Geometric cohomology frames on Hausmann-Holm-Puppe conjugation spaces

For certain manifolds with an involution the mod 2 cohomology ring of the set of fixed points is isomorphic to the cohomology ring of the manifold, up to dividing the degrees by two. Examples include complex projective spaces and Grassmannians with the standard antiholomorphic involution (with real projective spaces and Grassmannians as fixed point sets).

Hausmann, Holm and Puppe have put this observation in the framework of equivariant cohomology, and come up with the concept of conjugation spaces, where the ring homomorphisms arise naturally from the existence of what they call cohomology frames. Much earlier, Borel and Haefliger had studied the degree-halving isomorphism between the cohomology rings of complex and real projective spaces and Grassmannians using the theory of complex and real analytic cycles and cycle maps into cohomology.

The main result in the present note gives a (purely topological) connection between these two results and provides a geometric intuition into the concept of a cohomology frame. In particular, we see that if every cohomology class on a manifold with involution is the Thom class of an equivariant topological cycle of codimension twice the codimension of its fixed points (inside the fixed point set of ), these topological cycles will give rise to a cohomology frame.

     

K-theory tools for local and asymptotic cyclic cohomology

A generalization of the Connes-Thom isomorphism is given for stable, homotopy invariant, and split exact functors on separable -algebras. As examples of these functors, we concentrate on asymptotic and local cyclic cohomology, and the result is applied to improve some formulas in asymptotic and local cyclic cohomology of -algebras. As another application, it is shown that these cyclic theories are rigid under Rieffel's deformation quantizations.

     

Basic forms and Morse–Novikov cohomology of Lie groupoids

Given a basic closed 1‐form on a Lie groupoid , the Morse–Novikov cohomology groups are defined in this paper. They coincide with the usual de Rham cohomology groups when θ is exact and with the usual Morse–Novikov cohomology groups when is the unit groupoid generated by a smooth manifold M. We prove that the Morse–Novikov cohomology groups are invariant under Morita equivalences of Lie groupoids. On orbifold groupoids, we show that these groups are isomorphic to sheaf cohomology groups. Finally, when θ is not exact, we extend a vanishing theorem from smooth manifolds to orbifold groupoids.     

The local Gromov-Witten theory of curves

The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evaluation of basic integrals in the local Gromov-Witten theory of . A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a complete and effective solution.

The local Gromov-Witten theory of curves is equivalent to the local Donaldson-Thomas theory of curves, the quantum cohomology of the Hilbert scheme points of , and the orbifold quantum cohomology of the symmetric product of . The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.

     

On cohomology of ACH Kähler manifolds

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is a sharp vanishing theorem for the second cohomology of such manifolds under certain assumptions. The borderline case characterizes a Kähler-Einstein manifold constructed by Calabi.

     

The Nevo–Zimmer intermediate factor theorem over local fields

The Nevo–Zimmer theorem classifies the possible intermediate G-factors Y in Open image in new window , where G is a higher rank semisimple Lie group, P a minimal parabolic and X an irreducible G-space with an invariant probability measure. An important corollary is the Stuck–Zimmer theorem, which states that a faithful irreducible action of a higher rank Kazhdan semisimple Lie group with an invariant probability measure is either transitive or free, up to a null set. We present a different proof of the first theorem, that allows us to extend these two well-known theorems to linear groups over arbitrary local fields.     

Superperverse intersection cohomology: stratification (in)dependence

Within its traditional range of perversity parameters, intersection cohomology is a topological invariant of pseudomanifolds. This is no longer true once one allows superperversities, perversities with . In this case, intersection cohomology may depend on the choice of the stratification by which it is defined. Topological invariance also does not hold if one allows stratifications with codimension one strata. Nonetheless, both errant situations arise in important situations, the former in the Cappell-Shaneson superduality theorem and the latter in any discussion of pseudomanifold bordism. We show that while full invariance of intersection cohomology under restratification does not hold in this generality, it does hold up to restratifications that fix the the top stratum.     

Homology of pseudodifferential operators on manifolds with fibered cusps

The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals that generate the -dimensional Hochschild cohomology groups, the index of a fully elliptic fibered cusp operator is expressed as the sum of a local contribution of Atiyah-Singer type and a global term on the boundary. We announce a result relating this boundary term to the adiabatic limit of the eta invariant in a particular case.

     

Exponential sums on , II

We prove a vanishing theorem for the -adic cohomology of exponential sums on . In particular, we obtain new classes of exponential sums on that have a single nonvanishing -adic cohomology group. The dimension of this cohomology group equals a sum of Milnor numbers.

     

The l-component of the unipotent Albanese map

We prove a finiteness theorem for the local l ≠ p-component of the -unipotent Albanese map for curves. As an application, we refine the non-abelian Selmer varieties arising in the study of global points and deduce thereby a new proof of Siegel’s theorem for affine curves over of genus one with Mordell–Weil rank at most one.     

The cohomology of the moduli space of controllable linear systems

In this paper we explicitly describe, by generators and relations, the cohomology ring of the manifold _n,m (F) of controllable linear systems having m inputs and state-space dimension n. It is shown that the cohomology ring of _n,m (F) is isomorphic to the invariant cohomology ring of a product of projective spaces. Estimates for the cup length of the cohomology ring are obtained.     

