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a generalization of the strong separation result (to arbitrary oriented matroids, including centrally symmetric polygons) appears in Theorem 2.7 and Proposition 5.1 of arxiv.org/abs/1708.01329
As you explain, Freedman shows that every 5-dimensional h-cobordism between 4-manifolds is trivial (in the topological category), and Freedman+Perelman shows that every 4-dimensional h-cobordism between a 3-manifold and a 3-sphere is trivial (by gluing balls on both sides). But which reference shows that 4-dimensional h-cobordisms between general 3-manifolds are trivial?
Thanks! Actually, do you know if the topological h-Cobordism Theorem is even known in all dimensions? Specifically, we suspect it is open for n=4. So our "circular" argument is the only short proof we know that works for all n.
I've seen something along these lines for oriented matroids in the following book by Santos: personales.unican.es/santosf/Articulos/OMtriFinal.pdf see Definition 2.1, part "intersect properly". He treats the set of all extensions satisfying certain properties as a genuine convex hull of a simplex.
Also found the relevant place in the book: basically, my question can be rephrased as "when can a strong map be factored as an extension followed by a contraction", and the answer is "when $r(\mathcal{M}_1)\leq r(\mathcal{M}_2)+1$", see Proposition 7.7.4 and Exercise 7.30 here
@PerAlexandersson actually, I think there is an "easy" counterexample if you even replace $2$ with any number. Let $P(x_1,\dots,x_{k+1})=s_{21^{k-1}}-s_{1^{k+1}}$. Then fixing any monomial in $m\leq k$ first variables corresponds to basically skewing both shapes by a column $1^m$, so applying the L-R rule gives the result
Good question, I tried that but didn't succeed. If you take inner product of both sides with $s_\kappa$ and then use the fact that skewing by $\mu$ is adjoint to multiplying by $s_\mu$, you get that the sum of L-R coefficients $C^\lambda_{\mu,\kappa}$ over all $\lambda$ equals the sum of L-R coefficients $C^\lambda_{\mu^\vee,\kappa}$ over all $\lambda$ ($\mu^\vee$ denotes rotation). But even though the sums are the same, for certain values of $\mu$ and $\kappa$ you get different lists of numbers.
