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Feb 22, 2017 at 14:22 comment added Richard Stanley They would have the same Euler characteristic if $Y$ were the face poset of a triangulation of the torus, but we have adjoined an extra element $t$.
Feb 22, 2017 at 7:21 comment added მამუკა ჯიბლაძე Sorry for this question, but I got confused. Although I have checked what you say, and everything is correct, I still don't understand one thing. Geometrically, $Y\times Z$ must be the cylinder of $Y$, so be homotopy equivalent to $Y$, hence have the same Euler characteristic. How come they are different??
Feb 21, 2017 at 21:45 vote accept Nathaniel Bottman
Feb 21, 2017 at 21:42 history answered Richard Stanley CC BY-SA 3.0