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We introduce the wrapped Donaldson--Fukaya category of a (generalized) semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical divisor in a toric Calabi--Yau 3-fold.We compute the wrapped Floer cohomology of a Lagrangian section and find that it is the algebra of functions on the Hori--Vafa mirror. The latter result is the key step in proving homological mirror symmetry for this case. The techniques developed here allow the construction in general of the wrapped Fukaya category on an open Calabi--Yau manifold carrying an SYZ fibration with nice behavior at infinity. We discuss the relation of this to the algebraic vs analytic aspects of mirror symmetry. [ABSTRACT FROM AUTHOR]

Refining Yau's and Kołodziej's techniques, we establish very precise uniform a priori estimates for degenerate complex Monge--Ampère equations on compact Kähler manifolds, that allow us to control the blow up of the solutions as the cohomology class and the complex structure both vary. We apply these estimates to the study of various families of possibly singular Kähler varieties endowed with twisted Kähler--Einstein metrics, by analyzing the behavior of canonical densities, establishing uniform integrability properties, and developing the first steps of a pluripotential theory in families. This provides interesting information on the moduli space of stable varieties, extending works by Berman--Guenancia and Song, as well as on the behavior of singular Ricci-flat metrics on (log) Calabi--Yau varieties, generalizing works by Rong--Ruan--Zhang, Gross--Tosatti--Zhang, Collins--Tosatti and Tosatti--Weinkove--Yang. [ABSTRACT FROM AUTHOR]

We study the quantum sheaf cohomology of flag manifolds with deformations of the tangent bundle and use the ring structure to derive how the deformation transforms under the biholomorphic duality of flag manifolds. Realized as the OPE ring of A/2-twisted two-dimensional theories with (0,2) supersymmetry, quantum sheaf cohomology generalizes the notion of quantum cohomology. Complete descriptions of quantum sheaf cohomology have been obtained for abelian gauged linear sigma models (GLSMs) and for nonabelian GLSMs describing Grassmannians. In this paper we continue to explore the quantum sheaf cohomology of nonabelian theories. We first propose a method to compute the generating relations for (0,2) GLSMs with (2,2) locus. We apply this method to derive the quantum sheaf cohomology of products of Grassmannians and flag manifolds. The dual deformation associated with the biholomorphic duality gives rise to an explicit IR duality of two A/2-twisted (0,2) gauge theories. [ABSTRACT FROM AUTHOR]

In this paper we prove that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated with the mth symmetric power of the standard representation of SL2(C) grows exponentially in m2. We give upper and lower bounds for the growth rate. Our result extends a result of W. Müller and S. Marshall, who proved the corresponding statement for closed arithmetic 3-manifolds, to the finite-volume case. We also prove a limit multiplicity formula for combinatorial Reidemeister torsions on higher-dimensional hyperbolic manifolds. [ABSTRACT FROM AUTHOR]

Consider the holomorphic Hamiltonian action of a compact Lie group K on a compact Kähler manifold M with a moment map Φ : M → k ∗ . Assume that 0 is a regular value of the moment map. Weitsman raised the question of what we can say about the cohomology of the Kähler quotient M 0 ≔ Φ − 1 (0) ∕ K if all the ordinary cohomology of M is of type (p , p). In this paper, using the Cartan–Chern–Weil theory we show that in the above context there is a natural surjective Kirwan map from an equivariant version of the Dolbeault cohomology of M onto the Dolbeault cohomology of the Kähler quotient M 0. As an immediate consequence, this result provides an answer to the question posed by Weitsman. [ABSTRACT FROM AUTHOR]

Let M be a simple hyperkähler manifold. Kuga-Satake construction gives an embedding of H 2 (M , C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H • (M , C) → H • + l (T , C) for some l , which is compatible with the Hodge structures and the Poincaré pairing. Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz sl (2) -triples on M. [ABSTRACT FROM AUTHOR]

Abstract Let (M , ω , J , g) be a non-compact almost Kähler manifold. In this paper we provide various criteria that assure that ω k induces a non trivial class in the reduced L p maximal/minimal cohomology of (M , g). Furthermore in the last part we explore some topological applications of our results. [ABSTRACT FROM AUTHOR]

