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.

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

Article copyright:
© Copyright 1979
American Mathematical Society