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I keep getting $\chi=2$ for the solid in the picture. It's a prism with a hole joining two opposite sides. I remember reading that $\chi=0$ for such solids.

Help me find my error. I'd appreciate if someone could just point out which of V, E, F is wrong.

prism with a hole

$\text{Vertices} = 4\times 2\times 2=16$

$\text{Edges}=4\times2\times2+4\times 2=24$

$\text{Faces}=4\times2+2=10$

$\chi = V-E+F=16-24+10=2$.

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    See also the [duplicate question](http://math.stackexchange.com/questions/93208/the-euler-characteristic-a-cube-with-holes/93313#93313).2013-01-05

1 Answers 1

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The trouble is that not all your "faces" are simply-connected.

Draw segments connecting the corner points of each square to those of the inner square. Then $V=16$, $E=32$, and $F=16$, so $\chi = 0$.

Hope this helps!

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    "Simply connected" means that any two points can be joined by a path _and_ every _closed_ path in the space can be _continuously_ deformed into a point without leaving the space. The former of these conditions means that it is "path connected", which is a weaker condition. The punctured plane is path connected but not simply connected. Think of it as "connected in a particularly simple manner".2011-12-01