# A short exposition of the Patak-Tancer theorem on non-embeddability of k-complexes in 2k-manifolds

@article{Kogan2021ASE, title={A short exposition of the Patak-Tancer theorem on non-embeddability of k-complexes in 2k-manifolds}, author={Elena Kogan and Arkadiy Skopenkov}, journal={ArXiv}, year={2021}, volume={abs/2106.14010} }

In 2019 P. Patak and M. Tancer obtained the following higher-dimensional generalization of the Heawood inequality on embeddings of graphs into surfaces. We expose this result in a short well-structured way accessible to non-specialists in the field. Let ∆n be the union of k-dimensional faces of the n-dimensional simplex. Theorem. (a) If ∆n PL embeds into the connected sum of g copies of the Cartesian product S × S of two k-dimensional spheres, then g ≥ n− 2k − 1

## 2 Citations

### To the Kühnel conjecture on embeddability of k-complexes in 2k-manifolds

- MathematicsArXiv
- 2022

. The classical Heawood inequality states that if the complete graph K n on n vertices is embeddable in the sphere with g handles, then g > ( n − 3)( n − 4) 12 . A higher-dimensional analogue of the…

### Embeddings of k-complexes in 2k-manifolds and minimum rank of partial symmetric matrices

- MathematicsArXiv
- 2021

This work proves that for k ≥ 3 odd K embeds into M if and only if there are a skew-symmetric n × n-matrix A with Z-entries whose rank over Q does not exceed rkHk(M ;Z), and a collection of orientations on k-faces of K such that for any nonadjacent k- faces σ, τ the element Aσ,τ equals to the algebraic intersection of fσ and fτ.

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This work proves that for k ≥ 3 odd K embeds into M if and only if there are a skew-symmetric n × n-matrix A with Z-entries whose rank over Q does not exceed rkHk(M ;Z), and a collection of orientations on k-faces of K such that for any nonadjacent k- faces σ, τ the element Aσ,τ equals to the algebraic intersection of fσ and fτ.

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