# Whitney topologies

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.[1]

### Whitney Ck-topology

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:

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

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

### Whitney C∞-topology

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]

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

## Dimensionality

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

${\displaystyle \dim \left\{\mathbb {R} ^{k}[x_{1},\ldots ,x_{m}]\right\}=\sum _{i=1}^{k}{\frac {(m+i-1)!}{(m-1)!\cdot i!}}=\left({\frac {(m+k)!}{m!\cdot k!}}-1\right).}$

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:

${\displaystyle B_{m,n}^{k}=\bigoplus _{i=1}^{n}\mathbb {R} ^{k}[x_{1},\ldots ,x_{m}],\implies \dim \left\{B_{m,n}^{k}\right\}=n\dim \left\{A_{m}^{k}\right\}=n\left({\frac {(m+k)!}{m!\cdot k!}}-1\right).}$

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]

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

## Topology

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

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

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

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.