Thank you for your interest in the question. You said that the kind of singularities that occur may be complicated. But now I'm very interested to know the extent to which surgery can fix a singularity (in the sense that, after surgery, the flow can continue). So concretely: is p-surgery (where one cuts out some $S^p\times B^{d-p}$ and glues in $B^{p+1}\times S^{d-p-1}$ along the common boundary), for any $p$, always guaranteed to fix a singularity, however complicated?

@AndyPutman So in a sense, you're considering the metric topology derived from $g$; and this happens to coincide with the topology of $M$ unless $g$ is singular in which case the two topologies are different?

Hi Robert, can you say what the extra condition $Q_{\mathcal{R}}(g)=0$ is explicitly?