Secondary homotopy groups

Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of homotopy types with homotopy groups concentrated in two consecutive dimensions.Comment: We added further commets and references to make the paper more easily readabl

Virtually abelian K\"ahler and projective groups

We characterise the virtually abelian groups which are fundamental groups of compact K\"ahler manifolds and of smooth projective varieties. We show that a virtually abelian group is K\"ahler if and only if it is projective. In particular, this allows to describe the K\"ahler condition for such groups in terms of integral symplectic representations

Automorphism groups of polycyclic-by-finite groups and arithmetic groups

We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(\Gamma, 1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory

Representation theory of some infinite-dimensional algebras arising in continuously controlled algebra and topology

In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame cases. The algebra of row-column-finite (or locally finite) matrices over an arbitrary field is one of the algebras considered in this paper, its representation type is shown to be finite.Comment: 33 page

Global Dimension of Polynomial Rings in Partially Commuting Variables

For any free partially commutative monoid $M(E,I)$, we compute the global dimension of the category of $M(E,I)$-objects in an Abelian category with exact coproducts. As a corollary, we generalize Hilbert's Syzygy Theorem to polynomial rings in partially commuting variables.Comment: 11 pages, 2 figure

DG-algebras and derived A-infinity algebras

A differential graded algebra can be viewed as an A-infinity algebra. By a theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a minimal A-infinity algebra. We introduce the notion of a derived A-infinity algebra and show that any dga A over an arbitrary commutative ground ring k is equivalent to a minimal derived A-infinity algebra. Such a minimal derived A-infinity algebra model for A is a k-projective resolution of the homology algebra of A together with a family of maps satisfying appropriate relations. As in the case of A-infinity algebras, it is possible to recover the dga up to quasi-isomorphism from a minimal derived A-infinity algebra model. Hence the structure we are describing provides a complete description of the quasi-isomorphism type of the dga.Comment: v3: 27 pages. Minor corrections, to appear in Crelle's Journa

Strominger--Yau--Zaslow geometry, Affine Spheres and Painlev\'e III

We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitz\'eica equation as reductions of real forms of SL(3, \C) anti--self--dual Yang--Mills equations by two translations, or equivalently as a special case of the Hitchin equation. We use the Loftin--Yau--Zaslow construction to give an explicit expression for a six--real dimensional semi--flat Calabi--Yau metric in terms of a solution to the affine-sphere equation and show how a subclass of such metrics arises from 3rd Painlev\'e transcendents.Comment: 38 pages. Final version. To appear in Communications in Mathematical Physic

Follow-up of the GHSG HD16 trial of PET-guided treatment in early-stage favorable Hodgkin lymphoma.

The primary analysis of the GHSG HD16 trial indicated a significant loss of tumor control with PET-guided omission of radiotherapy (RT) in patients with early-stage favorable Hodgkin lymphoma (HL). This analysis reports long-term outcomes. Overall, 1150 patients aged 18-75 years with newly diagnosed early-stage favorable HL were randomized between standard combined-modality treatment (CMT) (2x ABVD followed by PET/CT [PET-2] and 20 Gy involved-field RT) and PET-2-guided treatment omitting RT in case of PET-2 negativity (Deauville score [DS] < 3). The study aimed at excluding inferiority of PET-2-guided treatment and assessing the prognostic impact of PET-2 in patients receiving CMT. At a median follow-up of 64 months, PET-2-negative patients had a 5-year progression-free survival (PFS) of 94.2% after CMT (n = 328) and 86.7% after ABVD alone (n = 300; HR = 2.05 [1.20-3.51]; p = 0.0072). 5-year OS was 98.3% and 98.8%, respectively (p = 0.14); 4/12 documented deaths were caused by second primary malignancies and only one by HL. Among patients assigned to CMT, 5-year PFS was better in PET-2-negative (n = 353; 94.0%) than in PET-2-positive patients (n = 340; 90.3%; p = 0.012). The difference was more pronounced when using DS4 as cut-off (DS 1-3: n = 571; 94.0% vs. DS ≥ 4: n = 122; 83.6%; p < 0.0001). Taken together, CMT should be considered standard treatment for early-stage favorable HL irrespective of the PET-2-result