Question 1: A fiber bundle consists of the data (E, B, π, F), where E, B, and F are ________ and π : E → B is a continuous surjection satisfying a local triviality condition outlined below.  

Question 2: In the category of ________, fiber bundles arise naturally as submersions of one manifold to another.  

Question 3: The corresponding nontwisted (trivial) bundle is the ________, S^{1} × S^{1}.  

Question 4: For any x in B, the ________ π^{−1}({x}) is homeomorphic to F and is called the fiber over x.  

Question 5: From any vector bundle, one can construct the ________ of bases which is a principal bundle (see below).  

Question 6: Examples of nontrivial fiber bundles, that is, bundles twisted in the large, include the Möbius strip and ________, as well as nontrivial covering spaces.  

Question 7: In ________, and particularly topology, a fiber bundle (or fibre bundle) is intuitively a space E which locally "looks" like a product space B × F, but globally may have a different topological structure.  

Question 8: A similar nontrivial bundle is the ________ which can be viewed as a "twisted" circle bundle over another circle.  

Question 9: In the case n=1 the sphere bundle is called a circle bundle and the Euler class is equal to the first ________, which characterizes the topology of the bundle completely.  

Question 10: Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in ________ and differential topology, as do principal bundles.  

