Let $\gamma:[0,1]\to\mathbb{R}^3\setminus\{0\}$ be a simplex, with $\gamma(0)=\gamma(1)$. How can I show that exists a $2$-chain $\sigma$ on $\mathbb{R}^3\setminus\{0\}$ such that $\partial\sigma=\gamma$.
Can I use this result to prove that an closed $C^{\infty}$ $1$-form $\alpha$ on $\mathbb{R}^3\setminus\{0\}$ is exact?
For the first fact, it's enough to know that $\mathbb{R}^3 \setminus \{0\}$ is simply connected, thus it has a trivial first singular homology group: it follows that each 1-chain is the boundary of a 2-chain.