International Journal of Mathematics

Douglas and Rado showed that an embedded rectifiable curve bounds a disk of least area. It is shown that this result holds for all rectifiable curves, including singular ones.

We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is a group algebra G, the invariant counts homomorphisms from the fundamental group of the manifold to G. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic.

Any compact, connected Lie group which acts effectively on a closed aspherical manifold is a torus T^k with k ≤ rank of , the center of π₁ (M). When , the torus action is called a maximal torus action. The authors have previously shown that many closed aspherical manifolds admit maximal torus actions. In this paper, a smooth maximal torus action is constructed on each solvmanifold. They also construct smooth maximal torus actions on some double coset spaces of general Lie groups as applications.

We define K-theory for Fréchet algebras (assumed to be locally multiplicatively convex) so as to simultaneously generalize K-theory for σ-C*-algebras and K-theory for Banach algebras. The main results on K-theory of σ-C*-algebras, which are analogs of standard theorems on representable K-theory of spaces, carry over to the more general case. Our theory also gives the expected results in two other cases. If the invertible elements of a Fréchet algebra are an open set, as is the case for dense subalgebras of C*-algebras closed under holomorphic functional calculus, then our theory agrees with the result of applying the Banach algebra definition. For commutative unital Fréchet algebras, our K-theory is the same as the representable K-theory of the maximal ideal space with its compactly generated topology.