Locally free sheaves on complex supermanifolds

A classification of locally free sheaves $ \mathcal{E} $ of $ \mathcal{O} $ -modules which have a given retract gr $ \mathcal{E} $ in the terms of non-abelian 1-cohomology is given. In the case of $ \mathbb{C}{{\mathbb{P}}^{1|m }} $ , m > 0, we show that the Birkhoff–Grothendieck Theorem does not hold true. We obtain a result similar to the Barth–Van de Ven–Tyurin Theorem for projective superspaces. Furthermore, a spectral sequence which connects the cohomology with values in a locally free sheaf $ \mathcal{E} $ to the cohomology with values in its retract gr $ \mathcal{E} $ is constructed.     

A generalization of the non-triviality theorem of Serre

We generalize the classical theorem of Serre on the non-triviality of infinitely many homotopy groups of -connected finite CW-complexes to CW-complexes where the cohomology groups either grow too fast or do not grow faster than a certain rate given by connectivity. For example, this result can be applied to iterated suspensions of finite Postnikov systems and certain spaces with finitely generated cohomology ring. In particular, we obtain an independent, short proof of a theorem of R. Levi on the non-triviality of -invariants associated to finite perfect groups. Another application concerns spaces where the cohomology grows like a polynomial algebra on generators in dimension , , for a fixed number . We also consider spectra where we prove a non-triviality result in the case of fast growing cohomology groups.

     

Statistical properties for nonhyperbolic maps with finite range structure

We establish the central limit theorem and non-central limit theorems for maps admitting indifferent periodic points (the so-called intermittent maps). We also give a large class of Darling-Kac sets for intermittent maps admitting infinite invariant measures. The essential issue for the central limit theorem is to clarify the speed of -convergence of iterated Perron-Frobenius operators for multi-dimensional maps which satisfy Renyi's condition but fail to satisfy the uniformly expanding property. Multi-dimensional intermittent maps typically admit such derived systems. There are examples in section 4 to which previous results on the central limit theorem are not applicable, but our extended central limit theorem does apply.

     

A connectedness result in positive characteristic

Let be a complete local ring of dimension at least two, which contains a separably closed coefficient field of positive characteristic. Using a vanishing theorem of Peskine-Szpiro, Lyubeznik proved that the local cohomology module is Frobenius-torsion if and only if the punctured spectrum of is connected in the Zariski topology. We give a simple proof of this theorem and, more generally, a formula for the number of connected components in terms of the Frobenius action on .

     

Von Neumann Betti numbers and Novikov type inequalities

In this paper we show that Novikov type inequalities for closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a consequence we obtain a vanishing theorem for cohomology. We also prove that von Neumann Betti numbers coincide with the Novikov numbers for free abelian coverings.

     

The image of the Thom map for Eilenberg-MacLane spaces

Fundamental classes in cohomology of Eilenberg-MacLane spaces are defined. The image of the Thom map from cohomology to mod- cohomology is determined for arbitrary Eilenberg-MacLane spaces. This image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of Milnor primitives to the fundamental class in mod- cohomology. This subalgebra in mod cohomology is invariant under the action of the Steenrod algebra, and it is annihilated by all Milnor primitives. We also show that cohomology determines Morava cohomology for Eilenberg-MacLane spaces.

     

A splitting theorem for compact complex homogeneous spaces with a symplectic structure

In this note we prove a splitting theorem for compact complex homogeneous spaces with a cohomology 2 class [] such that the top power [ ⁿ ]0.Dedicated to Professor W. C. Hsiang on the occasion of his 60th birthdayPartially supported by NSF Grant DMS-9401755.     

CHARACTERISTIC CLASSES OF FLAGS OF FOLIATIONS AND LIE ALGEBRA COHOMOLOGY

We prove the conjecture by Feigin, Fuchs, and Gelfand describing the Lie algebra cohomology of formal vector fields on an n-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given ag at the origin. The latter encodes characteristic classes of ags of foliations and was used in the formulation of the local Riemann-Roch Theorem by Feigin and Tsygan.Feigin, Fuchs, and Gelfand described the first symmetric power and to do this they had to make use of a fearsomely complicated computation in invariant theory. By the application of degeneration theorems of appropriate Hochschild-Serre spectral sequences, we avoid the need to use the methods of FFG, and moreover, we are able to describe all the symmetric powers at once.     

On the Associated Primes of Matlis Duals of Local Cohomology Modules II

In continuation of [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] we study associated primes of Matlis duals of local cohomology modules (MDLCM). We combine ideas from Helmut Zöschinger on coassociated primes of arbitrary modules with results from [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] 4-6 Hellus , M. , Stückrad , J. ( 2008 ). On endomorphism rings of local cohomology modules . Proceedings of the American Mathematical Society 136 : 2333 – 2341 . Hellus , M. , Stückrad , J. ( 2008 ). Matlis duals of top local cohomology modules . Proceedings of the American Mathematical Society 136 : 489 – 498 . Hellus , M. , Stückrad , J. ( 2009 ). Artinianness of local cohomology . Journal of Commutative Algebra 1 : 269 – 274 . ], and obtain partial answers to questions which were left open in [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. These partial answers give further support for conjecture (*) from [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] on the set of associated primes of MDLCMs. In addition, and also inspired by ideas from Zöschinger, we prove some non-finiteness results of local cohomology.