Whitney topologies

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In mathematics, and especially differential topology, functional analysis and singularity theory, the Whitney topologies are a countably infinite family of topologies defined on the set of smooth mappings between two smooth manifolds. They are named after the American mathematician Hassler Whitney.

Construction[edit]

Let M and N be two real, smooth manifolds. Furthermore, let C(M,N) denote the space of smooth mappings between M and N. The notation C means that the mappings are infinitely differentiable, i.e. partial derivatives of all orders exist and are continuous.[1]

Whitney Ck-topology[edit]

For some integer k ≥ 0, let Jk(M,N) denote the k-jet space of mappings between M and N. The jet space can be endowed with a smooth structure (i.e. a structure as a C manifold) which make it into a topological space. This topology is used to define a topology on C(M,N).

For a fixed integer k ≥ 0 consider an open subset U ⊂ Jk(M,N), and denote by Sk(U) the following:

 S^k(U) = \{ f \in C^{\infty}(M,N) : (J^kf)(M) \subseteq U \} .

The sets Sk(U) form a basis for the Whitney Ck-topology on C(M,N).[2]

Whitney C-topology[edit]

For each choice of k ≥ 0, the Whitney Ck-topology gives a topology for C(M,N); in other words the Whitney Ck-topology tells us which subsets of C(M,N) are open sets. Let us denote by Wk the set of open subsets of C(M,N) with respect to the Whitney Ck-topology. Then the Whitney C-topology is defined to be the topology whose basis is given by W, where:[2]

 W = \bigcup_{k=0}^{\infty} W^k .

Dimensionality[edit]

Notice that C(M,N) has infinite dimension, whereas Jk(M,N) has finite dimension. In fact, Jk(M,N) is a real, finite-dimensional manifold. To see this, let k[x1,…,xm] denote the space of polynomials, with real coefficients, in m variables of order at most k and with zero as the constant term. This is a real vector space with dimension

 \dim\left\{\R^k[x_1,\ldots,x_m]\right\} = \sum_{i=1}^k \frac{(m+i-1)!}{(m-1)! \cdot i!} = \left( \frac{(m+k)!}{m!\cdot k!} - 1 \right) .

Writing a = dim{ℝk[x1,…,xm]} then, by the standard theory of vector spaces k[x1,…,xm] ≅ ℝa, and so is a real, finite-dimensional manifold. Next, define:

B_{m,n}^k = \bigoplus_{i=1}^n \R^k[x_1,\ldots,x_m], \implies \dim\left\{B_{m,n}^k\right\} = n \dim \left\{ A_m^k \right\} = n \left( \frac{(m+k)!}{m!\cdot k!} - 1 \right) .

Using b to denote the dimension Bkm,n, we see that Bkm,n ≅ ℝb, and so is a real, finite-dimensional manifold.

In fact, if M and N have dimension m and n respectively then:[3]

 \dim\!\left\{J^k(M,N)\right\} = m + n + \dim \!\left\{B_{n,m}^k\right\} = m + n\left( \frac{(m+k)!}{m!\cdot k!}\right).

Topology[edit]

Consider the surjective mapping from the space of smooth maps between smooth manifolds and the k-jet space:

\pi^k : C^{\infty}(M,N) \twoheadrightarrow J^k(M,N) \ \mbox{where} \ \pi^k(f) = (j^kf)(M) .

In the Whitney Ck-topology the open sets in C(M,N) are, by definition, the preimages of open sets in Jk(M,N). It follows that the map πk between C(M,N) given the Whitney Ck-topology and Jk(M,N) given the Euclidean topology is continuous.

Given the Whitney C-topology, the space C(M,N) is a Baire space, i.e. every residual set is dense.[4]

References[edit]

  1. ^ Golubitsky, M.; Guillemin, V. (1974), Stable Mappings and Their Singularities, Springer, p. 1, ISBN 0-387-90072-1 
  2. ^ a b Golubitsky & Guillemin (1974), p. 42.
  3. ^ Golubitsky & Guillemin (1974), p. 40.
  4. ^ Golubitsky & Guillemin (1974), p. 44.