The Odd-Dimensional Surgery Obstruction

It turns out that the problem in this case is both harder and easier than in the even-dimensional case. It is harder because the algebra that describes the surgery obstructions is more complicated. On the other hand it is easier, because the surgery obstruction groups turn out to vanish more often. This happens for example in the simply connected situation, so in that case there is no surgery obstruction and our problem always has a positive solution.