A Splitting for Right Angled Reflection Manifolds.
Bruce Jay Levy · Deep Blue (University of Michigan) · 1984
Let Q be a simply connected manifold whose boundary is a homology sphere. By using a Coxeter group G as a reflection group and Q as fundamental domain, one may construct another simply connected manifold (')M so that G(FDIAG)(')M = Q. If H is any torsion free subgroup of finite index in G, then M = H(FDIAG)(')M is a closed manifold. Michael Davis has recently used this construction to produce closed aspherical manifolds whose universal covers are not Euclidean space. In the present work it is shown that when the Coxeter group is presented with all torsion of order 2, (pi)(,1)(M) is accessible in the sense of Waldhausen. These right angled Coxeter groups can be employed to produce manifolds with universal covers not Euclidean space as well as manifolds with Euclidean universal covers. The result is obtained by establishing the existence of codimension one manifolds in M, whose fundamental groups inject, such that deletion of these submanifolds simplifies M and (pi)(,1)(M) in such a way as to fit the requirements of the splitting. It is also shown that if the choice of the torsion free subgroup is restricted, the splitting is square root closed, in that at each stage of building (pi)(,1)(M) if the square of an element lies in the amalgamating subgroup of an amalgamated free product or associated subgroup of an HNN construction then the element itself lies in the subgroup. This decomposition of (pi)(,1)(M) is of interest for several reasons. In particular, it provides information on the long exact surgery sequence of M by controlling the homotopy classes of maps of M into the universal space G/TOP.