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Dec 16, 2012 at 13:10 vote accept Janesis
Dec 15, 2012 at 15:49 comment added Igor Belegradek More examples of non-manifolds whose products with $\mathbb R$ are manifolds can be found on pp.33-34 of Dusan Repovs's slides at math.utk.edu/~dydak/00STS2012/Repovs7.pdf. It also has has some visualization of the dogbone space.
Dec 15, 2012 at 9:04 comment added David Roberts I know, I just forgot the OP asked for the action of a compact group...
Dec 15, 2012 at 3:50 comment added Misha @David: The whole point is that $X\times R$ is a manifold ($R^4$ is a manifold last time I checked), but the quotient space $X$ is not a manifold. To find out why $X$ is not a manifold you can read Bing's paper "The cartesian product of a certain nonmanifold and a line is $E^4$", Annals of Math., 1959.
Dec 15, 2012 at 3:35 comment added David Roberts So why isn't $X\times R$ a manifold? ;-)
Dec 15, 2012 at 2:01 history answered Misha CC BY-SA 3.0