Rational Cohomology Research Articles

On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms

Let $$\mathrm {R}$$ be a real closed field. We prove that for any fixed d, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $$\mathrm {R}^k$$ defined by polynomials of degrees bounded by d vanishes in dimensions d and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of $$d^{O(d)} s^d k^{\lfloor d/2 \rfloor -1} $$ on the equivariant Betti numbers of closed symmetric semi-algebraic subsets of $$\mathrm {R}^k$$ defined by quantifier-free formulas involving s symmetric polynomials of degrees bounded by d, where $$1 < d \ll s,k$$ . This bound is tight up to a factor depending only on d. These results significantly improve upon those obtained previously in Basu and Riener (Adv Math 305:803–855, 2017) which were proved using different techniques. Our new methods are quite general, and also yield bounds on the equivariant Betti numbers of certain special classes of symmetric definable sets (definable sets symmetrized by pulling back under symmetric polynomial maps of fixed degree) in arbitrary o-minimal structures over $$\mathrm {R}$$ . Finally, we utilize our new approach to obtain an algorithm with polynomially bounded complexity for computing these equivariant Betti numbers. In contrast, the problem of computing the ordinary Betti numbers of (not necessarily symmetric) semi-algebraic sets is considered to be an intractable problem, and all known algorithms for this problem have doubly exponential complexity.

The Galois group of the category of mixed Hodge–Tate structures

The category of rational mixed Hodge-Tate structures is a mixed Tate category. So thanks to the Tannakian formalism, it is equivalent to the category of finite dimensional graded comodules over a graded commutative Hopf algebra H over Q. Since the category has homological dimension 1, the Hopf algebra H is isomorphic to the commutative graded Hopf algebra provided by the tensor algebra of the graded vector space given by the direct sum of the groups C/Q(n) over n>0. However this isomorphism is not natural in any sense, e.g. does not work in families. We give a natural explicit construction of the Hopf algebra H. Generalizing this, we define a Hopf dg-algebra related to any dg-algebra R over a field k, equipped with an invertible line k(1). When R is the sheaf of algebras given by the holomorphic de Rham complex of a complex manifold X, and the line is Q(1), the related Hopf dg-algebra describes a dg-model of the derived category of variations of Hodge-Tate structures on X. Its cobar complex provides a dg-model for the rational Deligne cohomology of X. We consider a variant of our construction which starts from Fontaine's crystalline / semi-stable period rings and produces graded / dg Hopf algebras, which we relate to the p-adic Hodge theory.

On an inductive approach to the standard conjecture for a fibred complex variety with strong semistable degeneracies

We prove that Grothendieck's standard conjecture of Lefschetz type on the algebraicity of the operators and of Hodge theory holds for a 4-dimensional smooth projective complex variety fibred over a smooth projective curve provided that every degenerate fibre is a union of smooth irreducible components of multiplicity with normal crossings, the standard conjecture holds for a generic geometric fibre , there is at least one degenerate fibre and the rational cohomology rings and of the irreducible components of every degenerate fibre are generated by classes of algebraic cycles. We obtain similar results for -dimensional fibred varieties with algebraic invariant cycles (defined by the smooth part of the structure morphism ) or with a degenerate fibre all of whose irreducible components possess the property .

Stable cohomology of spaces of non-resultant polynomial systems in ℝ 3

The stabilization of cohomology rings of spaces of non-resultant homogeneous polynomial systems of growing degree in ℝ3 is studied. The rational stable cohomology rings are explicitly calculated, and the instant of stabilization is estimated.

An example using improved Lefschetz duality

A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S4n−1 → S2n ×S m as a module and demonstrate that it is not necessarily determined by the map induced on cohomology by the embedding, nor is it a trivial extension. This demonstrates that the theorem is an improvement on the classical Lefschetz duality.

Classification and characterization of rationally elliptic manifolds in low dimensions

We give a characterization of closed, simply connected, rationally elliptic 6-manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings. We classify rational, real and complex homotopy types of closed, simply connected, rationally elliptic 7-manifolds. We give partial results in dimensions 8 and 9.

A tropical approach to a generalized Hodge conjecture for positive currents

Demailly showed that the Hodge conjecture is equivalent to the statement that any (p,p)-dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents associated to subvarieties, and asked whether any strongly positive (p,p)-dimensional closed current with rational cohomology class can be approximated by positive linear combinations of integration currents associated to subvarieties. Using tropical geometry, we construct a (p,p)-dimensional current on a smooth projective variety that does not satisfy the latter statement.

