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

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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

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