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THE MINIMAL GENUS PROBLEM IN CP 2 MOHAMED AIT NOUH
 

Summary: THE MINIMAL GENUS PROBLEM IN CP 2
#CP2
MOHAMED AIT NOUH
Abstract. T. Lawson conjectured in [11] that the minimal genus of (m, n) H2(CP2#CP2)
is given by (
|m|-1
2 ) + (
|n|-1
2 ) -this is the genus realized by the connected sum of algebraic
curves in each factor. We answer this conjecture by the positive for the small pairs (3, 3)
and (6, 6). The proofs use twisting of knots in S3 and gauge theory. We also give an
explicite representative for (2, 2n) H2(CP2
#CP2
) for any n 1 whose genus is the
proposed Lawson's minimal genus value.
1. Introduction
Throughout this paper, we work in the smooth category. All orientable manifolds will be
assumed to be oriented unless otherwise stated. In particular, all knots are oriented. Let
K be a knot in S3, then the dual knot of K is the inverse of the mirror-image K of K ([8])
i.e. K = -K. Let X4 be a closed 4-manifold and K a knot in (X4 -intB4) = S3, where

  

Source: Ait Nouh, Mohamed - Mathematics Department, California State University Channel Islands

 

Collections: Mathematics