ChapterZero

I ended up not reading much algebraic topology on the way here (trying to wend your way through proofs on a bus, even ones that have already been worked out for you, turned out to be slow and unfruitful work), but I’ve started reading Fulton’s Algebraic Topology since I arrived.

One of the basic issues in algebraic topology seems to be the connection between integration (or more generally, any sort of analysis process) and the topology of a manifold. As an exemplar question: when can a given 1-form be written as the differential of a function , ?

I don’t know yet, and the suspense is killing me. In the meantime, here’s a problem: note that if , then in the right half plane . However, the one form cannot be written as the differential of a function if the domain of definition is . What if the domain of definition is the set
?

Possibly relevant posts:

• Exactness of differential forms (7/28/2007)
• Topology (4/28/2005)
• Cauchy’s theorem via Homotopy (8/8/2007)