Bulletin of the
Korean Mathematical Society BKMS

The Property $P$ conjecture states that the 3-manifold $Y_r$ obtained by Dehn surgery on a non-trivial knot in $S^3$ with surgery coefficient $r\in {\bf Q}$ 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 $r=\pm 2$ 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 fillings. 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.

Keywords: Dehn surgery, property $P$ conjecture, $A$-polynomials

MSC numbers: Primary 57R57