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Homotopy properties of curves. (English) Zbl 0962.54027

The authors investigate conditions that imply noncontractibility of curves. They give an example of a noncontractible plane dendroid which contains no homotopically fixed subset. Next they say, that a nonempty subset \(A\) of a space \(X\) is homotopically steady if and only if, for every deformation \(H:X\times I\rightarrow X\), and for every \(t\in I\), \(A\subset H(X\times \{ t\})\). The kernel of steadiness of \(X\) is the union of all sets \(A\) that are homotopically steady. They prove that each contractible space has an empty kernel of steadiness. The paper contains other results relating the introduced notions to other ones, as well as other examples, and open problems. In particular, does every noncontractible dendroid have a nonempty kernel of steadiness?

MSC:

54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
54F15 Continua and generalizations
54E40 Special maps on metric spaces
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