This little note came out of a lunch discussion with NYU grad student Alfredo Hubard earlier this week. I think the problem-solving process was quite amusing hence worth shearing.
Back in kindergarten, we all learned this theorem called ‘at any given time, there are two opposite places on Earth having exactly the same temperature and air pressure’. Yes that’s the Bosuk-Ulam theorem. I remember at some point I came across a much less famous theorem in some kind of discrete/combinatorial geometry, saying:
Theorem: Any map from a n-dimensional simplex to must have a pair of intersecting opposite faces.
Note: each k-dimensional face of the n-dimensional simplex has a unique, well-defined dimensional opposite face, as shown:
Some examples of maps from the 3-simplex to :
i.e. in general they can be quite a mess. I think it can be proved by Thurston’s simplex straightening argument. (haven’t checked carefully)
To me this is like Bosuk-Ulam except for instead of considering a large amount of antipodal pairs, we consider only finitely many such pairs. Hence a discrete analoge.
However, one should note that although they are of the same nature, neither follows from the other.
So somehow this theorem came up during the lunch and Alfredo mentioned to me that professor Guth wondered whether the theorem can be proved for mappings from the simplex to lower dimensional simplicial complexes (instead of ). i.e.
Question: Given where denote the -dimensional simplex and is a -dimensional simplicical complex. Then must there be a pair of opposite faces with intersecting image?
So we started to throw out random ideas.
First of all, although only boundary faces of the simplex has non-empty opposite faces (hence can possibly be intersecting pair), it is important that is defined on the solid simplex. (i.e. if one just map the boundary, then we may let be topologically a sphere and make the map a homeomorphism!) So the moral is, we kind of need to ‘claps’ the simplex to a simply connected lower dimensional thing first, then map it to our , hence having non-trivial topology won’t be of much help. Looks almost like the case, doesn’t it?
Perhaps the image on can be complicated and has non-trivial topology, but this is merely ‘wrapping’ a contractible, -dimensional thing around. But wrapping around can only cause more overlapping hence making the faces intersecting more, not less.
The above line of thoughts give an immediate proof in the case when is a surface and : Lift the map to the universal cover and apply the theorem for . (It’s a little more tricky when the surface is but you can work it out~ the map restricted to must be of even degree) Note this won’t generalize to higher dimensions (even for manifolds) since universal covers are no longer that similar to .
So what’s the main difference between complexes and manifolds? well, one can have more than two dimensional faces attached to the same dimensional face. I decided to first think of whether Bosuk-Ulam is still true if we further assume that the map extends to the solid ball (as seen above, it is true in the surface case).
After trying to generalize that ‘lifting’ proof for a while, we realized the Bosuk-Ulam does not work for simplicial complexes in all dimensions! For very simple reason, i.e. if we map the to by projecting down, the pre-image of the center point is the only pair of antipodal points that’s mapped together. Hence if instead of we have three -simplicies attached along a single -simplex (think of it as a piece of with a vertical simplex attached in the middle):
Now we can still project to that piece of , with the image of antipodal point lying on the middle simplex, all we need to do is to separate this pair while not creating new pairs! But this is easy, just ‘drug’ the upper sheet into the third -simplex a little bit:
(the region outside of the red neibourhood is unchanged) It’s easy to see that no new antipodal pair is moved together.
So now we turned back to the simplex and realized it’s even easier to argue: project in orthogonal direction to a -face, originally the only intersecting pair of opposite faces was the vertex and that -face. Now if we lift the vertex int the third sheet, nothing can intersect~
OK~ perhaps not a useful answer but problem solved!
So far I do not know the answer to the following:
1. Is this intersecting faces theorem true for mapping -simplexes to -dimensional manifolds?
2. Is the Bosuk-Ulam true if we consider maps from spheres to -dimensional manifolds which extends to the ball?
The later might be well-known. But so far I can only find a theorem by Conner and Floyd, stating that any map from to a lower dimensional manifold must have a pair of antipodal points mapped together.