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Abstract

We examine a construction of topological spaces over an arbitrary polyhedron and show that it subsumes the lattice construction of R. R. Douglas and A. R. Rutherford. A Simplicial Approximation Theorem is proven for the general construction, for maps from a polyhedron to one of our spaces lying over another polyhedron. A special case of our construction (a slight generalization of the lattice construction) is examined and a class of locally trivial bundles is constructed. These are used to examine neighbourhood structure in the special case. We also enumerate exactly which spheres can be constructed by a lattice construction on a product of real orthogonal, complex unitary or quaternionic symplectic groups. The fundamental group of the real complete flag manifolds is determined following a detailed exposition of Clifford algebras. Appendices are provided on the diagonalization of quaternionic Hermitean matrices and on a generalized mapping cylinder that can be regarded as an endofunctor on the category of locally trivial bundles over a fixed locally compact base.