Let X be a 2n-dimensional torus manifold with a locally standard T ≅ (S1)n action whose orbit space is a homology polytope. Smooth complete complex toric varieties and quasitoric manifolds are examples of torus manifolds. Consider a principal T-bundle p : E → B and let π : E(X) → B be the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E(X) as a H*(B)-algebra and the topological K-ring of E(X) as a K*(B)-algebra with generators and relations. These generalize the results in [17] and [19] when the base B = pt. These also extend the results in [20], obtained in the case of a smooth projective toric variety, to any smooth complete toric variety. [ABSTRACT FROM AUTHOR]

A simple polytope P is said to be B-rigid if its combinatorial structure is characterized by its Tor-algebra, and is said to be C-rigid if its combinatorial structure is characterized by the cohomology ring of a quasitoric manifold over P. It is known that a B-rigid simple polytope is C-rigid. In this paper, we show that the B-rigidity is not equivalent to the C-rigidity. [ABSTRACT FROM AUTHOR]

We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) manifold of the first kind. We prove that the two notions are equivalent if there exists a Riemannian metric such that the Lee vector field is unitary and parallel and its metric dual 1-form coincides with the Lee 1-form. Finally, we discuss several examples of compact l.c.s. manifolds of the first kind which do not admit compatible Vaisman metrics. [ABSTRACT FROM AUTHOR]

Any action of a group Γ on H3 by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to Γ. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends of hyperbolic 3-manifolds. That is, if two homotopy equivalent hyperbolic manifolds with infinite volume and without parabolic cusps have different geometrically infinite end invariants, then they define a 2 dimensional subspace of bounded cohomology. Our techniques apply to classes of hyperbolic 3-manifolds that have sufficiently different end invariants, and we give explicit bases for vector subspaces whose dimension is uncountable. We also show that these bases are uniformly separated in pseudo-norm, extending results of Soma. The technical machinery of the Ending Lamination Theorem allows us to analyze the geometrically infinite ends of hyperbolic 3-manifolds with unbounded geometry. [ABSTRACT FROM AUTHOR]

The basic cohomology of a complete Riemannian foliation with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing foliations): We show that the total basic Betti number of the union C of the closed leaves is smaller than or equal to the total basic Betti number of the foliated manifold, and we give sufficient conditions for equality. If there is a basic Morse–Bott function with critical set equal to C, we can compute the basic cohomology explicitly. Another case in which the basic cohomology can be determined is if the space of leaf closures is a simple, convex polytope. Our results are based on Molino's observation that the existence of non-closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set C. We introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of Lie group actions enabling us to transfer many results from the theory of Lie group actions to Riemannian foliations. The prominent role of the fixed point set in the theory of torus actions explains the relevance of the set C in the basic setting. [ABSTRACT FROM AUTHOR]

Abstract Let M be a closed n -dimensional manifold, n > 2 , whose first real cohomology group H 1 (M ; R) is non-zero. We present a general method for constructing a Morse 1-form α on M , closed but non-exact, and a pseudo-gradient X such that the differential ∂ X of the Novikov complex of the pair (α , X) has at least one incidence coefficient which is an infinite series. This is an application of our previous study of the homoclinic bifurcation of pseudo-gradients of multivalued Morse functions. [ABSTRACT FROM AUTHOR]

The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de facto cocycles, known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: The definition of equivariant characteristic forms in the Cartan model due to Nicole Berline and Michéle Vergne motivates a refinement of equivariant integral cohomology by all Cartan cocycles. In view of this, we will also review previous definitions critically, in particular the one given in work of Kiyonori Gomi. [ABSTRACT FROM AUTHOR]

We study the class of compact Kähler manifolds with trivial canonical bundle and the property that the cohomology of the trivial line bundle is generated by one element. If the square of the generator is zero, we get the class of strict Calabi-Yau manifolds. If the generator is of degree 2, we get the class of compact hyperkähler manifolds. We provide some examples and structure results for the cases where the generator is of higher nilpotency index and degree. In particular, we show that varieties of this type are closely related to higher-dimensional Enriques varieties. [ABSTRACT FROM AUTHOR]

Moser's theorem states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular, we obtain Moser's theorem on simplices. The proof is based on Banyaga's paper (1974), where Moser's theorem is proven for manifolds with boundary. A cohomological interpretation of Banyaga's operator is given, which allows a proof of Lefschetz duality using differential forms. [ABSTRACT FROM AUTHOR]

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem. [ABSTRACT FROM AUTHOR]

