Volume 10, Issue 5

We study the validity of Courant's nodal domain theorem for eigenfunctions of selfadjoint second order elliptic operators with low regularity assumptions on the coefficients. We prove that in two dimensions Courant's theorem holds also when the coefficients are just bounded and measurable. In the higher dimensional case, we prove a weakened version of Courant's theorem when the coefficients in the principal part are Hölder continuous.

We develop a theory of Spanier-Whitehead duality in categories with cofibrations and weak equivalences (Waldhausen categories, for short). This includes L -theory, the involution on K -theory introduced by [Vogell, W.: The involution in the algebraic K -theory of spaces. Proc. of 1983 Rutgers Conf. on Alg. Topology. Springer Lect. Notes in Math. 1126, pp. 277–317] in a special case, and a map Ξ relating L -theory to the Tate spectrum of ℤ/2 acting on K -theory. The map Ξ is a distillation of the long exact Rothenberg sequences [Shaneson, J.: Wall's surgery obstruction groups for G × ℤ. Ann. of Math. 90 (1969), 296–334], [Ranicki, A.: Algebraic L -theory I. Foundations. Proc. Lond. Math. Soc. 27 (1973), 101–125], [Ranicki, A.: Exact sequences in the algebraic theory of surgery. Mathematical Notes, Princeton Univ. Press, Princeton, New Jersey 1981], including analogs involving higher K -groups. It goes back to [Weiss, M., Williams, B.: Automorphisms of manifolds and algebraic K -theory, Part II. J. Pure and Appl. Algebra 62 (1988), 47–107] in special cases. Among the examples covered here, but not in [Weiss, M., Williams, B.: Automorphisms of manifolds and algebraic K -theory, Part II. J. Pure and Appl. Algebra 62 (1988), 47–107], are categories of retractive spaces where the notion of weak equivalence involves control.

In this paper we prove two technical theorems about the equivariant moduli space of ASD connections on a SU 2 or SO 3 bundle over a smooth oriented four-manifold X which is equipped with a smooth and orientation preserving action of a finite group π. The first theorem relates, in the case π = ℤ/ p and compact moduli spaces, the existence of a non empty fixed set in the moduli space to the value of a certain Donaldson polynomial invariant. The second theorem gives a criterion under which one can avoid fixed reducible ASD connections by slightly varying the metric on X within the class of equivariant metrics.

The purpose of this paper is to introduce a new structure into the representation theory of quantum groups. The structure is motivated by braid and knot theory. Representations of quantum groups associated to classical Lie algebras have an additional symmetry which cannot be seen in the classical limit. We first explain the general formalism of these symmetries (called cylinder forms ) in the context of comodules. Basic ingredients are tensor representations of braid groups of type B derived from standard R -matrices associated to socalled four braid pairs. These are applied to the Faddeev-Reshetikhin-Takhtadjian construction of bialgebras from R -matrices. As a consequence one obtains four braid pairs on all representations of the quantum group. In the second part of the paper we study in detail the dual situation of modules over the quantum enveloping algebra U q ( sl 2 ). The main result here is the computation of the universal cylinder twist .