Geometric Construction of the Covering Manifold

Abstract

AbstractIn this chapter, we provide a geometric construction of a manifold extending a given Galois cover to a wreath product, using composita and fiber products. For this to be possible, a certain assumption on the homology, previously called (∗), needs to be strengthened to a new condition (∗∗) (equivalent in most cases). To motivate and use this new condition, we first recall the connection between homology of a quotient and coinvariants. Apart from geometric tools, the construction is also based on the vanishing of certain group cohomology, which is used to prove the existence of certain isometries of manifolds. In the final section, we give a universal property of the wreath product in relation to coverings of manifolds, just like there is such a universal property in the theory of Galois extensions of fields.