totalspace

Total space is a term used in topology to denote the space E in a fiber bundle p: E → B, where B is the base space and F is a typical fiber. E is called the total space, and p is a continuous surjection whose preimage p−1(b) for each b in B is homeomorphic to F. The total space encodes the entire collection of fibers over every point of the base.

In a bundle, the total space often has a locally product structure: over each open set U⊆B, p−1(U) is homeomorphic to U × F in a way that respects projection to U. Such local trivializations imply that E is a fibration, though it need not be globally a product B × F.

Examples help illustrate the concept. The tangent bundle TM of a smooth manifold M has total space consisting of all tangent vectors at all points of M. A trivial bundle, such as M × F, has total space equal to the Cartesian product. Nontrivial examples include the Möbius strip, which is a nontrivial line bundle over the circle, and the Hopf fibration S^3 → S^2 with fiber S^1, a principal bundle whose total space is S^3.

Sections are maps s: B → E with p ∘ s = id_B, representing choice of a fiber element above each base point. The total space interacts with various areas of topology and geometry through constructions like the long exact sequence of homotopy groups of a fibration and through characteristic classes that classify how E twists over B.

In summary, the total space is central to the study of fiber bundles, vector bundles, and differentiable manifolds, capturing the aggregate structure of all fibers over the base.