We characterize the cyclic branched covers of the 2-sphere where every homeomorphism of the sphere lifts to a homeomorphism of the covering surface. This answers a question that appeared in an early version of the erratum of Birman and Hilden [2].

In this article, we are concerned with various aspects of arcs on surfaces. In the first part, we deal with topological aspects of arcs and their complements. We use this understanding, in the second part, to construct an interesting action of the mapping class group on a subgraph of the arc graph. This subgraph naturally emerges from a new characterisation of infinite-type surfaces in terms of homeomorphic subsurfaces.Résumé. -Cet article s'intéresse à plusieurs aspects des arcs sur les surfaces. La première partie s'occupe des aspects topologiques des arcs et de leurs compléments. Nous utilisons les résultats de la première partie pour définir ensuite une action du groupe modulaire sur un sousgraphe du graphe des arcs. Ce sous-graphe ressort naturellement d'une nouvelle caractérisation des surfaces de type infini en termes de sous-surfaces homéomorphes.

We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides with the entire mapping class group of the surface. As a consequence, we construct infinite families of non-geometric embeddings of the braid group into mapping class groups in the sense of Wajnryb. Indeed, our embeddings map standard braid generators to products of Dehn twists about curves forming chains of arbitrary length. As key tools, we use the Birman-Hilden theorem and the action of the mapping class group on a particular fundamental groupoid of the surface.

In this article, we are concerned with various aspects of arcs on surfaces. In the first part, we deal with topological aspects of arcs and their complements. We use this understanding, in the second part, to construct interesting actions of the mapping class group on a subgraph of the arc graph. This subgraph naturally emerges from a new characterisation of infinite-type surfaces in terms of homeomorphic subsurfaces.Date: March 31, 2020. 1 I.e. such that the inclusion map is not homotopic to a homeomorphism.

This paper gives necessary and sufficient conditions that the free product with amalgamation of circularly ordered groups admit a circular ordering extending the given orderings of the factors. Our result follows from establishing a categorical framework that allows the problem to be restated in terms of amalgamating certain left-ordered central extensions, where we are able to apply work of Bludov and Glass.Problem 1.3. Suppose that M is a 3-manifold with geometric pieces M 1 , . . . , M n , and that π 1 (M i ) admits a circular ordering c i . Determine necessary and sufficient conditions in terms of the gluing maps which recover M from the pieces M i , and the circular orderings c i , which guarantee the existence of an ordering c of π 1 (M ) extending each of the c i (cf.[4, Theorem 1.7(2)]).The paper is organized as follows. In Section 2, we review definitions and place them in a categorical framework. In Section 3, we expand the categories introduced in Section 2 to include † The authors of [1] pose the question in terms of a decomposition of π 1 (M ) arising from a Heegaard splitting.