REVIEW OF: "Howie James, Can Dehn surgery yield three connected summands?, Groups Geom. Dyn. 4, No. 4, 785-797 (2010)".[DE05880964X]

The {\it Cabling Conjecture} of Gonzales-Acuna and Short ([Math. Proc. Camb. Philos. Soc. 99, 89-102 (1986; Zbl 0591.57002)]) states that Dehn surgery on a knot in $\Bbb S^3$ can produce a reducible 3-manifold only if the knot is a cable knot and the surgery slope is that of a cabling annulus. Since the case of cable knot yields exactly two connected summands (see [Math. Proc. Camb. Philos. Soc. 102, No. 1, 97-101 (1987; Zbl 0655.57500)]), a weaker conjecture may be stated, asserting that a manifold obtained by Dehn surgery (i.e. $M^3=M(K, r)$, where $K$ is a knot in $\Bbb S^3$ and $r \in \Bbb Q \cup \{\infty\}$) cannot by expressed as a connected sum of three non-trivial manifolds. Note that results by [Topology Appl. 87, No.1, 73-78 (1998; Zbl 0926.57020)], [Topology Appl. 98, No.1-3, 355-370 (1999; Zbl 0935.57024)] and [J. Pure Appl. Algebra 173, No.2, 167-176 (2002; Zbl 1026.20019)] ensure that if $M(K, r)$ has three connected summands, than two of these must be lens spaces and the third must be a $\Bbb Z$-homology sphere. Moreover, in the same hypothesis, $r$ must be an integer: see [Math. Proc. Camb. Philos. Soc. 102, No. 1, 97-101 (1987; Zbl 0655.57500)]. \medskip The present paper faces the problem of three connected summands by making use of standard techniques of intersection graphs by Scharlemann and Gordon-Luecke. The main result proves that, if $K$ has bride-number $b$ and $M(K, r)= M_1 \#M_2\#M_3,$ $M_1$ and $M_2$ being lens spaces and $M_3$ being a homology sphere (but not a homotopy sphere), then $$|\pi_1(M_1)| + |\pi_1(M_2)| \le b+1.$$ As a consequence, the inequality $$|r| = |\pi_1(M_1)| \cdot |\pi_1(M_2)| \le \frac{b(b+2)}{4}$$ is obtained, which is a sharpening of a similar inequality due to Sayari (see [J. Knot Theory Ramifications 18, No. 4, 493-504 (2009; Zbl 1188.57004)]).