In dimensions n at least 4, S^2 x S^(n-2) is not cancellable, so connected sum decomposition is not unique; for simply connected manifolds the same failure is claimed for n at least 17.
A cancellation theorem for modules over integral group rings
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abstract
A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups $G$ for which the integral group ring $\mathbb{Z}G$ has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that $\mathbb{Z}G$ has SFC provided at most one copy of the quaternions $\mathbb{H}$ occurs in the Wedderburn decomposition of the real group ring $\mathbb{R}G$. This generalises the Eichler condition in the case of integral group rings.
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Connected sum decompositions of high-dimensional manifolds
In dimensions n at least 4, S^2 x S^(n-2) is not cancellable, so connected sum decomposition is not unique; for simply connected manifolds the same failure is claimed for n at least 17.