2 Barr questions

2 Barr questions

[Peter Freyd]

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.