Let $M$ be a Manifold, $\pi_1(M,p_0)$ the Fundamental Group of $M$ at a base-point $p_0$ and $\widetilde{M}$ the Universal Covering of $M$.
Then $\widetilde{M}$ can be defined as follows: $$\widetilde{M}=\{[\alpha]: \alpha:[0,1]\longrightarrow M, \alpha(0)=p_0\}$$ where $[\alpha]=[\beta]$ if and only if $\alpha(1)=\beta(1)$ and $\alpha$ is homotopic to $\beta$.
Let $[c]\in \pi_1(M,p_0)$, then the deck transformation associated to $[c]$ is defined as follows: $$c:\widetilde{M}\longrightarrow \widetilde{M}$$ $$c.[\alpha]:=[\alpha \star c^{-1}]$$ This is the natural action of $\pi_1(M,p_0)$ on $\widetilde{M}$.
Am I right?
Is everything correct?
Is there any mistake?