An example of a torsion-free nilpotent group having no outer automorphisms

A negative answer is given to a question posed by E. Schenkman and F. Haimo: does every torsion-free nilpotent group have an outer automorphism?     

The Hausmann-Weinberger 4-manifold invariant of abelian groups

The Hausmann-Weinberger invariant of a group is the minimal Euler characteristic of a closed orientable 4-manifold with fundamental group . We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.

     

The unique 3-neighborly 4-manifold with few vertices

Triangulated 4-dimensional manifolds with n vertices are considered such that any triple of vertices determines a triangle belonging to the triangulation. It is shown that for n ? 13 there is exactly one such triangulation (besides the boundary of a 5-simplex). It has nine vertices and it is a triangulation of the complex projective plane.     

Extremal graphs having no matching cuts

A graph G = (V, E) is matching immune if there is no matching cut in G. We show that for any matching immune graph G, |E|≥?3(|V|?1)/2?. This bound is tight, as we define operations that construct, from a given vertex, exactly the class of matching immune graphs that attain the bound. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:206‐222, 2012 .     

On codes having no finite completions

A family of non finitely completable finite codes is given.     

Compactification of a class of conformally flat 4-manifold

In this paper we generalize Huber’s result on complete surfaces of finite total curvature. For complete locally conformally flat 4-manifolds of positive scalar curvature with Q curvature integrable, where Q is a variant of the Chern-Gauss-Bonnet integrand; we first derive the Cohn-Vossen inequality. We then establish finiteness of the topology. This allows us to provide conformal compactification of such manifolds. Oblatum 3-III-1999 & 18-II-2000?Published online: 8 May 2000     

Simple singularities and topology of symplectically filling 4-manifold

Topological restrictions of symplectically filling 4-manifolds of links around simple singularities are studied by using the Seiberg-Witten monopole equations. In particular, the intersection form of minimal symplectically filling 4-manifolds of the singularity of type E ₈ is determined. Moreover, for the case of simply elliptic singularities, similar restrictions are obtained. In the proof, a vanishing theorem of the Seiberg-Witten invariant is discussed. Received: June 9, 1998.     

On the Weyl tensor of a self-dual complex 4-manifold

We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-geodesics. We also show that the projective structure of the -surfaces of a self-dual manifold is flat. All these results are illustrated in detail in the case of the complexification of .

     

Inner amenable groups having no stable action

We construct inner amenable and ICC groups having no ergodic, free, probability-measure-preserving and stable action. This solves a problem posed by Jones-Schmidt in 1987.     

Minimal Euler characteristics of 4-manifolds with 3-manifold groups

Let π = π₁(M) for a compact 3-manifold M, and χ₄, p and q*be the invariants of Hausmann and Weinberger(1985), Kotschick(1994) and Hillman(2002), respectively. For a certain class of compact 3-manifolds M, including all those not containing two-sided RP2’s, we determine χ₄(π). We address when p(π)equals χ₄(π) and when q^*(π) equals χ₄(π), and answer a question raised by Hillman(2002).     

Random polynomials having few or no real zeros

Consider a polynomial of large degree whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly real zeros with probability as through integers of the same parity as the fixed integer . In particular, the probability that a random polynomial of large even degree has no real zeros is . The finite, positive constant is characterized via the centered, stationary Gaussian process of correlation function . The value of depends neither on nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability one may specify also the approximate locations of the zeros on the real line. The constant is replaced by in case the i.i.d. coefficients have a nonzero mean.

     

Infinite-dimensional 4-manifold of finite cohomological dimension under the continuum hypothesis

Under the assumption of the continuum hypothesis, a differentiable 4-manifoldM of dimension dimM=∞ and cohomological dimension cA—dimM=4 is constructed. The spaceM is perfectly normal and hereditarily separable. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 664–670, November, 1999.