Real flag manifolds and a construction of spaces over a polyhedron: Mathematical Investigations arising from the Jahn-Teller effect
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 o...
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| Language: | English |
| Published: |
2009
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| Online Access: | http://hdl.handle.net/2429/6198 |
| Summary: | 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. |
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