Preface This book is an exploration of Seifert fiberings. These are mappings which extend the notion of fiber bundle mappings by allowing some of the fibers to be singular. Seifert fiberings are mappings whose typical fibers are homeomorphic to a fixed homogeneous space. The singular fibers are quotients of the homogeneous space by distinguished groups of homeomorphisms. In a remarkable paper, in 1933, Herbert Seifert, introduced a class of 3-manifolds which became known as Seifert manifolds. They play a very significant role in low dimensional topology and remain under intense scrutiny today. A Seifert manifold maps onto a 2-dimensional surface such that the inverse image of each point on the surface is homeomorphic to a circle. The set of singular fibers are isolated from each other and the typical fibers wind nontrivially around the singular fibers. We will describe in detail the Seifert 3-manifolds as a special case of the general con- struction of Seifert fiberings. Our major focus, however, is on higher dimensional phenomenon where the typical fiber is a homogeneous space. A major inspiration for a generalization to higher dimensions comes from trans- formation groups. Let (G,X) be a proper action of the connected Lie group G on a path-connected X and examine the homomorphism ev∗ x : π1(G,e) π1(X,x) induced by the evaluation map evx : g mapsto→ gx, g G. The image H of ev∗ x is a central subgroup of π1(X,x) independent of the base point x. The G-action on X can be lifted to the covering space XH of X associated with the subgroup H of π1(X,x) so that π1(XH,hatwide) = H. The covering transformations, Q = π1(X,x)/H, commute with the lifted G-action on XH and so induce a proper Q-action on W = G\XH. We get the following commutative diagram of orbit mappings: (G,XH) G\ τ d47 d47 ν′ Q\ d15 d15 (Q,W) ν Q\ d15 d15 (G,X) G\ τ d47 d47 G\X = Q\W What we discover is that the lifted action of G on XH is usually simpler than on the original X. The discrete action of Q on W can be used to describe the action of G on X locally. In fact, under the appropriate circumstances, the Q-action on W can be used to construct all the possible G-actions on X whose orbit space is Q\W. For example, if G = Tk, the k-dimensional torus, and evx is injective, then (Tk,X) is called an injective torus action. For this, XH splits into a product xiii
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