Fredholm operators in the Toeplitz Algebra $\mathcal{I}(QC)$

We will give a complete description of $\mathcal{I}$, the set of invertible quasicontinuous functions on the unit circle. After doing this, we will then classify the path-connected components of $\mathcal{I}$ and show that $\mathcal{I}$ has uncountably many path-connected components. We will then use the above classifications to characterize $\mathcal{F}$, the set of Fredholm operators of the C$^*$-algebra generated by the Toeplitz operators $T_\phi$ with quasicontinuous symbols $\phi$. Then we will classify the path-connected components of $\mathcal{F}$ and show that $\mathcal{F}$ also has uncountably many path-connected components.