In the early days of topology, manifolds were often studied via triangulations. The combinatorial structure makes the computation of various invariants possible, and theorems can be proved based on the assumption of a suitable triangulation.

Since the manifolds themselves, and not their combinatorial structure, are the real objects of interest in topology, there was a growing desire to get away from triangulations: In the 1930's and 40's, algebraic tools gradually replaced the combinatorial ones.

Although it turned out that not every manifold can be triangulated, from the emergence of computers on there has again been growing interest in the combinatorial aspects of manifolds. It is now possible (at least in principle) to study compact manifolds and compute their invariants on a machine. A necessary requirement for this is that corresponding triangulations are small in size.

In this project we have constructed explicit triangulations for various classes of 3- and higherdimensional manifolds. Moreover, we completely enumerated manifolds with few vertices. An atlas of the obtained triangulations can be found at The Manifold Page.

• Anders Björner, KTH Stockholm,
• Jürgen Bokowski, TU Darmstadt,
• Ulrich Brehm, TU Dresden,
• Michael Joswig, TU Berlin,
• Ekkehard G. Köhler, TU Berlin,
• Wolfgang Kühnel, Universität Stuttgart.