Euler Characteristic of MengerMesh

Question

When I ask Mathematica to give me the Euler characteristic of MengerMesh[1, 3], it returns 16. It seems to me the 1-step Menger sponge has genus 5 and should therefore have Euler characteristic -8. For MengerMesh[2, 3], I'm again off by a factor of -2, as Mathematica returns 320 while I predict -160 based on its genus, which is 81. For MengerMesh[0, 3], me and Mathematica agree.

Can anyone please explain to me what I'm missing here?

Answer 1

MengerMesh returns a full dimensional mesh, e.g. in 3D a solid, not a surface.

mr = MengerMesh[2, 3];

RegionDimension[mr]

3

MeshCellCount[mr]

{896, 2304, 1728, 400}

We can instead find the characteristic of its boundary:

EulerCharacteristic[mr]

320

EulerCharacteristic[RegionBoundary[mr]]

-160