Affordable Access

The very-strong \mathrm{C}^{\infty} topology on \mathrm{C}^{\infty} (M,N) and K-equivariant maps

Publication Date
  • Mathematics


Illman, S. Osaka J. Math. 40 (2003), 409–428 THE VERY-STRONG C∞ TOPOLOGY ON C∞(M, N) AND K -EQUIVARIANT MAPS S ¨OREN ILLMAN (Received September 28, 2001) The main purpose of this paper is to prove Theorem C, given below. Let us how- ever in this introduction first discuss the role of the very-strong C∞ topology and its relation to the strong C∞ topology. Let and be C∞ manifolds. By C ( ), where 1 ≤ ≤ ∞, we denote the set of all C maps from to . In the case when is finite, and may be non-compact, there is a well established, standard choice of a topology for C ( ), namely the strong C topology, also called the Whitney C topology, see e.g. [5], Sec- tion 2.1, and [11], Section 2. We denote C ( ) with the strong C topology by CS( ), 1 ≤ < ∞. In the case when is finite the strong C topology is clearly the right topology to use on C ( ). However, when = ∞ the question concerning the right topology for C∞( ) is more complex and interesting. One possible choice of a topology for C∞( ) is the strong C∞ topology, introduced by Mather in [11], Section 2, see also [5], Section 2.1. The strong C∞ topology on C∞( ) has as a basis the union of all strong C topologies on C∞( ) 1 ≤ < ∞. (The strong C topology on C∞( ) is the relative topology from CS( ) 1 ≤ < ∞.) Mather calls the strong C∞ topology the Whitney C∞ topology, but as we shall see this choice of terminology is not well founded. It is only in the case when is finite that the strong C topology should be named the Whitney topology. In fact du Plessis and Wall, see [15], p. 59, pro- pose that the strong C∞ topology on C∞( ) be named the Mather topology. Note that the strong C∞ topology is completely determined by the strong C topologies on C∞( ) 1 ≤ < ∞, and in this sense the strong C∞ topology on C∞( ) is not a genuine C∞ topology. We let C∞S ( ) denote C∞( ) with the strong C∞ topology. There is however a genuine C∞ topology on C∞( ), namely the very-strong C∞ topology. This topology was introduced by Cerf in [3], Definition I.4.3.1. We

There are no comments yet on this publication. Be the first to share your thoughts.


Seen <100 times

More articles like this

Tilt of N2molecules physintercalated into C24K and...

on Journal of Physics and Chemist... Jan 01, 1996

Synthesis of the quaternary one-dimensional triple...

on Journal of Solid State Chemist... Jan 01, 1992

Theoretical study of C24N4molecule

on Journal of Molecular Structure... Jan 01, 1996

Fixed points of C2maps

on Journal of Computational and A... Jan 01, 1979
More articles like this..