums of complex projective planes, Lecture Notes in Mathematics, Vol. 603, Springer-Verlag, Berlin-New York, 1977. With an appendix by R. Livne. MR 0491730
Jean-Pierre Serre, Cours d’arithmétique, Collection SUP: “Le Mathématicien”, vol. 2, Presses Universitaires de France, Paris, 1970 (French). MR 0255476
I. R. Shafarevich, Basic algebraic geometry, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch. MR 0366917
C. T. C. Wall, On the orthogonal groups of unimodular quadratic forms, Math. Ann. 147 (1962), 328–338. MR 138565, DOI 10.1007/BF01440955
John W. Wood, Removing handles from non-singular algebraic hypersurfaces in $CP_{n+1}$, Invent. Math. 31 (1975), no. 1, 1–6. MR 388417, DOI 10.1007/BF01389862
John W. Wood, A connected sum decomposition for complete intersections, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 191–193. MR 520536
J. W. Wood, Complete intersections as branched covers and the Kervaire invariant, Math. Ann. 240 (1979), no. 3, 223–230. MR 526845, DOI 10.1007/BF01362312
Anatoly S. Libgober and John W. Wood, Diffeomorphic complete intersections with different multidegrees, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 3, 459–461. MR 561533, DOI 10.1090/S0273-0979-1980-14767-9

Differentiable structures on complete intersections