Fiber bundles #

A topological fiber bundle with fiber F over a base B is a space projecting on B for which the fibers are all homeomorphic to F, such that the local situation around each point is a direct product. We define a predicate is_topological_fiber_bundle F p saying that p : Z → B is a topological fiber bundle with fiber F.

It is in general nontrivial to construct a fiber bundle. A way is to start from the knowledge of how changes of local trivializations act on the fiber. From this, one can construct the total space of the bundle and its topology by a suitable gluing construction. The main content of this file is an implementation of this construction: starting from an object of type topological_fiber_bundle_core registering the trivialization changes, one gets the corresponding fiber bundle and projection.

Main definitions #

Basic definitions #

• bundle_trivialization F p : structure extending local homeomorphisms, defining a local trivialization of a topological space Z with projection p and fiber F.

• is_topological_fiber_bundle F p : Prop saying that the map p between topological spaces is a fiber bundle with fiber F.

• is_trivial_topological_fiber_bundle F p : Prop saying that the map p : Z → B between topological spaces is a trivial topological fiber bundle, i.e., there exists a homeomorphism h : Z ≃ₜ B × F such that proj x = (h x).1.

Operations on bundles #

We provide the following operations on bundle_trivializations.

• bundle_trivialization.comap: given a local trivialization e of a fiber bundle p : Z → B, a continuous map f : B' → B and a point b' : B' such that f b' ∈ e.base_set, e.comap f hf b' hb' is a trivialization of the pullback bundle. The pullback bundle (a.k.a., the induced bundle) has total space {(x, y) : B' × Z | f x = p y}, and is given by λ ⟨(x, y), h⟩, x.

• is_topological_fiber_bundle.comap: if p : Z → B is a topological fiber bundle, then its pullback along a continuous map f : B' → B is a topological fiber bundle as well.

• bundle_trivialization.comp_homeomorph: given a local trivialization e of a fiber bundle p : Z → B and a homeomorphism h : Z' ≃ₜ Z, returns a local trivialization of the fiber bundle p ∘ h.

• is_topological_fiber_bundle.comp_homeomorph: if p : Z → B is a topological fiber bundle and h : Z' ≃ₜ Z is a homeomorphism, then p ∘ h : Z' → B is a topological fiber bundle with the same fiber.

Construction of a bundle from trivializations #

• bundle.total_space E is a type synonym for Σ (x : B), E x, that we can endow with a suitable topology.
• topological_fiber_bundle_core ι B F : structure registering how changes of coordinates act on the fiber F above open subsets of B, where local trivializations are indexed by ι.

Let Z : topological_fiber_bundle_core ι B F. Then we define

• Z.fiber x : the fiber above x, homeomorphic to F (and defeq to F as a type).
• Z.total_space : the total space of Z, defined as a Type as Σ (b : B), F, but with a twisted topology coming from the fiber bundle structure. It is (reducibly) the same as bundle.total_space Z.fiber.
• Z.proj : projection from Z.total_space to B. It is continuous.
• Z.local_triv i: for i : ι, a local homeomorphism from Z.total_space to B × F, that realizes a trivialization above the set Z.base_set i, which is an open set in B.

Implementation notes #

A topological fiber bundle with fiber F over a base B is a family of spaces isomorphic to F, indexed by B, which is locally trivial in the following sense: there is a covering of B by open sets such that, on each such open set s, the bundle is isomorphic to s × F.

To construct a fiber bundle formally, the main data is what happens when one changes trivializations from s × F to s' × F on s ∩ s': one should get a family of homeomorphisms of F, depending continuously on the base point, satisfying basic compatibility conditions (cocycle property). Useful classes of bundles can then be specified by requiring that these homeomorphisms of F belong to some subgroup, preserving some structure (the "structure group of the bundle"): then these structures are inherited by the fibers of the bundle.

Given such trivialization change data (encoded below in a structure called topological_fiber_bundle_core), one can construct the fiber bundle. The intrinsic canonical mathematical construction is the following. The fiber above x is the disjoint union of F over all trivializations, modulo the gluing identifications: one gets a fiber which is isomorphic to F, but non-canonically (each choice of one of the trivializations around x gives such an isomorphism). Given a trivialization over a set s, one gets an isomorphism between s × F and proj^{-1} s, by using the identification corresponding to this trivialization. One chooses the topology on the bundle that makes all of these into homeomorphisms.

For the practical implementation, it turns out to be more convenient to avoid completely the gluing and quotienting construction above, and to declare above each x that the fiber is F, but thinking that it corresponds to the F coming from the choice of one trivialization around x. This has several practical advantages:

• without any work, one gets a topological space structure on the fiber. And if F has more structure it is inherited for free by the fiber.
• In the case of the tangent bundle of manifolds, this implies that on vector spaces the derivative (from F to F) and the manifold derivative (from tangent_space I x to tangent_space I' (f x)) are equal.

A drawback is that some silly constructions will typecheck: in the case of the tangent bundle, one can add two vectors in different tangent spaces (as they both are elements of F from the point of view of Lean). To solve this, one could mark the tangent space as irreducible, but then one would lose the identification of the tangent space to F with F. There is however a big advantage of this situation: even if Lean can not check that two basepoints are defeq, it will accept the fact that the tangent spaces are the same. For instance, if two maps f and g are locally inverse to each other, one can express that the composition of their derivatives is the identity of tangent_space I x. One could fear issues as this composition goes from tangent_space I x to tangent_space I (g (f x)) (which should be the same, but should not be obvious to Lean as it does not know that g (f x) = x). As these types are the same to Lean (equal to F), there are in fact no dependent type difficulties here!

For this construction of a fiber bundle from a topological_fiber_bundle_core, we should thus choose for each x one specific trivialization around it. We include this choice in the definition of the topological_fiber_bundle_core, as it makes some constructions more functorial and it is a nice way to say that the trivializations cover the whole space B.

With this definition, the type of the fiber bundle space constructed from the core data is just Σ (b : B), F, but the topology is not the product one, in general.

We also take the indexing type (indexing all the trivializations) as a parameter to the fiber bundle core: it could always be taken as a subtype of all the maps from open subsets of B to continuous maps of F, but in practice it will sometimes be something else. For instance, on a manifold, one will use the set of charts as a good parameterization for the trivializations of the tangent bundle. Or for the pullback of a topological_fiber_bundle_core, the indexing type will be the same as for the initial bundle.

Tags #

Fiber bundle, topological bundle, vector bundle, local trivialization, structure group

General definition of topological fiber bundles #

Constructing topological fiber bundles #