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Topological classification of 4-dimensional complete intersections. (English) Zbl 0866.57015
The main result of the paper is: Two 4-dimensional complete intersections of a complex projective space are homeomorphic if and only if they have the same total degree, Pontryagin numbers and Euler number.
For dimension 1 and 3 the diffeomorphism classification is well known by surface theory and the classification of 1-connected 6-manifolds [A. V. Zhubr, Math. USSR, Izv. 9(1975), 793-812 (1976); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 39, 839-859 (1975; Zbl 0318.57019)], for dimension 2 the topological classification is known by M. Freedman’s results. For higher dimensions it is still open. In fact, the authors classify certain more general 1-connected 8-manifolds using bordism structure.
Remark: The numbers mentioned in the Theorem are computed from the total Chern class, not in the paper but presented there.

##### MSC:
 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 14M10 Complete intersections
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##### References:
 [1] W. Browder,Surgery on simply-connected manifolds, Springer Verlag, Berlin 1973 · Zbl 0239.57016 [2] W. Browder,Complete intersections and the Kervaire invariant, Springer LNM 763 (1979), 88–108 · Zbl 0429.57010 [3] E. H. Brown,Generalizations of the Kervaire invariant, Ann. of Math. 95 (1972), 368–383 · Zbl 0241.57014 [4] V. Giambalvo,On (8)-cobordism, Ill. Journ. Math 15 (1971), 533–541 · Zbl 0221.57019 [5] M. Kreck,Surgery and duality. An extension of results of Browder, Novikov and Wall. Johannes Gutenberg-Universität Mainz preprint (1985), to appear as a book in Vieweg [6] M. Kreck,Surgery and duality, Johannes Gutenberg-Universität Mainz preprint (Nr.3, 1996), to appear [7] A. S. Libgober and J. W. Wood,On the topological structure of even dimensional complete intersections, Trans. Amer. Math. Soc. 267 (1981), 637–660 · Zbl 0475.57013 [8] A. S. Libgober and J. W. Wood,Differential structures on complete intersections II, Proc. Symp. Pure Math. 40, Singularities (2) (1983), 123–133 · Zbl 0531.57029 [9] S. Stolz,Hochzusammenhängende Mannigfaltigkeiten und ihre Ränder, Springer LNM 1116 (1985) [10] C. Traving,Klassifikation vollständiger Durchschnitte, Diplomarbeit, Mainz (1985) [11] J. Wood,Complete Intersections as branched coverings and the Kervaire invariant, Math. Ann. 240 (1979), 223–230 · Zbl 0394.57005 [12] A. V. Zubr,Classification of simply connected six-dimensional spin manifolds, Math. U.S.S.R. Izv. 9 (1975), 793–812 · Zbl 0337.57004
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