Universal (Co) Homology Theories

It is shown that the homology and cohomology theories on separable C*–algebras given by nonstable E–theory are the universal such theories. By specializing to Abelian C*–algebras, we obtain a family of extraordinary Steenrod homology and cohomology theories on pointed compact metric spaces which are the universal such theories in the same way. For each of the extraordinary Steenrod (co)homology theories considered, we describe an –spectrum which represents the theory.     

Finite Group Actions with Fixed Points on Homology 3-Spheres

 If a finite group acts freely on a homology 3-sphere, then it has periodic cohomology. To say that a finite group F has periodic cohomology is equivalent to say that any Sylow subgroup of F of odd order is cyclic and a Sylow 2-subgroup of F is either cyclic or a quaternion group. In this paper we consider more generally smooth actions of finite groups G on homology 3-spheres which may have fixed points. We prove that any Sylow subgroup of G of odd order is either cyclic or the direct sum of two cyclic groups. Moreover, we show that if G has odd order, then it splits as a semidirect product of a subgroup A and a normal subgroup B such that B acts freely and there exist some simple closed curves in the homology 3-sphere which are fixed pointwise by some non-trivial element of A. We discuss the relation between these algebraic results and some classical constructions of the theory of 3-manifolds. Received 25 September 1997; in revised form 2 June 1998     

The Lyusternik–Shnirel'man category,¶vector bundles, and immersions of manifolds

One of our two main results exhibits for a vector bundle over a compact Hausdorff space X an interplay between its span, its possible splittings, and the Lyusternik–Shnirel'man category of X. The other main result, also on vector bundles and the Lyusternik–Shnirel'man category, enables us to derive certain inequalities connecting the immersion codimension, the stable span, and the Lyusternik–Shnirel'man category of a smooth closed manifold which is not stably parallelizable. Our results are applicable in various situations of general interest. Received: 4 September 1997     

A note on nonstable monomorphisms¶of vector bundles

In this paper we consider the question of the existence of a nonstable vector bundle monomorphism u:α→β over a closed, connected and smooth manifold M, when dimension of α= 3, dimension of β= dimension of M=n≡ 0(4). The singularity method provides the full obstruction to this problem and under some homological hypothesis we can compute it in terms of well known invariants. Received: 31 May 1999     

Coincidences of maps into homogeneous spaces

Let f,g:X→M be maps between two closed connected orientable n-manifolds where M=G/K is the homogeneous space of left cosets of a compact connected Lie group G by a finite subgroup K. In this note, we obtain a simple formula for the Lefschetz coincidence number L(f,g) in terms of topological degree, generalizing some previously known formulas for fixed points. Our approach, by means of Nielsen root theory, also allows us to give a simpler and more geometric proof of the fact that all coincidence classes of f and g have coincidence index of the same sign. Received: 3 March 1998 / Revised version: 29 June 1998     

Homotopy uniqueness of BG 2

We prove that the homotopy type of BG ₂ is determined by its mod 2 cohomology as well as by its Weyl group data. Received: 21 April 1997 / Revised version: 8 December 1997     

A global slice theorem for proper Hamiltonian actions

An analogue of a theorem of Abels [A] reducing the proper action of a non-compact Lie group to a compact transformation group is proved for the Hamiltonian setting. Received: 26 June 1997 / Revised version: 12 October 1998     

Note On linear consistency of anonymous values for TU-games

In the framework of values for TU-games, it is shown that a particular type of consistency, called linear consistency, together with some kind of standardness for two-person games, imply efficiency, anonymity, linearity, as well as uniqueness of the value. Among others, this uniform treatment generalizes Sobolev's axiomatization of the Shapley value. Revised version: December 2001     

Sets of Periods for Expanding Maps on Flat Manifolds

 It is proven that the sets of periods for expanding maps on n-dimensional flat manifolds are uniformly cofinite, i.e. there is a positive integer m ₀, which depends only on n, such that for any integer , for any n-dimensional flat manifold ℳ and for any expanding map F on ℳ, there exists a periodic point of F whose least period is exactly m. (Received 10 April 1998; in revised form 20 January 1999)     

On exotic characteristic classes for spherical fibrations

We define for every so-called admissible relation r in the Steenrod algebra $\text{A}$ and for every oriented spherical fibration ξ over a CW-space an exotic characteristic class (mod 2) ε(r)(ξ), which is primitive and vanishes for sphere bundles. The set of exotic classes associated with the universal spherical fibration and the admissible Adem relations are compared with the algebra generators of H¹(BSG;$\text{Z}$₂) due to Milgram. Moreover, their behaviour under the action of $\text{A}$ is computed. Finally, we give a secondary Wu formula for exotic classes of special Poincaré duality spaces.     

