Simplicial complexes embedded on a compact manifold

Question

Every finite graph can be embedded on some compact surface of sufficient genus such that no two edges cross. If $S$ is a finite simplicial complex of dimension $n$, can $S$ be embedded in some compact Riemann Manifold of dimension $n+1$ such that no two simplices intersect?

Answer 1

Here is the complete story:

1. Let $X$ be a (finite, although, it is not really necessary) $k$-dimensional simplicial complex. Then $X$ always embeds in some (smooth) manifold $M$ of dimension $2k$. To sketch a proof of this: $X$ can be immersed (one needs to define what this means) in $R^{2k}$ (this is a form of Whitney's immersion theorem). Then pull-back of a regular neighborhood of the image of this immersion yields the desired manifold $M$.

2. For each $k$, there exists a finite $k$-dimensional simplicial complex which does not embed in any manifold of dimension $2k-1$. A specific example is given by the $k$-fold product of copies of the "tripod" $T$, which is the unique finite tree with four vertices, one of which (the "center") has valence 3. Let $v\in T$ be this center. The key to the proof is that the link $L$ of a vertex of $X$ corresponding to $(v,...,v)$ is isomorphic to the $k$-fold join of the copies of the set on 3 elements. This complex is known not to admit embeddings in $S^{2k-2}$ (this result is due to van Kampen). However, if $X$ embeds in a manifold of dimension $n$, the links of vertices of $X$ embed in the sphere $S^{n-1}$. Hence, $X$ does not embed in any $2k-1$-dimensional manifold.

The simplest example that you can work out by hand is the product $X=T\times T$. The link of $(v,v)$ in $X$ is the complete bipartite graph $K_{3,3}$. The latter does not embed in $S^2$ (this is an application of the Jordan curve theorem), hence, $X$ does not embed in any 3-dimensional manifold.

Shapiro, Arnold, Obstructions to the imbedding of a complex in a euclidean space. I: The first obstruction, Ann. Math. (2) 66, 256-269 (1957). ZBL0085.37701.

Van Kampen, E. R., Komplexe in euklidischen Räumen, Abhandlungen Hamburg 9, 72-78 (1932). ZBL58.0615.02.

Wu, Wen-tsün, A theory of imbedding, immersion, and isotopy of polytopes in a Euclidean space, Peking: Science Press. xv, 291 pp. (1965). ZBL0177.26402.