mathlib documentation

topology.fiber_bundle.basic

Fiber bundles #

Mathematically, 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.

In our formalism, a fiber bundle is by definition the type bundle.total_space E where E : B → Type* is a function associating to x : B the fiber over x. This type bundle.total_space E is just a type synonym for Σ (x : B), E x, with the interest that one can put another topology than on Σ (x : B), E x which has the disjoint union topology.

To have a fiber bundle structure on bundle.total_space E, one should additionally have the following data:

If all these conditions are satisfied, we register the typeclass fiber_bundle F E.

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 fiber_bundle_core registering the trivialization changes, one gets the corresponding fiber bundle and projection.

Similarly we implement the object fiber_prebundle which allows to define a topological fiber bundle from trivializations given as local equivalences with minimum additional properties.

Main definitions #

Basic definitions #

Construction of a bundle from trivializations #

Let Z : fiber_bundle_core ι B F. Then we define

Implementation notes #

Data vs mixins #

For both fiber and vector bundles, one faces a choice: should the definition state the existence of local trivializations (a propositional typeclass), or specify a fixed atlas of trivializations (a typeclass containing data)?

In their initial mathlib implementations, both fiber and vector bundles were defined propositionally. For vector bundles, this turns out to be mathematically wrong: in infinite dimension, the transition function between two trivializations is not automatically continuous as a map from the base B to the endomorphisms F →L[R] F of the fibre (considered with the operator-norm topology), and so the definition needs to be modified by restricting consideration to a family of trivializations (constituting the data) which are all mutually-compatible in this sense. The PRs #13052 and #13175 implemented this change.

There is still the choice about whether to hold this data at the level of fiber bundles or of vector bundles. As of PR #17505, the data is all held in fiber_bundle, with vector_bundle a (propositional) mixin stating fibrewise-linearity.

This allows bundles to carry instances of typeclasses in which the scalar field, R, does not appear as a parameter. Notably, we would like a vector bundle over R with fibre F over base B to be a charted_space (B × F), with the trivializations providing the charts. This would be a dangerous instance for typeclass inference, because R does not appear as a parameter in charted_space (B × F). But if the data of the trivializations is held in fiber_bundle, then a fibre bundle with fibre F over base B can be a charted_space (B × F), and this is safe for typeclass inference.

We expect that this choice of definition will also streamline constructions of fibre bundles with similar underlying structure (e.g., the same bundle being both a real and complex vector bundle).

Core construction #

A 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 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:

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 fiber_bundle_core, we should thus choose for each x one specific trivialization around it. We include this choice in the definition of the 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 fiber_bundle_core, the indexing type will be the same as for the initial bundle.

Tags #

Fiber bundle, topological bundle, structure group

General definition of fiber bundles #

@[class]
structure fiber_bundle {B : Type u_2} (F : Type u_3) [topological_space B] [topological_space F] (E : B Type u_4) [topological_space (bundle.total_space E)] [Π (b : B), topological_space (E b)] :
Type (max u_2 u_3 u_4)

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.

Instances of this typeclass
Instances of other typeclasses for fiber_bundle
  • fiber_bundle.has_sizeof_inst
@[class]

Given a type E equipped with a fiber bundle structure, this is a Prop typeclass for trivializations of E, expressing that a trivialization is in the designated atlas for the bundle. This is needed because lemmas about the linearity of trivializations or the continuity (as functions to F →L[R] F, where F is the model fibre) of the transition functions are only expected to hold for trivializations in the designated atlas.

Instances of this typeclass
@[continuity]

The projection from a fiber bundle to its base is continuous.

The projection from a fiber bundle to its base is an open map.

The projection from a fiber bundle with a nonempty fiber to its base is a surjective map.

The projection from a fiber bundle with a nonempty fiber to its base is a quotient map.

If E is a fiber bundle over a conditionally complete linear order, then it is trivial over any closed interval.

Core construction for constructing fiber bundles #

@[nolint]
structure fiber_bundle_core (ι : Type u_4) (B : Type u_5) [topological_space B] (F : Type u_6) [topological_space F] :
Type (max u_4 u_5 u_6)

Core data defining a locally trivial bundle with fiber F over a topological space B. Note that "bundle" is used in its mathematical sense. This is the (computer science) bundled version, i.e., all the relevant data is contained in the following structure. A family of local trivializations is indexed by a type ι, on open subsets base_set i for each i : ι. Trivialization changes from i to j are given by continuous maps coord_change i j from base_set i ∩ base_set j to the set of homeomorphisms of F, but we express them as maps B → F → F and require continuity on (base_set i ∩ base_set j) × F to avoid the topology on the space of continuous maps on F.

Instances for fiber_bundle_core
@[nolint]
def fiber_bundle_core.index {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) :
Type u_1

The index set of a fiber bundle core, as a convenience function for dot notation

Equations
@[nolint, reducible]
def fiber_bundle_core.base {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) :
Type u_2

The base space of a fiber bundle core, as a convenience function for dot notation

Equations
@[nolint]
def fiber_bundle_core.fiber {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (x : B) :
Type u_3

The fiber of a fiber bundle core, as a convenience function for dot notation and typeclass inference

Equations
Instances for fiber_bundle_core.fiber
@[protected, instance]
Equations
@[nolint, reducible]
def fiber_bundle_core.total_space {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) :
Type (max u_2 u_3)

The total space of the fiber bundle, as a convenience function for dot notation. It is by definition equal to bundle.total_space Z.fiber, a.k.a. Σ x, Z.fiber x but with a different name for typeclass inference.

