### 15-vertex triangulations of an 8-manifold.

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To any compactly supported, area preserving, piecewise linear homeomorphism of the plane is associated a relation in ${K}_{2}$ of the smallest field whose elements are needed to write the homeomorphism.Using a formula of J. Morita, we show how to calculate the relation, in some simple cases. As applications, a “reciprocity” formula for a pair of triangles in the plane, and some explicit elements of torsion in ${K}_{2}$ of certain function fields are found.

We aim at constructing a PL-manifold which is cellularly equivalent to a given homology manifold ${M}^{n}$. The main theorem says that there is a unique obstruction element in ${H}_{n-4}(M,{\mathscr{H}}^{3})$, where ${\mathscr{H}}^{3}$ is the group of 3-dimensional PL-homology spheres modulo those which are the boundary of an acyclic PL-manifold. If the obstruction is zero and $M$ is compact, we obtain a PL-manifold which is simple homotopy equivalent to $M$.

The purpose of this article is to introduce a method for computing the homology groups of cellular complexes composed of cubes. We will pay attention to issues of storage and efficiency in performing computations on large complexes which will be required in applications to the computation of the Conley index. The algorithm used in the homology computations is based on a local reduction procedure, and we give a subquadratic estimate of its computational complexity. This estimate is rigorous in two...