Exercise 6, § 2

Identifying $S_1$ with the torus $T$, the quotient of $\mathbf{R}$ by the relation $x \equiv y \pmod{1}$ (no. 4, Proposition 4, Corollary 2), let $\varphi$ denote the canonical mapping of $\mathbf{R}$ onto $S_1$. A continuous mapping $f$ of a topological space $E$ into $S_1$ is inessential if there exists a continuous mapping $g$ of $E$ into $\mathbf{R}$ such that $f = \varphi \circ g$. (**) A mapping which is not inessential is said to be essential.

Show that the identity mapping of $S_1$ onto $S_1$ is essential (use Exercise 4).