I have learned several things in the last week of tying down fiber bundle concepts.
For groups we often speak of the group of one element, or the group of two elements.
This manner of speaking is strictly illegitimate but we could note that all groups of two elements are isomorphic and which two element group is irrelevant to the discussion at hand.
Not all 4 element groups are isomorphic to each other.
I find it hard to think of specific groups rather than equivalence classes when I ask about the group of a bundle but I suppose we can safely speak of a group of homeomorphisms of the fiber space with itself.
While group theory enables a notion of an equivalence class of groups, there is no common definition of equivalence between fiber bundles and thus no notion of an abstract fiber bundle.
One of the requirements of a definition of the group of a bundle would supposedly require that equivalent fiber bundles produce isomorphic groups; but we can’t even state that requirement!
I paraphrase part of the Wikipedia definition:
We require that for any point of the base space there is a neighborhood such that the product space of the neighborhood and fiber space is homeomorphic to [the inverse image of the neighborhood].
I see a fundamental problem with such definitions.
Must you specify these homomorphisms to specify a fiber bundle?
Such specs are omitted in all examples in the literature that I recall.
Whether Steenrod’s torus and twisted torus are different (whatever that means) depends on these homomorphisms.
Steenrod’s informal distinctions are clear enough to be able to offer such homomorphisms.
Steenrod is silent on the group of the twisted torus; there is thus no clue here.
If the specs for a particular fiber bundle must include specs for the homomorphisms then we have lost the desirable property that many different sets of neighborhoods may apply to describing a given bundle.
This would be ameliorated with an equivalence relation between two sets of neighborhoods and associated homomorphisms.
It is such hair that Steenrod introduces to explain coordinate bundles.
Thus the ‘examples’ such as the Möbeus strip given in about every development don’t really fully specify a fiber bundle.
Relying on this definition and taking notation from there, I attempt the simplest specification of the Möbeus fiber bundle.
We use the reals [0, 1] as coordinates on the circle, identifying 0 and 1.
The base space is the circle, B = [0, 1].
The fiber space is [0, 1] ⊂ ℝ
Two neighborhoods cover the base: {(.8, .2), (.1, .9)}.
The 1st homomorphism is (x, y) → (x, y) and the 2nd is
I think that I am foundering on the variety of definitions of fiber bundle.
Must a bundle have an Atlas?