I am wondering if there is any benefit for a rotating wheel space station to be based on mobius strip layout such as more rooms for growing population maybe because both side of the strips are populated. Else why would it be a bad idea for the design of space station that contains Euler characteristic?
Bad idea, it just puts a twist, complcating your up/down vector, meaning you'd have weird steps and jagged bits between rooms. You can build on both 'sides' of a ring too. The advantage with a ring is that every room experiences the same gravitational effect, and the difference in the up direction (to the center) between rooms is a constant. This makes a uniform, curved corridor possible. But with a moebius strip design the twist occurs near the center (I'm imagining a spinning figure 8 rotating around the crosover in the middle. If this is not your design, you should provide a sketch). This means that rooms there will experience little to no g (good for cargo handling, poor for living) and rooms further out have progressively more gravity. If you walk along the moebius corridor you'll get lighter and eventually depart from the floor. Inconvenient. The twist seems like it just adds unnecessary structural complications, with no gain (a ring design can have shafts to the zero-g center)
Where is the gravity coming from?
If you are using "centrifugal force" to fake gravity, then this is a horrible idea. Because everything on the wrong side of the Möbius step would be tossed into n pet space the rotating wheel.
This is if I understand your concept correctly.