I was wondering if there is a bag of tricks somewhere for differential topology and smooth manifold problems just like there is for analysis by prof. Tao
http://terrytao.wordpress.com/2010/10/21/245a-problem-solving-strategies/.
What are some strategies/arguments you find useful? Please share below.
Here are some I find useful:
1)Clopen argument
To show a property is global in a connected space X, consider $S=\{x:P(x)$ is true $\}$ and prove it's clopen. Thus, by uniqueness $S=X$.
2)Extending to show continuity/smoothness
Extend the map in a larger space and show is continuous/smooth. Then by restricting to submanifolds, continuity/smoothness is preserved.
3)Baire category theorem
Every countable union of nowhere dense sets has empty interior. This is true for locally compact spaces and thus manifolds.
PS. If this not the right tag (ie. if it a community post) feel free to move it.