We’re using two texts as our primary sources:
The (tentative) syllabus:
|0: Basic Geometric Notions|
|1: Fundamental Groups|
|1.2 Van Kampen Theorem|
|1.3 Covering Spaces|
|3: Euler characteristic|
Here are solutions to some of the exercises in Hatcher.
Hatcher does not do differential manifolds, and Ghrist barely provides definitions (although he offers some nice examples). Two online sources for more info:
- Three Takes on the Tangent and Cotangent Bundles. 15 pages I wrote on the absolute basics of differential manifolds. Not so much formal definitions, but suggestions on how to think about them. The three takes are: the formal viewpoint; the geometrical viewpoint; the computational viewpoint.
- Dundas, A Short Course in Differential Topology. Chapters 2 and 3 cover the basics.
Also, the “phone book” (Gravitation, by Misner, Thorne, and Wheeler) spends more pages and spills more ink on the geometric intuition for differential geometry, than any math book I know. Especially the diagrams! Examples: pp. 55–58 for differential forms, and p.100 for the wedge product.