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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Irrational connected sums and the topology of algebraic surfaces
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by Richard Mandelbaum PDF
Trans. Amer. Math. Soc. 247 (1979), 137-156 Request permission

Abstract:

Suppose W is an irreducible nonsingular projective algebraic 3-fold and V a nonsingular hypersurface section of W. Denote by ${V_m}$ a nonsingular element of $\left | {mV} \right |$. Let ${V_1}$, ${V_m}$, ${V_{m + 1}}$ be generic elements of $\left | V \right |$, $\left | {mV} \right |$, $\left | {(m + 1)V} \right |$ respectively such that they have normal crossing in W. Let ${S_{1m}} = {V_1} \cap {V_m}$ and $C = {V_1} \cap {V_m} \cap {V_{m + 1}}$. Then ${S_{1m}}$ is a nonsingular curve of genus ${g_m}$ and C is a collection of $N = m\left ( {m + 1} \right )V_1^3$ points on ${S_{1m}}$. By [MM2] we find that $( \ast ) {V_{m + 1}}$ is diffeomorphic to $\overline {{V_m} - T({S_{1m}})} { \cup _\eta } \overline {{V_1}’ - T({S_{1m}}’)}$, where $T\left ( {{S_{1m}}} \right )$ is a tubular neighborhood of ${S_{1m}}$ in ${V_m}$, ${V_1}’$ is ${V_1}$ blown up along C, ${S_{1m}}’$ is the strict image of ${S_{1m}}$ in ${V_1}’$, $T({S_{1m}}’)$ is a tubular neighborhood of ${S_{1m}}’$ in ${V_1}’$ and $\eta : \partial T\left ( {{S_{1m}}} \right ) \to \partial T({S_m}’)$ is a bundle diffeomorphism. Now ${V_1}’$ is well known to be diffeomorphic to ${V_1} \# N\left ( { - C{P^2}} \right )$ (the connected sum of ${V_1}$ and N copies of $C{P^2}$ with opposite orientation from the usual). Thus in order to be able to inductively reduce questions about the structure of ${V_m}$ to ones about ${V_1}$ we must simplify the “irrational sum” $( \ast )$ above. The general question we can ask is then the following: Suppose ${M_1}$ and ${M_2}$ are compact smooth 4-manifolds and K is a connected q-complex embedded in ${M_i}$. Let ${T_i}$ be a regular neighborhood of K in ${M_i}$ and let $\eta : \partial {T_1} \to \partial {T_2}$ be a diffeomorphism: Set $V = \overline {{M_1} - {T_1}} \cup \overline {{M_2} - {T_2}}$. How can the topology of V be described more simply in terms of those of ${M_1}$ and ${M_2}$. In this paper we show how surgery can be used to simplify the structure of V in the case $q = 1, 2$ and indicate some applications to the topology of algebraic surfaces.
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • 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