Uniqueness of connected sum decompositions in dimension 4
It is a well known fact that any 3-dimensional manifold decomposes uniquely as a connected sum. In 4-manifold topology this is not true as the example \(CP^ 2 \# (CP^ 2 \# -CP^ 2) = CP^ 2 \# (S^ 2 \times S^ 2)\) shows. A more subtle example is given by the homeomorphism \(^*CP^ 2 \# ^*CP^ 2 = CP^ 2 \# CP^ 2\), where \(^*CP^ 2\) is the nonsmoothable 4-manifold homotopy equivalent to \(CP^ 2\), see the book ``Topology of 4-manifolds'' (1990; Zbl 0705.57001) by \textit{M. Freedman} and \textit{F. Quinn}. To avoid such examples, the author studies decompositions with one factor fixed and inducing ``the same'' decompositions on \(\pi_ 2\). In this situation he asks whether a homeomorphism \(M \# W_ 1 = M \# W_ 2\) implies \(W_ 1 = W_ 2\)? This was proven in the above book for good fundamental groups under the assumption that \(M\) is not spherically characteristic. Assuming again that the fundamental group is good, the author extends this result to other cases, most notably to the case when \(M\) is a spin manifold. As a nice corollary he proves that the \(^*\)-construction of Freedman and Quinn (loc. cit.) is well defined.
scientific article; zbMATH DE number 567712