95 research outputs found
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 , we compute the global
dimension of the category of -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
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