Is an X-shaped Pipe a Double Torus? According to answers in this question, a X-shaped pipe should have a Euler characteristic of -2.
Looking up to examples at Wikipedia, we find that the Double torus also has an Euler characteristic of -2.
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Can both things be transformed into each other without cutting? I don't see how...
 A: No. Compact surfaces with boundary are classified by their Euler characteristic, orientability and number of boundary components. As the X-tube has 4 boundary components, and the 2-torus has 0 boundary components, they cannot be homotopy equivalent (and hence cannot be homeomorphic). You need all three characteristics to coincide in order to be homotopy equivalent.
A: It depends on your definition of an X-shaped pipe. Take, for example, a regular cylindrical pipe. You can define it to be a cylinder, or you can define it to be a torus (which, of course, can be deformed to look just like a regular cylindrical pipe).
If you define an X-shaped pipe by gluing together the boundaries of four cylinders, then Dan Rust's answer has got you covered.
However, you might think of an X-shaped pipe as a sphere whose surface has four tunnel entrances, with all tunnels connecting inside the sphere (that is, a sphere with an X-shaped hole). This can be deformed to look just like an X-shaped pipe. In this case, the X-shaped pipe is actually homotopic to a triple torus (like a torus, but with three holes).
To see this, try to imagine what happens when you drastically widen one of the tunnel entrances.
Hope this helps after nine months!
