Bounds in piecewise linear topology

Abstract:

The following types of results are obtained: Given a polyhedral $2$-sphere $P$ with rectilinear triangulation $T$ lying in the interior of a solid tetrahedron $G$ in ${E^3}$, then there is a simplicial isotopy $f:G \times [0,1] \to G$ taking $P$ onto a tetrahedron so that for $t$ in $[0,1],f(x,t) = x$ on $\text {Bd} (G)$ and ${f_t}$ is affine on each element of the triangulation $S$ of $G$, where card $(S)$ is a known function of card $(T)$. Also, given (1) $P$ as above, (2) polyhedral disks ${D_1}$ and ${D_2}$, where $\text {Bd} ({D_1}) = \text {Bd} ({D_2}) \subset P$ and $\operatorname {Int} ({D_1}) \cup \operatorname {Int} ({D_2}) \subset \operatorname {Int} (P)$ and (3) a triangulation $T$ of ${D_1} \cup {D_2} \cup P$, then analogous results are found for a simplicial isotopy $f$ which is fixed on $P$ and takes ${D_1}$ onto ${D_2}$. Given $G$ as above and a piecewise linear homeomorphism $h:G \to G$ which is fixed on ${\text {Bd(G)}}$ and affine on each $r \in R$, then analogous bounds are found for a simplicial isotopy $f:G \times [0,1] \to G$ so that ${f_0}(x) = x$ and ${f_1}(r) = h(r)$ for all $r$ in $R$. 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.