Abstract
The Property P Conjecture States that the 3-manifold <TEX>$Y_r$</TEX> obtained by Dehn surgery on a non-trivial knot in <TEX>$S^3$</TEX> with surgery coefficient <TEX>${\gamma}{\in}Q$</TEX> has the non-trivial fundamental group (so not simply connected). Recently Kronheimer and Mrowka provided a proof of the Property P conjecture for the case <TEX>${\gamma}={\pm}2$</TEX> that was the only remaining case to be established for the conjecture. In particular, their results show that the two phenomena of having a cyclic fundamental group and having a homomorphism with non-cyclic image in SU(2) are quite different for 3-manifolds obtained by Dehn filings. In this paper we extend their results to some other Dehn surgeries via the A-polynomial, and provide more evidence of the ubiquity of the above mentioned phenomena.
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