1,720,968 research outputs found
A MATSUMOTO–MOSTOW RESULT FOR ZIMMER’S COCYCLES OF HYPERBOLIC LATTICES
Full text linkFollowing the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer’s cocycle Γ × X → PO° (n, 1), where Γ is a torsion-free (non-)uniform lattice in PO° (n, 1), with n > 3, and X is a suitable standard Borel probability Γ-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor–Wood type inequality in terms of the volume of the manifold ΓHn. Additionally this invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to define a family of measurable cocycles with vanishing volume. The same interpretation enables us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map X → PO° (n, 1) with essentially constant sign. As a by-product of our rigidity result for the volume of cocycles, we give a different proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles. In dimension n = 2, we introduce the notion of Euler number of measurable cocycles associated to a closed surface group and we show that it extends the classic Euler number of representations. Our Euler number agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. Imitating the techniques developed in the case of the volume, we show a Milnor–Wood type inequality whose upper bound is given by the modulus of the Euler characteristic of the associated closed surface. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, using the interpretation of the Euler number as a multiplicative constant between bounded cohomology classes, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization
Multiplicative constants and maximal measurable cocycles in bounded cohomology
Full text linkMultiplicative constants are a fundamental tool in the study of maximal representations. In this paper, we show how to extend such notion, and the associated framework, to measurable cocycles theory. As an application of this approach, we define and study the Cartan invariant for measurable PU(m, 1)-cocycles of complex hyperbolic lattices
Ideal simplicial volume of manifolds with boundary
Full text linkWe define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold M measures the minimal size of possibly ideal triangulations of M “with real coefficients”, thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, the main difference being that ideal simplices are now allowed to appear in representatives of the fundamental class. We show that the ideal simplicial volume is bounded above by the ordinary simplicial volume, and that it vanishes if and only if the ordinary simplicial volume does. We show that, for manifolds with amenable boundary, the ideal simplicial volume coincides with the classical one, whereas for hyperbolic manifolds with geodesic boundary it can be strictly smaller. We compute the ideal simplicial volume of an infinite family of hyperbolic 3-manifolds with geodesic boundary, for which the exact value of the classical simplicial volume is not known, and we exhibit examples where the ideal simplicial volume provides sharper bounds on mapping degrees than the classical simplicial volume
Abstract sectional category in model structures on topological spaces
No full textWe study the behavior of the abstract sectional category in the Quillen, the Strøm and the Mixed proper model structures on topological spaces and prove that, under certain reasonable conditions, all of them coincide with the classical notion. As a result, the same conclusions hold for the abstract Lusternik-Schnirelmann category and the abstract topological complexity of a space
On volumes of truncated tetrahedra with constrained edge lengths
No full textTruncated tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of low-dimensional topology, like the computation of quantum invariants of 3-manifolds and the use of variational methods in the study of circle packings on surfaces. The Lobachevsky–Schläfli formula neatly describes the behaviour of the volume of truncated tetrahedra with respect to dihedral angles, while the dependence of volume on edge lengths is worse understood. In this paper we prove that, for every l< l, where l is an explicit constant, the regular truncated tetrahedron of edge length l maximizes the volume among truncated tetrahedra whose edge lengths are all not smaller than l. This result provides a fundamental step in the computation of the ideal simplicial volume of an infinite family of hyperbolic 3-manifolds with geodesic boundary
Simplicial volume via normalised cycles
Full text linkWe show that the Connes-Consani semi-norm on singular homology with real coefficients, defined via s-modules, coincides with the ordinary l(1)-semi-norm on singular homology in all dimensions
Bounded acyclicity and relative simplicial volume
No full textIn this paper, we provide new vanishing and glueing results for relative simplicial volume, following up on two current themes in bounded cohomology: The passage from amenable groups to boundedly acyclic groups and the use of equivariant topology. More precisely, we consider equivariant nerve pairs and relative classifying spaces for families of subgroups. Typically, we apply this to uniformly boundedly acyclic families of subgroups. Our methods also lead to vanishing results for & ell;2-Betti numbers of aspherical CW-pairs with small relative amenable category and to a relative version of a result by Dranishnikov and Rudyak concerning mapping degrees and the inheritance of freeness of fundamental groups
Topological volumes of fibrations: A note on open covers
Full text linkWe establish a straightforward estimate for the number of open sets with fundamental group constraints needed to cover the total space of fibrations. This leads to vanishing results for simplicial volume and minimal volume entropy, e.g., for certain mapping tori
Stable integral simplicial volume of 3-manifolds
Full text linkWe show that non-elliptic prime 3-manifolds satisfy integral approximation for the simplicial volume, that is, that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial volume and tools from ergodic theory
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