S-spaces from free extensions

We prove that there exist S-spaces containing an arbitrary number of non-isomorphic affine planes of any order. The proof is obtained by constructing some new S-spaces in two different ways. In one case we obtain S-spaces of finite order containing an infinite number of points, while in the other case we obtain S-spaces of infinite order

Manifolds homotopy equivalent to certain torus bundles over lens spaces

We compute the topological simple structure set of closed manifolds which occur as total spaces of flat bundles over lens spaces S^l/(Z/p) with fiber an n-dimensjional torus T^n for an odd prime p and l greater or equal to 3, provided that the induced Z/p-action on pi_1(T^n) = Z^n is free outside the origin. To the best of our knowledge this is the first computation of the structure set of a topological manifold whose fundamental group is not obtained from torsionfree and finite groups using amalgamated and HNN-extensions. We give a collection of classical surgery invariants such as splitting obstructions and rho-invariants which decide whether a simple homotopy equivalence from a closed topological manifold to M is homotopic to a homeomorphism.Comment: 40 pages, to appear in Communications on Pure and Applied Mathematic

Cohomological Hall algebra of a symmetric quiver

In the paper \cite{KS}, Kontsevich and Soibelman in particular associate to each finite quiver $Q$ with a set of vertices $I$ the so-called Cohomological Hall algebra \cH, which is $\Z_{\geq 0}^I$-graded. Its graded component \cH_{\gamma} is defined as cohomology of Artin moduli stack of representations with dimension vector $\gamma.$ The product comes from natural correspondences which parameterize extensions of representations. In the case of symmetric quiver, one can refine the grading to $\Z_{\geq 0}^I\times\Z,$ and modify the product by a sign to get a super-commutative algebra (\cH,\star) (with parity induced by $\Z$-grading). It is conjectured in \cite{KS} that in this case the algebra (\cH\otimes\Q,\star) is free super-commutative generated by a $\Z_{\geq 0}^I\times\Z$-graded vector space of the form V=V^{prim}\otimes\Q[x], where $x$ is a variable of bidegree $(0,2)\in\Z_{\geq 0}^I\times\Z,$ and all the spaces $\bigoplus\limits_{k\in\Z}V^{prim}_{\gamma,k},$ $\gamma\in\Z_{\geq 0}^I.$ are finite-dimensional. In this paper we prove this conjecture (Theorem 1.1). We also prove some explicit bounds on pairs $(\gamma,k)$ for which $V^{prim}_{\gamma,k}\ne 0$ (Theorem 1.2). Passing to generating functions, we obtain the positivity result for quantum Donaldson-Thomas invariants, which was used by S. Mozgovoy to prove Kac's conjecture for quivers with sufficiently many loops \cite{M}. Finally, we mention a connection with the paper of Reineke \cite{R}.Comment: 16 pages, no figures; a reference adde

Categorical aspects of bivariant K-theory

This survey article on bivariant Kasparov theory and E-theory is mainly intended for readers with a background in homotopical algebra and category theory. We approach both bivariant K-theories via their universal properties and equip them with extra structure such as a tensor product and a triangulated category structure. We discuss the construction of the Baum-Connes assembly map via localisation of categories and explain how this is related to the purely topological construction by Davis and Lueck

De Branges spaces and Krein's theory of entire operators

This work presents a contemporary treatment of Krein's entire operators with deficiency indices $(1,1)$ and de Branges' Hilbert spaces of entire functions. Each of these theories played a central role in the research of both renown mathematicians. Remarkably, entire operators and de Branges spaces are intimately connected and the interplay between them has had an impact in both spectral theory and the theory of functions. This work exhibits the interrelation between Krein's and de Branges' theories by means of a functional model and discusses recent developments, giving illustrations of the main objects and applications to the spectral theory of difference and differential operators.Comment: 37 pages, no figures. The abstract was extended. Typographical errors were corrected. The bibliography style was change

Hilbert space compression for free products and HNN-extensions

Given the Hilbert space compression of two groups, we find bounds on the Hilbert space compression of their free product. We also investigate the Hilbert space compression of an HNN-extension of a group relative to a finite normal subgroup or a finite index subgroup.Comment: 18 page

Rational families of vector bundles on curves, I

Let C be a smooth complex projective curve of genus at least 2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1. For any k>1, we find two irreducible components of the space of rational curves of degree k on M. One component, which we call the nice component has the property that the general element is a very free curve if k is sufficiently large. The other component has the general element a free curve. Both components have the expected dimension and their maximal rationally connected fibration is the Jacobian of the curve C.Comment: 23 page