Whitney topologies explained

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^k-topology

For some integer, 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 consider an open subset and denote by S^k(U) the following:

S ^{k(U)} = \{ f \in C ^{infty} (M,N) : (J ^{kf)(M)} \subseteq U \} . The sets S^k(U) form a basis for the Whitney C^k-topology on C^∞(M,N).^[1]

Whitney C^∞-topology

For each choice of, 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:^[1]

W = c up ^{inf ty} _{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 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 ^{k} _{i =1} ((m + i -1 )!)/((m -1 )! ⋅ i!) = \left( ((m + k)!)/(m! ⋅ k!) - 1 \right) . Writing then, by the standard theory of vector spaces and so is a real, finite-dimensional manifold. Next, define:B ^{k} _{m,n} = o plus ^{n} _{i =1} \R ^{k[x} _{1 ,\ldots,x} _{m],} \implies \dim\left\{B ^{k\right\}} _{m,n} = n \dim \left\{ A ^{k} _{m} \right\} = n \left( ((m + k)!)/(m! ⋅ k!) - 1 \right) . Using b to denote the dimension B^k_m,n, we see that, and so is a real, finite-dimensional manifold.

In fact, if M and N have dimension m and n respectively then:^[2]

\dim\left\{J ^{k(M,N)\right\}} = m + n + \dim \left\{B ^{k\right\}} _{n,m} = m + n\left( ((m + k)!)/(m! ⋅ k!) \right).

Topology

Given the Whitney C^∞-topology, the space C^∞(M,N) is a Baire space, i.e. every residual set is dense.^[3]

Notes and References

1. , p. 42.
2. , p. 40.
3. , p. 44.