Inessential mapping

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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].

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


[a1] P.S. Aleksandrov, "Topologie" , 1 , Springer (1974)
[a2] R. Engelking, "Dimension theory" , PWN (1977)
[a3] W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)
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