Twisty cross notation

Question

I read Hatcher's 3-manifold paper earlier this year and I ran into several twisty cross notations which denote manifolds that I understand, but I am not exactly sure what the twisty cross is supposed to mean in general. For example $S^1\tilde{\times}S^2$ is the mapping torus of the antipodal map $S^2\to S^2$. At least, I think it is. Other examples in hatcher I was able to explain away similarly, but now that I've read Kirby Calculus I've encountered $S^2\tilde{\times}S^2$. But what is $S^2\tilde{\times}S^2$? More precisely my question is what does $\tilde{\times}$ mean in general? Is it saying "there's only two bundle structures on the things you're looking at, and this is the non-orientable one"?

Answer 1

First, I would say that the notation is not entirely standard, so whenever you encounter it, look for an explanation of the notation somewhere nearby.

That said, my first thought about $M\tilde{\times}N$ is that it denotes the total space of some fixed non-trivial bundle. It just so happens that among bundles $S^2\rightarrow E\rightarrow S^1$ and $S^2\rightarrow E\rightarrow S^2$, there is a unique non-trivial bundle.

Finally, note that the non-trivial $S^2$-bundle over $S^2$ is orientable - the base is simply connected to every bundle over it is orientable as a bundle. Further if the fiber is orientable, then the total space will also be orientable.