Related papers: Embedding periodic maps on surfaces into those on …

A periodic automorphism of a surface $\Sigma$ is said to be extendable over $S^3$ if it extends to a periodic automorphism of the pair $(S^3,\Sigma)$ for some possible embedding $\Sigma\to S^3$. We classify and construct all extendable…

It is known that any periodic map of order $n$ on a closed oriented surface of genus $g$ can be equivariantly embedded into $S^m$ for some $m$. In the orientable and smooth category, we determine the smallest possible $m$ when $n\geq 3g$.…

Let $G$ be a finite group acting on a connected compact surface $\Sigma$, and $M$ be an integer homology 3-sphere. We show that if each element of $G$ is extendable over $M$ with respect to a fixed embedding $\Sigma\rightarrow M$, then $G$…

In this paper, we give a classification of orientation reversing periodic maps on closed surfaces which generalizes the theory of Nielsen for the orientation preserving periodic maps. On one hand, we give a group of data for each…

We consider orientation-preserving actions of finite groups $G$ on pairs $(S^3, \Sigma)$, where $\Sigma$ denotes a compact connected surface embedded in $S^3$. In a previous paper, we considered the case of closed, necessarily orientable…

We restrict our discussion to the orientable category. For $g > 1$, let $OE_g$ be the maximum order of a finite group $G$ acting on the closed surface $\Sigma_g$ of genus $g$ which extends over $(S^3, \Sigma_g)$, where the maximum is taken…

Let $F_g$ be the closed orientable surface of genus $g$. We address the problem to extend torsion elements of the mapping class group ${\mathcal{M}}(F_g)$ over the 4-sphere $S^4$. Let $w_g$ be a torsion element of maximum order in…

Let $OE_g$ (resp. $CE_g$ and $AE_g$) and resp. $OE^o_g$ be the maximum order of finite (resp. cyclic and abelian) groups $G$ acting on the closed orientable surfaces $\Sigma_g$ which extend over $(S^3, \Sigma_g)$ among all embeddings…

An automorphism $f$ of a closed orientable surface $\Sigma$ is said to be extendable over the 3-sphere $S^3$ if $f$ extends to an automorphism of the pair $(S^3, \Sigma)$ with respect to some embedding $\Sigma \hookrightarrow S^3$. We prove…

Kulkarni showed that, if g is greater than 3, a periodic map on an oriented surface S_g of genus g with order more than or equal to 4g is uniquely determined by its order, up to conjugation and power. In this paper, we show that, if g is…

It is known that an automorphism of $F_g$, the oriented closed surface of genus $g$, is extendable over the 4-sphere $S^4$ if and only if it has a bounding invariant spin structure \cite{WsWz}. We show that each automorphism of $F_g$ has an…

The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, \Sigma)$…

We investigate the iterative behaviour of continuous order preserving subhomogeneous maps that map a polyhedral cone into itself. For these maps we show that every bounded orbit converges to a periodic orbit and, moreover, that there exists…

Suppose an orientation preserving action of a finite group $G$ on the closed surface $\Sigma_g$ of genus $g>1$ extends over the 3-torus $T^3$ for some embedding $\Sigma_g\subset T^3$. Then $|G|\le 12(g-1)$, and this upper bound $12(g-1)$…

We study in detail mirror symmetry for the quartic K3 surface in P3 and the mirror family obtained by the orbifold construction. As explained by Aspinwall and Morrison, mirror symmetry for K3 surfaces can be entirely described in terms of…

We prove that, under a mild condition on the hyperbolicity of its periodic points, a map $g$ which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map).…

For any fixed surface Sigma of genus g, we give an algorithm to decide whether a graph G of girth at least five embedded in Sigma is colorable from an assignment of lists of size three in time O(|V(G)|). Furthermore, we can allow a subgraph…

We consider the orientation-preserving actions of finite groups $G$ on pairs $(S^3, \Gamma)$, where $\Gamma$ is a connected graph of genus $g>1$, embedded in $S^3$. For each $g$ we give the maximum order $m_g$ of such $G$ acting on $(S^3,…

In this work we study homeomorphisms of closed orientable surfaces homotopic to the identity, focusing on the existence of non-contractible periodic orbits. We show that, if $g$ is such a homeomorphism, and if $\hat g$ is its lift to the…

Extension dimension is characterized in terms of $\omega$-maps. We apply this result to prove that extension dimension is preserved by refinable maps between metrizable spaces. It is also shown that refinable maps preserve some…