Möbius strip with multiple twists The Möbius strip is obtained from a closed band by fixing one end, taking a half twist of the other end and gluing the ends together. 
I wonder what happens, if one doesn't just take a half twist but multiple (half) twists of the one end. 
Questions: 


*

*Is the resulting space homeomorphic to the classical Möbius strip ? 

*If they are not homeomorphic, are they homotopy equivalent ? 

 A: Any space constructed in this fashion deformation retracts to the core circle, so they are all homotopy equivalent to $S^1$.
I think the study of homeomorphism classes of these objects is an interesting exercise, so I'll leave it as one for you with the following hints as a guide:


*

*Homeomorphism doesn't depend on any embeddings; it's an intrinsic property. So "number of twists," which is definitely an extrinsic property (in this case, it comes from the relationship between the space and its immersion in $\mathbb{R}^3$) isn't going to be as helpful as you might want in classifying up to homeomorphism.

*Instead, it may be helpful to think of these spaces intrinsically, in particular as quotients of the square $[0,1]\times [0,1]$. Do this by thinking about how you would "glue" the square to produce these multiply-twisted strips.

*Notice that the square does not "know" how many times you've twisted it. All it knows is how you glue it. Can you formalize this into a proof about homeomorphisms between differently-twisted strips?

*Based on the result of (3), can you classify all such twisted bands into homeomorphism types?
