This section on fiber bundles is in disrepair for the following problems. This section started when I had a bone to pick with a comment by Penrose about his tangent bundle groups.

I want to be able to say that the classic Möbius strip has ‘the cyclic group of 2 elements’ as the group of the fiber bundle.

In support of that aim I want to be able to say that a topological line segment has the same symmetry group in some sense.

In order to find such a sense I think I need to find an equivalence relation between homomorphisms of the segment with itself, that relates maps that don’t swap the ends of the line segment.

Continuously deformable homomorphisms are not a valid notion for lack of a topology on some space of such homomorphisms.

The definition of fiber bundle that I remember requires: I was appalled when I discovered that explicitly providing such φ’s for a two element covering of b did not provide information that the Möbius strip was indeed twisted.

Only then did I notice that there are no equivalence relations proposed between fiber bundles. Groups may be isomorphic which allows us the talk of the group of three elements (up to isomorphism). Perhaps there is no way of talking about ‘the abstract Klein bottle’ as we talk about the abstract 3 group or the ‘three sphere’.

I noticed that Steenrod calls his initial ‘definition’ of fiber bundles preliminary and the replaces that definition on page 9 with a definition that relies on coordinate bundles. The punch line is that fiber bundles are probably more complex than I recall.