For arbitrary field coefficients K , we show that K -perverse sheaves on a complex affine torus satisfy the so-called propagation package , i.e., the generic vanishing property and the signed Euler characteristic property hold, and the corresponding cohomology jump loci satisfy the propagation property and codimension lower bound. The main ingredient used in the proof is Gabber–Loeser's Mellin transformation functor for K -constructible complexes on a complex affine torus, and the fact that it behaves well with respect to perverse sheaves. As a concrete topological application of our sheaf-theoretic results, we study homological duality properties of complex algebraic varieties, via abelian duality spaces . We provide new obstructions on abelian duality spaces by showing that their cohomology jump loci satisfy a propagation package. This is then used to prove that complex abelian varieties are the only complex projective manifolds which are abelian duality spaces. We also construct new examples of abelian duality spaces. For example, we show that if a smooth quasi-projective variety X satisfies a certain Hodge-theoretic condition and it admits a proper semi-small map (e.g., a closed embedding or a finite map) to a complex affine torus, then X is an abelian duality space. [ABSTRACT FROM AUTHOR]

We prove several basic ring-theoretic results about tautological rings of manifolds W, that is, the rings of generalised Miller-Morita-Mumford classes for fibre bundles with fibre W. Firstly we provide conditions on the rational cohomology of W which ensure that its tautological ring is finitely-generated, and we show that these conditions cannot be completely relaxed by giving an example of a tautological ring which fails to be finitely-generated in quite a strong sense. Secondly, we provide conditions on torus actions on W which ensure that the rank of the torus gives a lower bound for the Krull dimension of the tautological ring of W. Lastly, we give extensive computations in the tautological rings of CP2 and S2×S2. [ABSTRACT FROM AUTHOR]

Abstract: Given a basic closed 1‐form on a Lie groupoid G, the Morse–Novikov cohomology groups H θ n ( G ) are defined in this paper. They coincide with the usual de Rham cohomology groups H d R n ( G ) when θ is exact and with the usual Morse–Novikov cohomology groups H θ n ( M ) when G is the unit groupoid M ⇉ M 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. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to provide a detailed description of the spaces that can be specified as L2 domains for the operators of a first order elliptic complex on a compact manifold with conical singularities. This entails an analysis of the nature of the minimal domain and of a complementary space in the maximal domain of each of the operators. The key technical result is the nondegeneracy of a certain pairing of cohomology classes associated with the indicial complex. It is further proved that the set of choices of domains leading to Hilbert complexes in the sense of Brüning and Lesch form a variety, as well as a theorem establishing a necessary and sufficient condition for the operator in a given degree to map its maximal domain into the minimal domain of the next operator. [ABSTRACT FROM AUTHOR]

Starting from a so-called flat exact semisimple bihamiltonian structure of hydrodynamic type, we arrive at a Frobenius manifold structure and a tau structure for the associated principal hierarchy. We then classify the deformations of the principal hierarchy which possess tau structures. [ABSTRACT FROM AUTHOR]

We give a sufficient condition for a class of tame compactified Landau-Ginzburg models in the sense of Katzarkov-Kontsevich-Pantev to satisfy some versions of their conjectures. We also give examples that satisfy the condition. The relations to the quantum D-modules of Fano manifolds and the original conjectures are explained in the appendices. [ABSTRACT FROM AUTHOR]

We extend the Donaldson-Corlette-Hitchin-Simpson correspondence between Higgs bundles and flat connections on compact Kähler manifolds to compact quasi-regular Sasakian manifolds. A particular consequence is the translation of restrictions on Kähler groups proved using the Donaldson-Corlette-Hitchin-Simpson correspondence to fundamental groups of compact Sasakian manifolds (not necessarily quasi-regular). [ABSTRACT FROM AUTHOR]

We compute the Betti numbers of two-fold coverings of small covers with some special properties; in particular, we use the results of Davis and Januszkiewicz on the cohomology of small covers. It turned out that their proof contains some gap that we describe in detail and fill in. [ABSTRACT FROM AUTHOR]

In this paper we investigate the spectral sequence associated with a Riemannian foliation which arises naturally on a Vaisman manifold. Using the Betti numbers of the underlying manifold we establish a lower bound for the dimension of some terms of this cohomological object. This way we obtain cohomological obstructions for two-dimensional foliations to be induced from a Vaisman structure. We show that if the foliation is quasi-regular the lower bound is realized. In the final part of the paper we discuss two examples. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of J-holomorphic curves on almost Hermitian manifolds. Let (M, g, J, F) be an almost Hermitian manifold and f:Σ→M be an injective immersion. We prove that if the Lp functional has a critical point or a stable point in the same almost Hermitian class, then the immersion is J-holomorphic. [ABSTRACT FROM AUTHOR]

