1 Preliminaries
1.1 Jets spaces
Let be a locally trivial fibration. A local section of is a (smooth) map such that .
Let and be two local section defined near . We say that and are tangent up to order at if in a local trivialization where becomes , so that and , and have the same Taylor expansion at up to order .
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 . The equivalence class of is denoted by . The set of all equivalence classes at all points is denoted by , and its obvious projection to is denoted by . For any section , is a section of . We equip with the quotient topology coming from the topology (any variant is fine since we only care about neighborhoods of points). The next three questions endow with a well defined and functorial smooth structure.
Prove that, for every (local) section defined near some , there is a neighborhood of and a local homeomorphism such that and each is a holonomic section.
Prove that there is a unique smooth structure on such that is smooth and every smooth family of sections (i.e. is smooth and each is a section of ) is smooth.
For every local diffeomorphism of and every we set . Prove that is a local diffeomorphism of .
1.2 Polyhedra and triangulations
An affine cell in is the convex hull of a finite set of points. Its dimension is the dimension of its affine span . Its interior is its topological interior inside . Its boundary is .
Prove that is a finite union of cells of dimension (they are called faces of ).
A cell which is the convex hull of exactly points is called a simplex. A cube is a cell which is an affine transform of .
A cell complex is a countable collection of cells whose interiors are pairwise disjoint, and whose faces are in , and such that every point of is in finitely many cells. The union of these cells, equipped with the induced topology, is denoted by .
A cell complex is a subdivision of if and every cell of is contained in a cell of . The -skeleton of is the collection of all cells in with dimension at most . The barycentric subdivision of a cell is the subdivision obtained by adding as vertices the barycenters of and all its faces, and then adding all simplices of dimension at most determined by the old and new vertices. The barycentric subdivision of a cell complex is obtained by barycentric subdivisions of all its cells.
Explain why the above definition of a barycentric subdivision of a cell complex makes sense.
Prove that every cell complex has a subdivision whose cells are all simplices (resp. cubes).
A map from to a manifold is called piecewise smooth if, for every cell in , there exists which is smooth and restricts to on .
A smooth polyhedron in a smooth manifold is where is a cell complex, and is a topological embedding that is piecewise smooth. A cell decomposition of is where is a piecewise smooth homeomorphism from to . 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 , there is a polyhedron with and, for every neighborhood of , an isotopy of embeddings such that , and for all .
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 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.
Let be a triangulation of a connected non-compact -manifold. Let be the one-skeleton of the dual triangulation, seen as a sub-complex of the (first) barycentric subdivision of . Prove that contains a special tree which contains all vertices of .
Let be the sub-complex of the -skeleton of made of cells which do not intersect . Let be a smooth hypersurface separating into a regular neighborhood of and a regular neighborhood of (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 exists and that is diffeomorphic to (where ). It may help to stare at the second barycentric subdivision of .
For every positive , consider defined by
Prove that is an isotopy of embeddings and maps to .
Conclude.