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Embeddings of k-complexes into 2k-manifolds

Banff International Research Station for Mathematical Innovation and Discovery

Let $K$ be a simplicial $k$-complex and $M$ be a closed PL $2k$-manifold. Our first aim during the talk is to describe an obstruction for embeddability of $K$ into $M$ via the intersection form on $M$. For description of the obstruction, we need a technical condition which is satisfied, in particular, either if $M$ is $(k-1)$-connected or if $K$ is the $k$-skeleton of $n$-simplex, for some $n$. Under the technical condition, if $K$ (almost) embeds in $M$, then our obstruction vanishes. In addition, if $M$ is $(k-1)$-connected and $k \geq 3$, then the obstruction is complete, that is, we get the reverse implication. Modulo a recent hard Lefschetz theorem of Adiprasito, a consequence of our results on the existence and completeness of the obstruction are very good bounds on the Helly number in a certain Helly type theorem where the ambient space is a (suitable) manifold. The details will be explained during the talk. The talk is based on a joint work with Pavel Paták.

Mathematics; Geometry; Convex and discrete geometry; Discrete mathematics

video/mp4

Author affiliation: Charles University in Prague

2021-01-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/77063

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/