In the present paper we completely classify locally free sheaves of rank 2 with vanishing intermediate cohomology modules on the image of the Segre embedding P 2 × P 2 ⊆ P 8 and its general hyperplane sections. Such a classification extends similar already known results regarding del Pezzo varieties with Picard numbers 1 and 3 and dimension at least 3. [ABSTRACT FROM AUTHOR]

We show how certain topological properties of co-Kähler manifolds derive from those of the Kähler manifolds which construct them. In particular, we show that the existence of parallel forms on a co-Kähler manifold reduces the computation of cohomology from the de Rham complex to certain amenable sub-cdga's defined by geometrically natural operators derived from the co-Kähler structure. This provides a simpler proof of the formality of the foliation minimal model in this context. [ABSTRACT FROM AUTHOR]

We apply the recent results of Galkin et al. [Derived categories of Keum's fake projective planes, Adv. Math. 278 (2015) 238-253] to study some geometrical features of Keum's fake projective planes. Among other things, we show that the bicanonical map of Keum's fake projective planes is always an embedding. Moreover, we construct a nonstandard exceptional collection on the unique fake projective plane with . [ABSTRACT FROM AUTHOR]

We establish a formality theorem for smooth dg manifolds. More precisely, we prove that, for any finite-dimensional dg manifold ( M , Q ) , there exists an L ∞ quasi-isomorphism of dglas from ( T poly • ⊕ ( M ) , [ Q , − ] , [ − , − ] ) to ( D poly • ⊕ ( M ) , 〚 m + Q , − 〛 , 〚 − , − 〛 ) whose first Taylor coefficient (1) is equal to the composition hkr ∘ ( td ( M , Q ) ∇ ) 1 2 : T poly • ⊕ ( M ) → D poly • ⊕ ( M ) of the action of ( td ( M , Q ) ∇ ) 1 2 ∈ ∏ k ≥ 0 ( Ω k ( M ) ) k on T poly • ⊕ ( M ) (by contraction) with the Hochschild–Kostant–Rosenberg map and (2) preserves the associative algebra structures on the level of cohomology. As an application, we prove the Kontsevich–Shoikhet conjecture: a Kontsevich–Duflo-type theorem holds for all finite-dimensional smooth dg manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we classify b m -Nambu structures via b m -cohomology. The complex of b m -forms is an extension of the De Rham complex, which allows us to consider singular forms. b m -Cohomology is well understood thanks to Scott (2016) [12] , and it can be expressed in terms of the De Rham cohomology of the manifold and of the critical hypersurface using a Mazzeo–Melrose-type formula. Each of the terms in b m -Mazzeo–Melrose formula acquires a geometrical interpretation in this classification. We also give equivariant versions of this classification scheme. [ABSTRACT FROM AUTHOR]

The problem of finding new lower bounds for the degree of a branched covering of a manifold in terms of the cohomology rings of this manifold is considered. This problem is close to M. Gromov’s problem on the domination of manifolds, but it is more delicate. Any branched (finite-sheeted) covering of manifolds is a domination, but not vice versa (even up to homotopy). The theory and applications of the classical notion of the group transfer and of the notion of transfer for branched coverings are developed on the basis of the theory of n-homomorphisms of graded algebras.The main result is a lemma imposing conditions on a relationship between the multiplicative cohomology structures of the total space and the base of n-sheeted branched coverings of manifolds and, more generally, of Smith-Dold n-fold branched coverings. As a corollary, it is shown that the least degree n of a branched covering of the N-torus TN over the product of k 2-spheres and one (N − 2k)-sphere for N ≥ 4k + 2 satisfies the inequality n ≥ N − 2k, while the Berstein-Edmonds well-known 1978 estimate gives only n ≥ N/(k + 1). [ABSTRACT FROM AUTHOR]

We discuss the question of geometric formality for rationally elliptic manifolds of dimension 6 and 7. We prove that a geometrically formal six-dimensional biquotient with b2 = 3 has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with b2 ≥ 2 and b3 = 0 can not be geometrically formal. As it follows from their real homotopy classification, the seven-dimensional geometrically formal rationally elliptic manifolds have the real cohomology of symmetric spaces as well. [ABSTRACT FROM AUTHOR]

