What are the Euler characteristics of fractals?

Question

I have been led to believe that all shapes, surfaces and polyhedra have Euler characteristics. Does this apply to fractals as well? If so, is this linked in any way to fractal dimensions? How would such an Euler characteristic be calculated? If fractals do not have an Euler characteristic, why is that the case?

Answer 1

The usual way to define an Euler characteristic for a topological space is $$ \chi = b_0 - b_1 + b_2 - \ldots$$ where $b_n$ is the $n$'th Betti number. For a subspace of $\mathbb R^n$, $b_k = 0$ for $k > n$. However, some of the Betti numbers could be $\infty$.

Answer 2

This is too long for a comment.

I am a bit confused by the question, as I imagine fractals as being more of a metric property, and the Euler characteristic as a topological one. For instance, the von Koch snowflake is homeomorphic to the circle, hence has Euler characteristic $0$, any rough curve with Hausdorff dimension $>1$ homeomorphic to a segment will have Euler characteristc $1$ (for instance the Osgood curve, of dimension $2$), the Mandelbrot set has Euler characteristic $1$ (it is path connected and all its homotopy group vanish)... I understand that some examples might be more topological in nature (for instance the Sierpinski triangle), but considering the examples above, I doubt there might be a direct link to the Hausdorff dimension.