I've got well over a thousand e-mails waiting to be looked at. I'm working on the pile from both ends. Two recent postings from Mike Barr: 1) The duality he describes sounds like Spanier-Whitehead duality. It's usually defined geometrically. Embed a finite complex into a higher-dimensional sphere, take the complement, contract it to a finite complex. Do this in the stable-homotopy category (obtained by forcing the suspension functor on the homotopy category to become an automorphism) and normalize the dimensions so that the dual of a space has the negative of its dimension. Show that the embeddings can be chosen in a coherent fashion to obtain a self-duality for stable homotopy. (The last step is non-trivial and the only person I know who verified it is Frank Adams, and that wasn't published. Does anyone have a citation?) 2) If every n-1 cell of an n-dimensional finite complex is the face of _exactly_ 2 n-dimensional cells then there's a fairly easy argument that the n-dimensional Z_2 cohomology is non-trivial. But if the condition is changed to "_at least_ 2 n-dimensional cells" then the space can be contractible. Start with the closed unit disk in the complex plane and glue the boundary onto the closed unit interval (which constitutes half of the intersection of the disk and the real line) by identifying a point x on the interval with e^(2(pi)xi) on the boundary. If one triangulates this space, each edge will be a face of either 2 or 3 triangles. One way of seeing that this is contractible is to consider first the space obtained by identifying just the upper half of the boundary with the unit interval by identifying a point x on the interval with e^((pi)xi) on the boundary. Recalling that the homotopy type of a space is unchanged when reasonable closed contractible subsets are collapsed to points, note that if the lower half of the boundary is collapsed to a point we obtain the first space. On the other hand, the upper half (together with the unit interval) is contractible, and when it's collapsed to a point we obtain the unit disk. By taking the suspension of this example we can get an example in every larger finite dimension. This example is a mapping cone. Take the map from the circle to the circle that wraps the first third of the circle around the circle, wraps the second third around in the opposite direction, and the last third in the original direction. This map is, of course, homotopic to the identity map. The mapping-cone construction depends only on the homotopy type of the map. The mapping cone of the identity map on the circle is, of course, the disk.