Whitney topologies - Alchetron, The Free Social Encyclopedia

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

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.

Whitney Ck-topology

For some integer k ≥ 0, let J^k(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 ⊂ J^k(M,N), and denote by S^k(U) the following:

S k ( U ) = { f ∈ C ∞ ( M , N ) : ( J k f ) ( M ) ⊆ U } .

The sets S^k(U) form a basis for the Whitney C^k-topology on C^∞(M,N).

Whitney C∞-topology

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

W = ⋃ k = 0 ∞ W k .

Dimensionality

Notice that C^∞(M,N) has infinite dimension, whereas J^k(M,N) has finite dimension. In fact, J^k(M,N) is a real, finite-dimensional manifold. To see this, let ℝ^k[x₁,…,x_m] 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 ⁡ { R k [ x 1 , … , x m ] } = ∑ i = 1 k ( m + i − 1 ) ! ( m − 1 ) ! ⋅ i ! = ( ( m + k ) ! m ! ⋅ k ! − 1 ) .

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

B m , n k = ⨁ i = 1 n R k [ x 1 , … , x m ] , ⟹ dim ⁡ { B m , n k } = n dim ⁡ { A m k } = n ( ( m + k ) ! m ! ⋅ k ! − 1 ) .

Using b to denote the dimension B^k_m,n, we see that B^k_m,n ≅ ℝ^b, and so is a real, finite-dimensional manifold.

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

dim { J k ( M , N ) } = m + n + dim { B n , m k } = m + n ( ( m + k ) ! m ! ⋅ k ! ) .

Topology

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

π k : C ∞ ( M , N ) ↠ J k ( M , N ) where π k ( f ) = ( j k f ) ( M ) .

In the Whitney C^k-topology the open sets in C^∞(M,N) are, by definition, the preimages of open sets in J^k(M,N). It follows that the map π^k between C^∞(M,N) given the Whitney C^k-topology and J^k(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.

References

Whitney topologies Wikipedia

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