Very special framed links for a homotopy 3-sphere (Q5930114)

A framed link \(K\) in a closed 3-manifold \(M\) is a system of disjoint simple closed curves \(K=K_1\cup K_2\cup\cdots\cup K_n\) equipped with another system of simple closed curves \(\widetilde K=\widetilde K_1\cup \widetilde K_2\cup \cdots\cup \widetilde K_n\) such that each component \(\widetilde K_j\), called the framing curve of \(K_j\), lies on the boundary of a regular neighborhood \(V_j\) of \(K_j\) in \(M\) and meets the meridian curve of \(V_j\) exactly once. Given a framed link \(K\) in \(M\), a 3-manifold \(\chi(M;K)\) obtained by a Dehn surgery along \(K\) is defined as follows: NEWLINE\[NEWLINE\chi(M;K)=\Bigl(M \setminus \bigcup^n_{j=1}\text{int}(V_j)\Bigr)\cup\Bigl( \bigcup^n_ {j=1}V_j' \Bigr)NEWLINE\]NEWLINE where each \(V_j'\) is a solid torus glued back by a homeomorphism \(h_j: \partial V_j'\to\partial V_j\) which takes a meridian curve of \(V_j'\) onto the framing curve \(\widetilde K_j\). \textit{J. S. Birman} and \textit{J. Powell} [Geometric topology, Proc. Conf., Athens/Ga. 1977, 23-51 (1979; Zbl 0471.57002)] proved that any closed 3-manifold \(M\) has a framed link such that \(\chi(M;K)\) is homeomorphic to a 3-sphere, and showed that such framed link of \(M\) is closely related to a Heegaard diagram of \(M\). The author defines the concept of very special framed link and investigates it. The main result of the paper under review states that a closed 3-manifold \(M\) has a very special framed link if and only if \(M\) is a homotopy 3-sphere.