We introduce some machinery that will be useful in the proof of the Ornstein isomorphism theorem.

Using geometric homology and cohomology we give a simple conceptual proof of the Thom isomorphism theorem.

As is well-known, two equivalent total numberings are computably isomorphic, if at least one of them is precomplete. Selivanov asked whether a result of this type is true also for partial numberings. As has been shown by the author, numberings of this kind appear by necessity in studies of effectively given topological spaces like the computable real numbers. In the present paper it is demonstr...

We show that two Alexander biquandles M and M ′ are isomorphic iff there is an isomorphism of Z[s, t]-modules h : (1− st)M → (1− st)M ′ and a bijection g : Os(A) → Os(A ) between the s-orbits of sets of coset representatives of M/(1 − st)M and M /(1 − st)M ′ respectively satisfying certain compatibility conditions.

We show that all sets complete for NC1 under AC0 reductions are isomorphic under AC0computable isomorphisms. Although our proof does not generalize directly to other complexity classes, we do show that, for all complexity classes C closed under NC1-computable many-one reductions, the sets complete for C under NC0 reductions are all isomorphic under AC0-computable isomorphisms. Our result showin...

Coxeter groups have presentations 〈S : (st)st∀s, t ∈ S〉 where for all s, t ∈ S, mst ∈ {1, 2, . . . ,∞}, mst = mts and mst = 1 if and only if s = t. A fundamental question in the theory of Coxeter groups is: Given two such “Coxeter” presentations, do they present the same group? There are two known ways to change a Coxeter presentation, generally referred to as twisting and simplex exchange. We ...

We show that every problem in the complexity class SZK (Statistical Zero Knowledge) is efficiently reducible to the Minimum Circuit Size Problem (MCSP). In particular Graph Isomorphism lies in RP. This is the first theorem relating the computational power of Graph Isomorphism and MCSP, despite the long history these problems share, as candidate NP-intermediate problems.

The purpose of this note is to show that any order isomorphism between noncommutative L2-spaces associated with von Neumann algebras is decomposed into a sum of a completely positive map and a completely copositive map. The result is an L2 version of a theorem of Kadison for a Jordan isomorphism on operator algebras.

This paper provides a full controlled version of algebraic K-theory. This includes a rich array of assembly maps; the controlled assembly isomorphism theorem identifying the controlled group with homology; and the stability theorem describing the behavior of the inverse limit as the control parameter goes to 0. There is a careful treatment of spectral cosheaf homology and related tools, includi...

The principal theme of the present paper is to consider isomorphism classes of binary matroids as orbits of a suitable group action . This interpretation is based on a theorem of Brylawski – Lucas . A refinement of the Burnside Lemma is used in order to enumerate these orbits . Ternary matroids are dealt with in much the same way (Section 2) . Counting regular matroids is more dif ficult , but ...