## Extending isotopies of planar continua.(English)Zbl 1221.57035

The paper treats the isotopy extension problem for planar continua, i.e. if $$Z$$ is a continuum in the complex plane $$\mathbb{C}$$ and $$h^t:Z\to \mathbb{C}, t\in [0,1]$$ an isotopy starting at the identity, can $$h^t$$ be extended to an isotopy of the plane starting at the identity? In order to solve this question the authors consider the plane as a subset of the complex sphere $$\mathbb{C^*} = \mathbb{C} \cup \{\infty\}$$. Using known results related to $$\mathbb{C^*},$$ improving and adapting appropriate techniques, in Theorem 7.3 of the paper they obtain:
Suppose that $$h^t:Z\to \mathbb{C}$$ is an isotopy of a planar continuum $$Z$$, which we consider as a subset of the sphere $$\mathbb{C^*}$$ with $$h^0= id|Z$$. Then there exists an extension to an isotopy $$H^t:\mathbb{C^*}\to \mathbb{C^*}$$ such that $$H^0=id|\mathbb{C^*}.$$
To return to the plane $$\mathbb{C}$$ let $$U$$ denote the component of $$\mathbb{C^*} \setminus Z$$ containing the point at infinity. By composing the isotopy $$H^t$$ with an isotopy $$K^t$$ of the sphere such that $$K^0=id|\mathbb{C^*}$$ and $$K^t|\mathbb{C^*}\setminus U=id|\mathbb{C^*}\setminus U,$$ and $$K^t(H^t(\infty))=\infty$$ for all $$t\in [0,1]$$ one obtains:
Suppose that $$h^t:Z\to \mathbb{C}$$ is an isotopy of a planar continuum $$Z\subset \mathbb{C}$$ with $$h^0= id|Z$$. Then there exists an extension to an isotopy $$H^t:\mathbb{C}\to \mathbb{C}$$ such that $$H^0=id|\mathbb{C}.$$

### MSC:

 57N37 Isotopy and pseudo-isotopy 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 54C20 Extension of maps 54F15 Continua and generalizations
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