In the plane, Euler's Polyhedral formula tells us that $V - E + F = \chi$, where for graph embeddings we have that $\chi = 1$. Alternatively, we can think of a graph embedding as a simplicial $1$-complex embedded in the plane.
My question is about if there exists a straightforward generalization of the value of the Euler characteristic to higher dimensions. In particular, for a graph embedded in 3-space, or equivalently a simplicial 2-complex or simplicial 3-complex embedded in $\mathbb{E}^3$, is there a simple expression that defines the Euler characteristic as a constant value, as there is for the plane? (namely $\chi =1$).
Thank you!
EDIT: I am aware that there is a general formula for $\chi$ if you know the number of faces of a given dimension, I am looking to determine those faces by knowing what the Euler characteristic is from external information.