On the fundamental group of an orbit space

Abstract

Introduction. Let K be a connected simplicial complex, finite or infinite, its polyhedron ((2), page 45) being the space X. Then X is connected. Suppose further that X is simply connected. For any group G of simplicial transformations of X, H will denote the normal subgroup generated by elements which have a non-empty fixed-point set.

A presentation for the baseleaf preserving mapping class group of the punctured solenoid

“…By a standard result of Armstrong [1], we get the following Theorem. Acknowledgements We are grateful to John Milnor for useful comments.We also thank the referee whose careful reading has led to many improvements in our paper.…”

“…There is a partial ordering on the covers as follows. If one cover 1 can be factored as the composition of two covers 1 D ı 2 , where ; 2 are also finite unbranched covers, then 1 2 . The system of covers is inverse directed, and there is thus an inverse limit.…”

“…Theorem (Armstrong [1]) Let G act simplicially on a connected and simply connected complex K , and let H be the normal subgroup generated by those elements of G which leave some point fixed. Then the fundamental group of the orbit space jKj=G is isomorphic to G=H .…”

Cutting and pasting in the Torelli group

“…Our proof of Theorem 1.3 makes use of a basic result of Armstrong [1] (see Section 2.2) which says that if a group G acts nicely on a simply-connected simplicial complex X , then G is generated by elements which stabilize vertices of X if and only if X=G is simply connected. We prove that C .…”

“…Mod. † g;n 1 / is induced by "gluing a disc to a boundary component", and the map 1 .U † g;n 1 / ! Mod.…”