According to Wikipedia,

For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected, χ = 2.

Is it really necessary to specify here, that the faces are also simply connected?

I can imagine a simply connected body with a not simply connected face (for example a cylinder), but can't imagine a simply connected polyhedron with a not simply connected face. Does such thing exist at all?

  • Sure, fatten up a can (cylinder with one end capped off) and adjust it to be polyhedral. – aes May 3 '15 at 6:59
  • 2
    Or stick a small cube on top of a big one. – Gerry Myerson May 3 '15 at 7:03
up vote 1 down vote accepted

Consider the following construction. Take a solid tetrahedron, and attach a very small tetrahedron to the middle of one of the faces. The boundary of this object is simply connected (it's a sphere, topologically), but one of the faces looks like a triangle with a small triangle removed from its center, which is not simply connected (it's an annulus, topologically).

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    And the resulting shape has the number of vertices of two tetrahedra (8), the number of edges of two tetrahedra (12), and one less than the number of faces of two tetrahedra (7), and so V - E + F = 3. – Alex Zorn May 3 '15 at 7:43

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