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A friend told me that the $\mathbf{C^1}$-topology on the set $C^\infty(M,N)$ of smooth functions between two smooth manifolds $M$ and $N$ can be defined as the coarsest topology making the map $$ C^\...

Suppose we are given an open cover $\mathcal{U}=(U_{i})_{i \in I}$ of a smooth manifold $M$, a cocycle of smooth transition functions $g_{ij}: U_{ij} \to G$ where $G$ is a Lie group, and a (not ...

In trying to prove Cerf's theorem about homotopies between Morse-functions I ended up thinking about the following problem. For $n>2$, $a= (a_{i,j})\in GL(n-2)$, we define the polynomial map $C_a:\...