Boy's
surface, a counterintuitive immersion of $\mathbb{RP}^2$ into $\mathbb{R}^3$.
Future:
I wish I had used The
Geometry of Physics as our textbook. This is an excellent reference for
most of the material we covered. It is extremely broad.
Differential Forms in Algebraic Topology is a masterpiece, and probably
my favorite textbook ever. I'm proud to say that you are now prepared to
tackle this, should you choose to journey deeper into algebraic topology.
The big picture
Where have we been?
Intro (M&T Chapter 1)
Tensor algebra and tensor calculus (Hitchin Chapter 2, Section 5)