Morse functions on a closed manifold M need not realize the minimal number of critical points of smooth functions on M since several critical points often may be fused into a single degenerate one. We study conditions under which critical points of smooth functions on a manifold M of dimension 3 can be fused. We say that a function \(f:M\rightarrow [a, c]\) on a compact manifold is regular if the boundary of M is the union of \(f^{-1}(a)\) and \(f^{-1}(c)\) , and f has no critical points on \(\partial M\) . We prove that if \(f^{-1}(a)\) is a sphere and \(f^{-1}(c)\) is a non-empty surface, then the Conley index is a complete obstruction to fusing critical points of a regular function. On the other hand, we prove that the Conley index is not a complete obstruction to fusing critical points if, for example, M is \(F_{0,3}\times S^1\) , where \(F_{0,3}\) is a 2-sphere with the interior of three disjoint disks removed.
