Torsten Ekedahl: Orientations and spannings of posets

In joint work with Antonio Díaz Ramos I have introduced the notion of spanning and orientation of a poset. A spanning is a choice of combinatorial data on a poset which has as a consequence that its nerve (as well as the nerves of all
of its downwards intervals) have the homotopy type of a wedge of spheres (of appropriate equal dimension), we say that the poset then is (homotopically) wedgical. This is to be somewhat contrasted with the various types of shellings that imply that all the intervals have this property. Also the
number of spheres involved gets a combinatorial description. An orientation is a compatible choice of spannings for all the downwards intervals of a poset. The nerve of the poset is then shown to be homotopic to a simplicial set associated to the orientation and the chain complex of that simplicial set is
the chain complex that always exist for wedgical posets and which is normally much smaller than the chain complex for the nerve, yet computes its homology.