Bounds in Piecewise Linear Topology

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Abstract
The following types of results are obtained: Given a polyhedral -sphere with rectilinear triangulation lying in the interior of a solid tetrahedron in , then there is a simplicial isotopy taking onto a tetrahedron so that for in on Bd and is affine on each element of the triangulation of , where card is a known function of card . Also, given (1) as above, (2) polyhedral disks and , where Bd Bd and and (3) a triangulation of , then analogous results are found for a simplicial isotopy which is fixed on and takes onto . Given as above and a piecewise linear homeomorphism which is fixed on and affine on each , then analogous bounds are found for a simplicial isotopy so that and for all in . In the second half of this paper the normal surface and normal equation theory of Haken is briefly explained and extended slightly. Bounds are found in connection with nontrivial integer entried solutions of normal equations. Also bounds are found for the number of Simplexes used in triangulating normal surfaces associated with certain solutions of the extended normal equations.

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