Now let’s cut out part of it:
And now we have two circles! Stated a bit more technically, we’ve cut out a copy of S^0xD^1 from S^1, and then glued it back in a different way to produce two S^1s. This process is called surgery, and one can do it on general manifolds as well; in particular, it lets us generate a lot of 3 dimensional manifolds! Come to this talk to see more (hopefully somewhat pretty) pictures, more examples of surgery, and hear about why it’s such a vital tool in geometric topology.
I’ll be assuming some basic topology (e.g. intuitive definition of a manifold, idea of what a homeomorphism is, and ideally being able to visualize S^3 as 3 dimensional Eucildean space but with an extra point, though don’t worry if you can’t— I’ll talk more about this at the beginning of my talk).