Introduction to the h -principle

1 Preliminaries

1.1 Jets spaces

Let π \! :M → X be a locally trivial fibration. A local section of π is a (smooth) map s \! :\operatorname{Op}\{ m_0\} → X such that π ∘ s = \operatorname{Id} .

Let s and s’ be two local section defined near m_0 . We say that s and s’ are tangent up to order r at m_0 if in a local trivialization where π becomes ℝ^ n × ℝ^ p → ℝ^ n , so that s(m) = (m, f(m)) and s’(m) = (m, f’(m)) , f and f’ have the same Taylor expansion at m_0 up to order r .

  1. Prove that this condition is independent of the choice of local trivialization, hence well-defined.

This condition defines an equivalence relation on the set of local sections near m_0 . The equivalence class of s is denoted by j^ r s(m_0) . The set of all equivalence classes at all points m_0 is denoted by X^{(r)} , and its obvious projection to M is denoted by π^ r . For any section s \! :U ⊂ M → X , m ↦ j^ r s(m) is a section of π^ r . We equip X^{(r)} with the quotient topology coming from the C^ r topology (any variant is fine since we only care about neighborhoods of points). The next three questions endow X^{(r)} with a well defined and functorial smooth structure.

  1. Prove that, for every (local) section s defined near some m_0 , there is a neighborhood U of m_0 and a local homeomorphism Φ \! :U × ℝ^ N ↪ X^{(r)} such that j^ rs = Φ_0 and each Φ_ p is a holonomic section.

  2. Prove that there is a unique smooth structure on X^{(r)} such that π^ r is smooth and every smooth family of sections s_ p (i.e. s \! :U × ℝ^ k → X is smooth and each s_ p is a section of π ) (m, p) ↦ j^ r s_ p(m) is smooth.

  3. For every local diffeomorphism φ of M and every s we set φ_*(j^ r(m)) := j^ r(s ∘ φ)(m) . Prove that φ_* is a local diffeomorphism of X^{(r)} .

1.2 Polyhedra and triangulations

An affine cell σ in ℝ^ N is the convex hull of a finite set of points. Its dimension \dim σ is the dimension of its affine span ⟨σ⟩ . Its interior \operatorname{Int}σ is its topological interior inside ⟨σ⟩ . Its boundary ∂σ is σ ∖ \operatorname{Int}σ .

  1. Prove that ∂σ is a finite union of cells of dimension \dim (σ) - 1 (they are called faces of σ ).

A cell σ which is the convex hull of exactly \dim (σ) + 1 points is called a simplex. A cube is a cell which is an affine transform of [0, 1]^ d × \{ 0\} ^{N-d} .

A cell complex is a countable collection K of cells σ_ i whose interiors are pairwise disjoint, and whose faces are in K , and such that every point of ℝ^ N is in finitely many cells. The union of these cells, equipped with the induced topology, is denoted by |K| .

A cell complex K’ is a subdivision of K if |K’| = |K| and every cell of K’ is contained in a cell of K . The l -skeleton of K is the collection of all cells in K with dimension at most l . The barycentric subdivision of a cell σ is the subdivision obtained by adding as vertices the barycenters of σ and all its faces, and then adding all simplices of dimension at most \dim σ determined by the old and new vertices. The barycentric subdivision of a cell complex K is obtained by barycentric subdivisions of all its cells.

  1. Explain why the above definition of a barycentric subdivision of a cell complex makes sense.

  2. Prove that every cell complex has a subdivision whose cells are all simplices (resp. cubes).

A map f from |K| to a manifold M is called piecewise smooth if, for every cell σ in K , there exists f_σ \! :\operatorname{Op}σ ⊂ ⟨σ⟩ → M which is smooth and restricts to f on σ .

A smooth polyhedron in a smooth manifold M is f(|K|) where K is a cell complex, and f is a topological embedding that is piecewise smooth. A cell decomposition of M is (K, f) where f is a piecewise smooth homeomorphism from |K| to M . It is called a smooth triangulation (resp. cubulation) if all cells are simplices (resp. cubes).

Whitehead proved that every smooth manifold has a smooth triangulation (hence also a cubulation). There is also a uniqueness result up to suitable equivalence but stating it requires more definitions, and we won’t need it.

A manifold is called closed if it’s compact and has empty boundary. An open manifold is a manifold having no closed connected component. The goal of the next series of questions is to prove that, in every open manifold M , there is a polyhedron A with \dim (A) < \dim (M) and, for every neighborhood U of A , an isotopy of embeddings φ_ t \! :M ↪ M such that φ_0 = \operatorname{Id} , φ_1(M) ⊂ U and φ_ t|_{A} = \operatorname{Id}_ A for all t .

The beginning happens purely on the combinatorial topology side. A graph is a cell complex whose cells have dimension zero or one. A graph isomorphic to \{ [i, i + 1], i ∈ ℕ \} is called a ray. A special tree is a graph which is the union of a ray and finitely many disjoint tree having one vertex in the ray.

  1. Let K be a triangulation of a connected non-compact n -manifold. Let G be the one-skeleton of the dual triangulation, seen as a sub-complex of the (first) barycentric subdivision of K . Prove that G contains a special tree G’ which contains all vertices of G .

  2. Let L be the sub-complex of the (n-1) -skeleton of K made of cells which do not intersect G’ . Let Σ be a smooth hypersurface separating M into a regular neighborhood U of L and a regular neighborhood U’ of G’ (here regular means there is an isotopy of self-embeddings contracting the given neighborhood into an arbitrarily small one). Convince yourself that such an hypersurface Σ exists and that U’ is diffeomorphic to ℝ^{n-1} × ℝ^+ (where ℝ^+ = [0, +∞) ). It may help to stare at the second barycentric subdivision of K .

  3. For every positive ε , consider φ^ε_ t \! :ℝ^+ → ℝ^+ defined by

    φ^ε_ t(x) = \frac{x}{1 + t/ε x}.

    Prove that t ↦ φ^ε_ t is an isotopy of embeddings and φ^ε_1 maps ℝ^+ to [0, 1/ε) .

  4. Conclude.