N ov 2 00 5 Theory of valuations on manifolds , IV . Further properties of the multiplicative structure

This is the fourth part in the series of articles [4], [5], [6] (see also [3]) where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations introduced in [5] is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations introduced in [5] is an automorphism of the algebra of valuations. Then an integration functional on valuations with compact support is introduced, and a property of selfduality of valuations is proved. Next a space of generalized valuations is defined, and some basic properties of it are proved. Finally a canonical imbedding of the space of constructible functions on a real analytic manifold into the space of generalized valuations is constructed, and various structures on valuations are compared with known structures on constructible functions.