We study conditions under which sub-complexes of a double complex of vector spaces allow to compute the Bott-Chern cohomology. We are especially aimed at studying the Bott-Chern cohomology of special classes of solvmanifolds, namely, complex parallelizable solvmanifolds and solvmanifolds of splitting type. More precisely, we can construct explicit finite-dimensional double complexes that allow to compute the Bott-Chern cohomology of compact quotients of complex Lie groups, respectively, of some Lie groups of the type $$\mathbb {C}^n\ltimes _\varphi N$$ where N is nilpotent. As an application, we compute the Bott-Chern cohomology of the complex parallelizable Nakamura manifold and of the completely solvable Nakamura manifold. In particular, the latter shows that the property of satisfying the $$\partial \overline{\partial }$$ -Lemma is not strongly closed under deformations of the complex structure. [ABSTRACT FROM AUTHOR]

Let X be a resolvable generalized n -manifold and M be a (PL) n -manifold, where n ≥ 6 . We claim every resolution from M to X can be approximated, arbitrarily closely, by cell-like maps with ( n − 3 ) -dimensional point preimages. The key instrument of the proof is the disjoint Pontryagin disks property. [ABSTRACT FROM AUTHOR]

The authors compute the (rational) Betti number of real toric varieties associated to Weyl chambers of type B, and furthermore show that their integral cohomology is p-torsion free for all odd primes p. [ABSTRACT FROM AUTHOR]

We give explicit formulas for the Betti numbers, both stable and unstable, of the unordered configuration spaces of an arbitrary surface of finite type. [ABSTRACT FROM AUTHOR]

We introduce a property of compact complex manifolds under which the existence of balanced metric is stable by small deformations of the complex structure. This property, which is weaker than the ∂ ∂ ‾ -Lemma, is characterized in terms of the strongly Gauduchon cone and of the first ∂ ∂ ‾ -degree measuring the difference of Aeppli and Bott–Chern cohomologies with respect to the Betti number b 1 . [ABSTRACT FROM AUTHOR]

We introduce a property of compact complex manifolds under which the existence of balanced metric is stable by small deformations of the complex structure. This property, which is weaker than the ∂ ∂ ‾ -Lemma, is characterized in terms of the strongly Gauduchon cone and of the first ∂ ∂ ‾ -degree measuring the difference of Aeppli and Bott–Chern cohomologies with respect to the Betti number b 1 . [ABSTRACT FROM AUTHOR]

The classification of Seifert manifolds was given in terms of numeric data by Seifert (1933), and then generalized by Raymond (1968) and Orlik and Raymond (1968) to circle actions on closed 3d manifolds. In this paper, we further generalize the classification to circle actions on 3d manifolds with boundaries by adding a numeric parameter and a graph of cycles. Then, we describe the rational equivariant cohomology of 3d manifolds with circle actions in terms of ring, module and vector-space structures. We also compute equivariant Betti numbers and Poincaré series for these manifolds and discuss the equivariant formality. [ABSTRACT FROM AUTHOR]

We use the mapping cone for the relative deRham cohomology of a manifold with boundary in order to show that the Chern–Gauss–Bonnet Theorem for oriented Riemannian vector bundles over such manifolds is a manifestation of Lefschetz Duality in any of the two embodiments of the latter. We explain how Thom isomorphism fits into this picture, complementing thus the classical results about Thom forms with compact support. When the rank is odd , we construct, by using secondary transgression forms introduced here, a new closed pair of forms on the disk bundle associated to a vector bundle, pair which is Lefschetz dual to the zero section. [ABSTRACT FROM AUTHOR]

We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree n of compact congruence p-dimensional hyperbolic manifolds "of simple type" as long as n is strictly smaller than p/3. We also prove that for connected Shimura varieties associated to O(p,2) the Hodge conjecture is true for classes of degree < p+1/3. The proof of our general theorem makes use of the recent endoscopic classification of automorphic representations of orthogonal groups by [6]. As such our results are conditional on the hypothesis made in this article, whose proofs have only appeared in preprint form so far; see the second paragraph of Section 1.3.1 below. [ABSTRACT FROM AUTHOR]

