# Inessential mapping

homotopically-trivial mapping

A continuous mapping $f : X \to Q^n$ of a topological space $X$ into the $n$-dimensional ball $Q^n$ such that there is a continuous mapping $g : X \to Q^n$ that coincides with $f$ on the inverse image $f^{-1} S^{n-1}$ of the boundary $S^{n-1}$ of $Q^n$ and takes $X$ into $S^{n-1}$ (that is, $gX \subseteq S^{n-1}$). When $X$ is a normal Hausdorff space, then $\dim X < n$ if and only if every continuous mapping $f : X \to Q^n$, $n = 1, 2, \dots ,$ is inessential (Aleksandrov's theorem).

A continuous mapping of a topological space into the $n$-dimensional sphere is called inessential if it is homotopic to the constant mapping.

The term "homotopically-trivial mapping" is not used in the context $f : X \to Q^n$. See [a1], [a2] and [a3].