View source for 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.



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

Generally, a mapping which is homotopic to a constant mapping is called nullhomotopic or homotopically trivial; in [[#References|[a3]]] they are called inessential. See also [[Essential mapping|Essential mapping]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Topologie" , '''1''' , Springer  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Engelking,   "Dimension theory" , PWN  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Hurevicz,   G. Wallman,   "Dimension theory" , Princeton Univ. Press  (1948)</TD></TR></table>

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