A remark on kinks and time machines (Q1897621)

The author gives a simple proof that a manifold with the topology of the Politzer time machine does not admit a nonsingular, asymptotically flat metric (cf. \textit{A. Chamblin}, \textit{G. W. Gibbons}, and \textit{H. R. Steif} [Phys. Rev. D. 50, No. 4, R 2353--R 2355 (1994)]\ for a more complicated proof). He notes that Politzer space has topology \((S^1 \times S^{n- 1}) \setminus p_\infty\), where \(p_\infty\) is a single point representing infinity. He claims without proof that if Politzer space admitted an asymptotically flat metric, one could replace the infinity by an \(n\)-torus (minus a point) and still get a smooth metric (the referee does not find this statement obvious). Then he calculates the Euler number of the connected sum of \(S^1 \times S^{n-1}\) with the \(n\)- torus which gives \(-2\neq 0\). Hence this manifold cannot admit a smooth Lorentzian metric (for \(n\) even), yielding the desired contradiction.