This talk will begin with a discussion of why one might desire to produce an orientation-reversing automorphism of a closed three-manifold. The answer is related to Freedman's classification of simply-connected four-manifolds (which we will state in passing). As soon as we find ourselves wishing for an orientation-reversing automorphism, we will lose all hope of producing one, at least for non-trivial circle bundles on orientable surfaces.

The elements at play will include a sophisticated application of Van Kampen's theorem and a magical tale involving the Euler class of a bundle. If a rug is found, small amounts of algebra may be swept under it.