Homology of Linear Groups with Coefficients in the Adjoint Action and K-Theory

The main purpose of this paper is to compute some homology groups of linear groups with coefficients in the adjoint action which have appeared in different works. Our method uses various weight decompositions in group homology, algebraic K–theory, and Hochschild homology. This paper originated in a question posed by Cathelineau which is related to Hilbert's third problem on the scissors congruence of polyhedra.     

Lower bounds¶for the relative Lusternik–Schnirelmann category

We give estimates of numerical homotopy invariants of the pair (X,X×S ^p ) in terms of homotopy invariants of X. More precisely, we prove that σ ^{p +1} cat(X) + 1 ≤ cat(X,X×S ^p }), that and that e(X,X×S< ^p )=e(X)+1, where σ ^{p +1} cat is the (relative) σ category of Vandembroucq and e is the (relative) Toomer invariant. The proof is based on an extension of Milnor's construction of the classifying space of a topological group to a relative setting (due to Dold and Lashof). Received: 14 October 1998 / Revised version: 5 November 1999     

Coisotropic actions on compact homogeneous Kähler manifolds

Abstract. The main result of the paper is that a compact homogeneous K?hler manifold admitting an isometric and coisotropic action with a fixed point is isometric to a Hermitian symmetric space. Received: 28 December 2001; in final form: 19 March 2002 / Published online: 14 February 2003 Part of the work on this paper was done during a visit of the second author at the University of Florence that was financially supported by G.N.S.A.G.A. – I.N.d.A.M.     

A dual graph-norm refinement indicator for finite volume approximations of the Euler equations

Summary. We generalise and apply a refinement indicator of the type originally designed by Mackenzie, Süli and Warnecke in [15] and [16] for linear Friedrichs systems to the Euler equations of inviscid, compressible fluid flow. The Euler equations are symmetrized by means of entropy variables and locally linearized about a constant state to obtain a symmetric hyperbolic system to which an a posteriori error analysis of the type introduced in [15] can be applied. We discuss the details of the implementation of the refinement indicator into the DLR--Code which is based on a finite volume method of box type on an unstructured grid and present numerical results. Received May 15, 1995 / Revised version received April 17, 1996     

On the geography of symplectic 6-manifolds

The article investigates the geography of closed, connected and simply connected, six-dimensional manifolds. It is proved that any triple of integers satisfying some necessary arithmetical restrictions occurs as the Chern triple of such a manifold. The main tools used for producing the examples are the symplectic connected sum and the symplectic blow-up. Received: 28 May 1998 / Revised version: 22 January 1999     

Equations in free groups and coincidence of mappings on surfaces

Let be a continuous map and a constant map between closed orientable surfaces of genus h,g, respectively. By definition the pair (f,c) has the Wecken property if f can be deformed into a map such that every coincidence classes of (f',c) is essential and consists of exactly one point. The main result is that (f,c) has the Wecken property if and only if where . Certain quadratic equations in free groups closely related to the coincidence problem are solved. Received January 26, 1999; in final form December 10, 1999 / Published online March 12, 2001     

The minimal number of roots of surface mappings and quadratic equations in free groups

Let be a continuous mapping between orientable closed surfaces of genus h and g and let c denote the constant map with . Let be the minimal number of roots of f' among all maps f' homotopic to f, i.e. . We prove that where and denotes the Euler characteristic. In addition, certain quadratic equations in free groups closely related to the coincidence problem are solved. Received January 26, 1999; in final form November 8, 1999 / Published online February 5, 2001