Before revision: If K is a simplicial k-complex , the standard van Kampen obstructions tells you whether K can be embedded into \mathbb R^{2k or not (provided k\neq 2).
After revision: Let K be a simplicial k-complex , the standard van Kampen obstructions tells you whether K can be embedded into \mathbb R^{2k or not (provided k\neq 2).