A differential characteristic class of a geometric quantity (e.g., Riemannian or Kähler metric, connection, etc.) on a smooth manifold is a closed differential form whose components are expressed in the components of the given geometric quantity and in their partial derivatives in local coordinates via algebraic formulas independent of the choice of coordinates, and whose cohomology class is stable under deformations of the given quantity. In this note, we present a short proof of the theorem of P. Gilkey on characteristic classes of Riemannian metrics, which is based on the method of invariant-theoretic reduction developed by P. I. Katsylo and D. A. Timashev, and generalize this result to Kähler metrics and connections. [ABSTRACT FROM AUTHOR]

Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M. Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida's formula on -invariants in a -extension of -fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa's second proof, with use of -adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched -cover as an inverse system of cyclic branched -covers of 3-manifolds, generalize the Iwasawa type formula, and compute the Tate cohomology of 2-cycles explicitly. [ABSTRACT FROM AUTHOR]

With any g -manifold M are associated two dglas tot ( Λ • g ∨ ⊗ k T poly • ( M ) ) and tot ( Λ • g ∨ ⊗ k D poly • ( M ) ) , whose cohomologies H CE • ( g , T poly • ( M ) → 0 T poly • + 1 ( M ) ) and H CE • ( g , D poly • ( M ) → d H D poly • + 1 ( M ) ) are Gerstenhaber algebras. We establish a formality theorem for g -manifolds: there exists an L ∞ quasi-isomorphism Φ : tot ( Λ • g ∨ ⊗ k T poly • ( M ) ) → tot ( Λ • g ∨ ⊗ k D poly • ( M ) ) whose first ‘Taylor coefficient’ (1) is equal to the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd cocycle of the g -manifold M , and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the g -manifold M is an isomorphism of Gerstenhaber algebras from H CE • ( g , T poly • ( M ) → 0 T poly • + 1 ( M ) ) to H CE • ( g , D poly • ( M ) → d H D poly • + 1 ( M ) ) . [ABSTRACT FROM AUTHOR]

We show that under Ricci curvature integral assumptions the dimension of the first cohomology group can be estimated in terms of the Kato constant of the negative part of the Ricci curvature. Moreover, this provides quantitative statements about the cohomology group, expanding results by Elworthy and Rosenberg. [ABSTRACT FROM AUTHOR]

We show that the group cohomology of torsion-free virtually polycyclic groups and the continuous cohomology of simply connected solvable Lie groups can be computed by the rational cohomology of algebraic groups. Our results are generalizations of certain results on the cohomology of solvmanifolds and infra-solvmanifolds. Moreover as an application of our results, we give a new proof of the surprising cohomology vanishing theorem given by Dekimpe-Igodt. [ABSTRACT FROM AUTHOR]

On a projective complex manifold, the Abelian group of divisors maps surjectively onto that of holomorphic line bundles (the Picard group). On a G 2 -manifold we use coassociative submanifolds to define an analogue of the divisors, and a gauge theoretical equation for a connection on a gerbe to define an analogue of the Picard group. Then, we construct a map from the former to the later. We also prove that the canonical map from our analogue of the Picard group to the third cohomology group with integer coefficients is surjective. As a side remark we make an observation relating the topological type of coassociative submanifolds and the cohomology classes they represent. [ABSTRACT FROM AUTHOR]

For an irreducible orientable compact 3-manifold N with empty or incompressible toral boundary, the full $$L^2$$ -Alexander torsion $$\tau ^{(2)}(N,\phi )(t)$$ associated to any real first cohomology class $$\phi $$ of N is represented by a function of a positive real variable t. The paper shows that $$\tau ^{(2)}(N,\phi )$$ is continuous, everywhere positive, and asymptotically monomial in both ends. Moreover, the degree of $$\tau ^{(2)}(N,\phi )$$ equals the Thurston norm of $$\phi $$ . The result confirms a conjecture of J. Dubois, S. Friedl, and W. Lück and addresses a question of W. Li and W. Zhang. Associated to any admissible homomorphism $$\gamma :\pi _1(N)\rightarrow G$$ , the $$L^2$$ -Alexander torsion $$\tau ^{(2)}(N,\gamma ,\phi )$$ is shown to be continuous and everywhere positive provided that G is residually finite and $$(N,\gamma )$$ is weakly acyclic. In this case, a generalized degree can be assigned to $$\tau ^{(2)}(N,\gamma ,\phi )$$ . Moreover, the generalized degree is bounded by the Thurston norm of $$\phi $$ . [ABSTRACT FROM AUTHOR]