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Whitney topologies

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Topologies defined on the set of smooth mappings between manifolds

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 C-topology

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

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

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

Whitney C-topology

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

W = ⋃ k = 0 ∞ W k . {\displaystyle W=\bigcup _{k=0}^{\infty }W^{k}.}

Dimensionality

Notice that C(M,N) has infinite dimension, whereas J(M,N) has finite dimension. In fact, J(M,N) is a real, finite-dimensional manifold. To see this, let ℝ 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 ) . {\displaystyle \dim \left\{\mathbb {R} ^{k}\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{ℝ} then, by the standard theory of vector spaces ℝ ≅ ℝ, 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 ) . {\displaystyle B_{m,n}^{k}=\bigoplus _{i=1}^{n}\mathbb {R} ^{k},\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 Bm,n, we see that Bm,n ≅ ℝ, 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 ! ) . {\displaystyle \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

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

References

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