Irrational connected sums and the topology of algebraic surfaces

Author:
Richard Mandelbaum

Journal:
Trans. Amer. Math. Soc. **247** (1979), 137-156

MSC:
Primary 57R15; Secondary 14J99

DOI:
https://doi.org/10.1090/S0002-9947-1979-0517689-2

MathSciNet review:
517689

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Abstract: Suppose *W* is an irreducible nonsingular projective algebraic 3-fold and *V* a nonsingular hypersurface section of *W*. Denote by a nonsingular element of . Let , , be generic elements of , , respectively such that they have normal crossing in *W*. Let and . Then is a nonsingular curve of genus and *C* is a collection of points on . By [**MM2**] we find that is diffeomorphic to , where is a tubular neighborhood of in , is blown up along *C*, is the strict image of in , is a tubular neighborhood of in and is a bundle diffeomorphism.

Now is well known to be diffeomorphic to (the connected sum of and *N* copies of with opposite orientation from the usual). Thus in order to be able to inductively reduce questions about the structure of to ones about we must simplify the ``irrational sum'' above.

The general question we can ask is then the following: Suppose and are compact smooth 4-manifolds and *K* is a connected *q*-complex embedded in . Let be a regular neighborhood of *K* in and let be a diffeomorphism: Set . How can the topology of *V* be described more simply in terms of those of and .

In this paper we show how surgery can be used to simplify the structure of *V* in the case and indicate some applications to the topology of algebraic surfaces.

**[GR]**Robert C. Gunning and Hugo Rossi,*Analytic functions of several complex variables*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR**0180696****[Ht]**Robin Hartshorne,*Ample subvarieties of algebraic varieties*, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR**0282977****[H]**J. F. P. Hudson,*Piecewise linear topology*, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0248844****[KM]**Michel A. Kervaire and John W. Milnor,*Groups of homotopy spheres. I*, Ann. of Math. (2)**77**(1963), 504–537. MR**0148075**, https://doi.org/10.2307/1970128**[MM1]**R. Mandelbaum and B. Moishezon,*On the topological structure of non-singular algebraic surfaces in 𝐶𝑃³*, Topology**15**(1976), no. 1, 23–40. MR**0405458**, https://doi.org/10.1016/0040-9383(76)90047-1**[MM2]**-,*On the topology of algebraic surfaces*, Trans. Amer. Math. Soc. (to appear).**[Mz]**Barry Mazur,*Differential topology from the point of view of simple homotopy theory*, Inst. Hautes Études Sci. Publ. Math.**15**(1963), 93. MR**0161342****[Mi]**John Milnor,*On simply connected 4-manifolds*, Symposium internacional de topología algebraica International symposi um on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 122–128. MR**0103472****[RS]**C. P. Rourke and B. J. Sanderson,*Piecewise linear topology*, Springer-Verlag, Berlin, 1974.**[Sh]**I. R. Shafarevich,*Basic algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch; Die Grundlehren der mathematischen Wissenschaften, Band 213. MR**0366917****[S]**N. E. Steenrod,*Topology of fiber bundles*, Princeton Univ. Press, Princeton, N. J., 1951.**[W1]**C. T. C. Wall,*Diffeomorphisms of 4-manifolds*, J. London Math. Soc.**39**(1964), 131–140. MR**0163323**, https://doi.org/10.1112/jlms/s1-39.1.131**[W2]**C. T. C. Wall,*On simply-connected 4-manifolds*, J. London Math. Soc.**39**(1964), 141–149. MR**0163324**, https://doi.org/10.1112/jlms/s1-39.1.141**[Wk]**John J. Wavrik,*Deformations of Banach [branched] coverings of complex manifolds*, Amer. J. Math.**90**(1968), 926–960. MR**0233384**, https://doi.org/10.2307/2373491

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DOI:
https://doi.org/10.1090/S0002-9947-1979-0517689-2

Article copyright:
© Copyright 1979
American Mathematical Society