Let $K$ be a $k$-dimensional simplicial complex having $n$ faces of dimension $k$, and $M$ a closed $(k-1)$-connected PL $2k$-dimensional manifold. We prove that for $k\ge3$ odd $K$ embeds into $M$ if and only if there are $\bullet$ a skew-symmetric $n\times n$-matrix $A$ with integer entries, whose rank over $\mathbb Q$ does not exceed $rk H_k(M;\mathbb Z)$, $\bullet$ a general position PL map $f:K\to\mathbb R^{2k}$, and $\bullet$ orientations on $k$-faces of $K$ \newline such that for any nonadjacent $k$-faces $σ,τ$ of $K$ the entry $A_{σ,τ}$ equals to the algebraic intersection of $fσ$ and $fτ$. We prove some analogues of this result (for any parity of $k$), including those for $\mathbb Z_2$- and $\mathbb Z$-embeddability. Our results generalize the Bikeev-Fulek-Kyn\v cl criteria for the $\mathbb Z_2$- and $\mathbb Z$-embeddability of graphs to surfaces, and are related to the Harris-Krushkal-Johnson-Paták-Tancer criteria for the embeddability of $k$-complexes into $2k$-manifolds. The main novelty of this paper is passing from the cohomology condition of Paták-Tancer to the simpler extendability of some intersection function to a low-rank matrix (defined in the paper using the idea of Fulek-Kyn\v cl).
Using dehydration signatures as an encoding, we train GPT-2-style transformers to generate triangulations of 3-dimensional manifolds. Our main runs each use approximately 3 exaFLOPs, spanning a 200 million parameter model trained on 60 million sphere triangulations to an 11 million parameter model trained on 1.1 billion sphere triangulations. The optimal model has 10 blocks, width 640 and generates dehydration signatures of closed manifolds of various topological types at a rate of 6.9%, 82% of which are spheres. To investigate how our method scales, we use the technique of IsoFLOP curves on 15-tetrahedron triangulations of the 3-sphere, over a compute range of 6 to 3000 petaFLOPs, and observe neural scaling behaviour. We extract a parameter scaling exponent of $α= 0.49$ ($[0.48, 0.50]$ stat, $\pm 0.02$ syst), and an optimal ratio of 234 tokens per parameter for training. In addition, we present a number of smaller experiments, including evidence that transformers are able to learn to count degree-one edges near-perfectly, but get progressively worse when counting edges with higher degrees.