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.

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:
:$S^k(U) = \ .$
The sets ''S^k''(''U'') form a basis for the Whitney C^''k''-topology on C^∞(''M'',''N'')., p. 42.

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:
:$W = \bigcup_^ W^k .$

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
:$\dim\left\ = \sum_^k \frac = \left( \frac - 1 \right) .$
Writing then, by the standard theory of vector spaces and so is a real, finite-dimensional manifold. Next, define:
:B_^k = \bigoplus_^n \R^k _1,\ldots,x_m \implies \dim\left\ = n \dim \left\ = n \left( \frac - 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:
:$\dim\!\left\ = m + n + \dim \!\left\ = m + n\left( \frac\right).$

Topology

Given the Whitney C^∞-topology, the space C^∞(''M'',''N'') is a Baire space, i.e. every residual set is dense., p. 44.

References

{{Reflist

Differential topology
Singularity theory