Let's say you've learned everything there is to know about surfaces and now you want to try your hand at some 3-manifolds. Like a good mathematician, the first thing you want to do is build some examples. Since you know all about surfaces, to start generating examples, a nice thing to do is to look at first the product of circles with surfaces and then circle bundles over surfaces. If you play with circle bundles for long enough, you realize that they're classified by the surface you build them over and a single number measuring how twisty they are. Now you're bored and you want more examples, so you start to wonder if there's a way to build something like a circle bundle that twists in as many independent ways as you want. You work out a construction, but what do these strange new spaces even look like?
Over the course of this talk I hope to answer this in detail and carefully tell this story. Spoiler warning: there’s a twist ending!