Are there intersection theoretic proofs for Ham-Sandwich type theorems?

Concrete Question:

Let f:SnRn be a Z/2 smooth equivariant map where the action on the sphere is antipodal, and Rn is multiplication of co-ordinates.

Can one show that

0Hn(ni=1Xi)Hn(Sn)

where
Xi:=f1(R×{0}ith component××R).

It’s not immediately clear that such a question is well-posed (transverse intersections for example.) One might need to take a subset of the intersection in order to get the Borsuk-Ulam theorem. It’s not clear to me how such a proof would go. See the update section for how one can recover the continuous version from the smooth one.

The rest of this question is just expanding on this.

Background

In equivariant topology there is a configuration space/test map paradigm to solve a geometric problem.

Specifically, given a geometric problem P, we define the configuration space, X, which parametrizes all associated solutions to the problem (such as points, lines, or arcs.) Additionally, we consider a test subspace ZV and a continuous map f:XV where pX is a solution to a problem if and only if f(p)Z. With this setup in mind, we further require that f be G-equivariant, where G acts on both X and Y. From this, the typical method of proof is to show the nonexistence of maps f:XGYZ, ensuring the existence of a geometric solution.

More concretely, in the Ham Sandwich theorem, we let X=Sn parametrize all possible configurations of n1 dimensional hyperplanes (this is done by scaling coefficients in a hyperplane formula and adding two hyperplanes at infinity), we let Y be Rn, and let f:XY be a function assigning a volume to each “mass” above and below a hyperplane and subtracts them. Here, G is Z2, and it acts on the sphere antipodally and Rn in the standard way. f is equivariant with respect to these actions.

A solution is Z=0, since this would imply a bisection of all the masses. Of course, the Borsuk–Ulam theorem tells us that such an equivariant map cannot exist.

Motivation

A different way of thinking about the above set up (and I believe it is used in a popular youtube video) is to consider the problem co-ordinate wise:

For example, in the Ham-Sandwich theorem, we have that a solution is an intersection ni=1Xi

where
Xi:=f1(R×{0}ith component××R)

My Question

Thinking about the above as an n1-dimensional manifold. Are there theorems guaranteeing that such intersections are not empty when f is equivariant ?(and for my discussion smooth, although I do not know If this is necessary for a proof)

More specifically:

  1. Can one show that Xi intersects all Xj with ji transversely?

  2. (If 1 is true) Regarding Xi as submanifolds, can one show that the homology class of their intersection is nonempty, i.e 0[XiXj]H0(Sn)?

A Naive Guess

The map f:SnRn that is Z2 equivariant can be associated with a G-bundle (vector bundles in this case) RPn×GRnRPn given by [x,y][y]. A section of this bundle is given by [x][x,f(x)] and the nonvanishing of f gives a nonvanishing section in this bundle. One can then show that this is impossible on the level of cohomology by considering Stiefel–Whitney classes (and in particular the top Stiefel–Whitney class.)

My hope is to maybe use something about the cup product structure on RPn (understood as the Poincare dual to intersection) in the above decomposition to get a similar theorem/result.

For example, in the two dimensional version of the ham-sandwich theorem, if we let [Xi]H1(Sn), then generically [X1X2]H0(X) (I’m not sure how to show that they intersect transversally.) Then, using the fact that [X1X2]H2(S2), one can then use
[X1][X2]=[X1X2]
to deduce that [X1X2] is nontrivial, so there is an intersection.

In order to use more information about equivariance, I would consider the images of X1,X2 in the covering map S2RP2, and actually use cohomology classes H(RP2,Z2).

From here, the problem is essentially just computing [Xi]H1(RP2) and showing that it is indeed nontrivial.

A hopeful, but unfounded idea: The induced map by inclusion i:XiX, gives that 0w1(i(Sn×GRn))=i(w1(X))=[Xi], where w1 is the first Stiefel–Whitney class.

From this, we can conclude that [X1][X2]=x2Z2[x]/(x3) is nonzero, and hence H0(X1X2) and deduce the ham sandwich theorem from this.

If there is something drastically different from this, I’d love to hear it as well, this is just kind of where my brain went.

Update

In the paper “geometric proofs of the two-dimensional Borsuk–Ulam theorem”, it is shown that

  1. one can recover the continuous version of the theorem from the smooth version (the proof here easily generalizes to higher dimensions.)

  2. Theorem 2.2 in the paper states that a smooth equivariant map with 0 as a regular value must contain an equator in f1(0), from which one can deduce the theorem (for two dimensions.)

Therefore, the smooth approach is equivalent, and there is a “geometric” proof showing that the intersection must be nonempty. Part 1 of my question is unanswered, but it has been shown that each Xi (for two dimensions) contain a subset of manifolds that do intersect transversely.

Answer

If the question is more generally “can one use cohomology rings to prove the ham sandwich theorem”, the answer is yes. Here is one route. Suppose, for the sake of contradiction, there were a map SnRn{0} as stated. Then the map would pass to the quotients RPnRPn1. The condition that antipode in Sn becomes antipode in Rn means that a path from x to x in Sn is sent to such a path in Rn, and thus a noncontractible loop in RPn is sent to a noncontractible loop in RPn1. So we would have a map f:RPnRPn1 which was nonzero on π1.

Now, look at cohomology with (Z/2)-coefficients. We get a map f:(Z/2)[η]/ηn(Z/2)[η]/ηn+1. The generator ηH1(RP,Z/2) corresponds to the nontrivial map π1Z/2, so f must carry η to η. But we have ηn=0 on the left hand side and not the right, a contradiction.


An idea which seems closer to your original approach would be to use (Z/2)-equivariant cohomology with (Z/2)-coefficients. The space Rn is (Z/2)-equivariantly contractible, so
HZ/2(Rn,Z/2)HZ/2(point,Z/2)H(RP,Z/2)(Z/2)[η]
where the generator η is in degree 1.

The action on Sn is free, so
HZ/2(Sn,Z/2)H(RPn,Z/2)(Z/2)[η]/ηn+1.
As before, the condition that the map is equivariant means that fη=η. I believe that the class of a hyperplane through 0 is η. If I am right, then the computation is that ηn0 in HnZ/2(Sn,Z/2), which is true.

Attribution
Source : Link , Question Author : Andres Mejia , Answer Author : David E Speyer

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