On Generalized Heawood Inequalities for Manifolds: a van Kampen-Flores-type Nonembeddability Result

Wednesday, 03 May 2017
401 Moscow center for continuous mathematical education

The fact that the complete graph $K_5$ does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph $K_n$ embeds in a closed surface M (other than the Klein bottle) if and only if $(n-3)(n-4) \leq 6b_1(M)$, where $b_1(M)$  is the first $Z_2$-Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the high-dimensional analogue of $K_{n+1}$) embeds in $\mathbb{R}_{2k}$ if and only if $n\leq 2k+1$.  Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k−1)-connected 2k-manifold with kth $Z_2$-Betti number $b_k$ only if the following generalized Heawood inequality holds, which is a common generalization of the case of graphs on surfaces as well as the van Kampen-Flores theorem (the special cases k=1 and $b_k =0$, respectively), and also closely related to the theory of face numbers of triangulated manifolds. In the spirit of Kühnel's conjecture, we prove that if the k-skeleton of the n-simplex embeds in a 2k-manifold with kth $Z_2$-Betti number $b_k$, then $n\leq 2k_k \binom{2k+2}{k}+2k+4$. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k−1)-connected. Our results generalize to maps without q-covered points, in the spirit of Tverberg's theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.