Article history: Received 2 November 2011 Available online xxxx Communicated by Luchezar L. Avramov MSC: primary 16E40, 55P35 secondary 16E45, 16T10, 16T15, 18G55, 55P40, 55U10, 55U15, 57T05, 57T30

1. Introduction. In [5] Mazur constructed a homotopy 4-sphere which looked like one of the strongest candidates for a counterexample to the 4-dimensional Poincaré Conjecture. In this paper we show that Mazur's example is in fact a true 4-sphere after all. This raises the odds in favour of the 4-dimensional Poincaré Conjecture. The proof involves a smooth knot of S2 in S4 with unusual properties.

We generalize graded Hecke algebras to include a twisting twococycle for the associated finite group. We give examples where the parameter spaces of the resulting twisted graded Hecke algebras are larger than that of the graded Hecke algebras. We prove that twisted graded Hecke algebras are particular types of deformations of a crossed product.

Edwards’ elliptic curve form is popular in modern cryptographic implementations thanks to their fast, strongly unified addition formulas. Twisted Edwards curves with a = −1 are slightly faster, but their addition formulas are not complete over Fp where p ≡ 3 (mod 4). In this short note, we propose that designers specify Edwards curves, but implement scalar multiplications and the like using an ...

A novel adaptive-gain twisting controller that is robust to the bounded disturbances with the unknown boundaries is proposed. The control algorithm is derived using the Lyapunov function technique. It is shown that an ideal or real second order sliding mode is established, and the adaptation algorithm does not overestimate the control gain. The efficacy of the proposed adaptive twisting control...

We introduce a new class of 2-cocycles defined explicitly on the generators of certain multiparameter standard quantum groups. These allow us, through the process of twisting the familiar standard quantum groups, to generate new as well as previously known examples of non-standard quantum groups. In particular we are able to construct generalisations of both the Cremmer-Gervais deformation of S...

We show that certain twisting deformations of a family of supersolvable groups are simple as Hopf algebras. These groups are direct products of two generalized dihedral groups. Examples of this construction arise in dimensions 60 and pq, for prime numbers p, q with q|p − 1. We also show that certain twisting deformation of the symmetric group is simple as a Hopf algebra. On the other hand, we p...

Let G = HoQ be a split extension of nite groups. A theorem of Charlap and Vasquez gives an explicit description of the di erentials d2 in the Lyndon-Hochschild-Serre spectral sequence of the extension with coe cients in a eld k. We generalize this to give an explicit description of all the dr (r 2) in this case. The generalization is obtained by associating to the group extension a new twisting...

The goal of this paper is to express the Bach tensor of a four– dimensional conformal geometry of an arbitrary signature by the Cartan normal conformal (CNC) connection. We show that the Bach tensor can be identified with the Yang–Mills current of the connection. It follows from that result that a conformal geometry whose CNC connection is reducible in an appropriate way has a degenerate Bach t...

This paper gives a short summary of the central role played by Ed Brown’s “twisting cochains” in higher FranzReidemeister (FR) torsion and higher analytic torsion. Briefly, any fiber bundle gives a twisting cochain which is unique up to fiberwise homotopy equivalence. However, when they are based, the difference between two of them is a higher algebraic K-theory class measured by higher FR tors...