Equations
@[simp, reducible]
def fiber_bundle_core.proj {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) :

The projection from the total space of a fiber bundle core, on its base.

Equations
def fiber_bundle_core.triv_change {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (i j : ι) :
local_homeomorph (B × F) (B × F)

Local homeomorphism version of the trivialization change.

Equations
@[simp]
theorem fiber_bundle_core.mem_triv_change_source {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (i j : ι) (p : B × F) :
def fiber_bundle_core.local_triv_as_local_equiv {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (i : ι) :

Associate to a trivialization index i : ι the corresponding trivialization, i.e., a bijection between proj ⁻¹ (base_set i) and base_set i × F. As the fiber above x is F but read in the chart with index index_at x, the trivialization in the fiber above x is by definition the coordinate change from i to index_at x, so it depends on x. The local trivialization will ultimately be a local homeomorphism. For now, we only introduce the local equiv version, denoted with a prime. In further developments, avoid this auxiliary version, and use Z.local_triv instead.

Equations

The composition of two local trivializations is the trivialization change Z.triv_change i j.

@[protected, instance]

Topological structure on the total space of a fiber bundle created from core, designed so that all the local trivialization are continuous.

Equations
def fiber_bundle_core.local_triv {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (i : ι) :

Extended version of the local trivialization of a fiber bundle constructed from core, registering additionally in its type that it is a local bundle trivialization.

Equations

Preferred local trivialization of a fiber bundle constructed from core, at a given point, as a bundle trivialization

Equations
@[simp]
theorem fiber_bundle_core.local_triv_at_def {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (b : B) :
theorem fiber_bundle_core.continuous_const_section {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (v : F) (h : (i j : ι) (x : B), x Z.base_set i Z.base_set j Z.coord_change i j x v = v) :
continuous (show B Z.total_space, from λ (x : B), x, v⟩)

If an element of F is invariant under all coordinate changes, then one can define a corresponding section of the fiber bundle, which is continuous. This applies in particular to the zero section of a vector bundle. Another example (not yet defined) would be the identity section of the endomorphism bundle of a vector bundle.

@[simp]
theorem fiber_bundle_core.base_set_at {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (i : ι) :
@[simp]
theorem fiber_bundle_core.local_triv_apply {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (i : ι) (p : Z.total_space) :
(Z.local_triv i) p = (p.fst, Z.coord_change (Z.index_at p.fst) i p.fst p.snd)
@[simp]
theorem fiber_bundle_core.local_triv_at_apply {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (p : Z.total_space) :
(Z.local_triv_at p.fst) p = (p.fst, p.snd)
@[simp]
theorem fiber_bundle_core.local_triv_at_apply_mk {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (b : B) (a : F) :
(Z.local_triv_at b) b, a⟩ = (b, a)
@[simp]
theorem fiber_bundle_core.mem_source_at {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (b : B) (a : F) :
@[simp]
@[simp]
theorem fiber_bundle_core.local_triv_symm_apply {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (i : ι) (p : B × F) :
@[simp]
theorem fiber_bundle_core.mem_local_triv_at_base_set {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) (b : B) :
@[continuity]

The inclusion of a fiber into the total space is a continuous map.

@[protected, instance]

A fiber bundle constructed from core is indeed a fiber bundle.

Equations
theorem fiber_bundle_core.continuous_proj {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) :

The projection on the base of a fiber bundle created from core is continuous

theorem fiber_bundle_core.is_open_map_proj {ι : Type u_1} {B : Type u_2} {F : Type u_3} [topological_space B] [topological_space F] (Z : fiber_bundle_core ι B F) :

The projection on the base of a fiber bundle created from core is an open map

Prebundle construction for constructing fiber bundles #

@[nolint]
structure fiber_prebundle {B : Type u_2} (F : Type u_3) (E : B Type u_4) [topological_space B] [topological_space F] :
Type (max u_2 u_3 u_4)

This structure permits to define a fiber bundle when trivializations are given as local equivalences but there is not yet a topology on the total space. The total space is hence given a topology in such a way that there is a fiber bundle structure for which the local equivalences are also local homeomorphism and hence local trivializations.

Instances for fiber_prebundle
  • fiber_prebundle.has_sizeof_inst
def fiber_prebundle.fiber_topology {B : Type u_2} {F : Type u_3} {E : B Type u_4} [topological_space B] [topological_space F] (a : fiber_prebundle F E) (b : B) :

Topology on the fibers E b induced by the map E b → E.total_space.

Equations
@[continuity]
@[continuity]

Make a fiber_bundle from a fiber_prebundle. Concretely this means that, given a fiber_prebundle structure for a sigma-type E -- which consists of a number of "pretrivializations" identifying parts of E with product spaces U × F -- one establishes that for the topology constructed on the sigma-type using fiber_prebundle.total_space_topology, these "pretrivializations" are actually "trivializations" (i.e., homeomorphisms with respect to the constructed topology).

Equations

For a fiber bundle E over B constructed using the fiber_prebundle mechanism, continuity of a function total_space E → X on an open set s can be checked by precomposing at each point with the pretrivialization used for the construction at that point.