Representation stability for the cohomology of arrangements associated to root systems

From a root system, one may consider the arrangement of reflecting hyperplanes, as well as its toric and elliptic analogues. The corresponding Weyl group acts on the complement of the arrangement and hence on its cohomology. We consider a sequence of linear, toric, or elliptic arrangements which arise from a family of root systems of type A, B, C, or D, and we show that the rational cohomology stabilizes as a sequence of Weyl group representations. Our techniques combine a Leray spectral sequence argument similar to that of Church in the type A case along with FI$_W$-module theory which Wilson developed and used in the linear case. A key to the proof relies on a combinatorial description, using labelled partitions, of the poset of connected components of intersections of subvarieties in the arrangement.

Simplicial Abel-Jacobi maps and reciprocity laws

We describe an explicit morphism of complexes that induces the cycle-class maps from (simplicially described) higher Chow groups to rational Deligne cohomology. The reciprocity laws satisfied by the currents we introduce for this purpose are shown to provide a clarifying perspective on functional equations satisfied by complex-valued di- and trilogarithms.

The stable cohomology of the Satake compactification of 𝒜g

Charney and Lee have shown that the rational cohomology of the Satake-Baily-Borel compactification the moduli space of principally polarized abelian varieties of dimension g stabilizes as g grows and they computed this stable cohomology as a Hopf algebra. We give a relatively simple algebro-geometric proof of their theorem that also takes into account the mixed Hodge structure that is present here. We find the latter to be impure.

On the Chow ring of certain rational cohomology tori

Let f:X→A be an abelian cover from a complex algebraic variety with quotient singularities to an abelian variety. We show that f⁎ induces an isomorphism between the rational cohomology rings H•(A,Q) and H•(X,Q) if and only if f⁎ induces an isomorphism between the Chow rings with rational coefficients CH•(A)Q and CH•(X)Q.

Rational cohomology tori

We study normal compact K\"ahler spaces whose rational cohomology ring is isomorphic to that of a complex torus. We call them rational cohomology tori. We classify, up to dimension three, those with rational singularities. We then give constraints on the degree of the Albanese morphism and the number of simple factors of the Albanese variety for rational cohomology tori of general type (hence projective) with rational singularities. Their properties are related to the birational geometry of smooth projective varieties of general type, maximal Albanese dimension, and with vanishing holomorphic Euler characteristic. We finish with the construction of series of examples.

Cohomology of automorphism groups of free groups with twisted coefficients

We compute the groups H^*(mathrm {Aut}(F_n); M) and H^*(mathrm {Out}(F_n); M) in a stable range, where M is obtained by applying a Schur functor to H_mathbb {Q} or H^*_mathbb {Q}, respectively the first rational homology and cohomology of F_n. The answer may be described in terms of stable multiplicities of irreducibles in the plethysm mathrm {Sym}^k circ mathrm {Sym}^l of symmetric powers. We also compute the stable integral cohomology groups of mathrm {Aut}(F_n) with coefficients in H or H^*.

On The Growth of the Homology of a Free Loop Space II

Controlled exponential growth is a stronger version of exponential growth. We prove that the homology of the free loop space ℒX has controlled exponential growth in two important situations : (1) when X is a connected sum of manifolds whose rational cohomologies are not monogenic, (2) when the rational homotopy Lie algebra L X contains an inert element and ρ(L X )<ρ(L X /[L X ,L X ]), where ρ(V) denotes the radius of convergence of V.

Intersection Spaces, Equivariant Moore Approximation and the Signature

We generalize the first author's construction of intersection spaces to the case of stratified pseudomanifolds of stratification depth 1 with twisted link bundles, assuming that each link possesses an equivariant Moore approximation for a suitable choice of structure group. As a by-product, we find new characteristic classes for fiber bundles admitting such approximations. For trivial bundles and flat bundles whose base has finite fundamental group these classes vanish. For oriented closed pseudomanifolds, we prove that the reduced rational cohomology of the intersection spaces satisfies global Poincare duality across complementary perversities if the characteristic classes vanish. The signature of the intersection spaces agrees with the Novikov signature of the top stratum. As an application, these methods yield new results about the Goresky-MacPherson intersection homology signature of pseudomanifolds. We discuss several nontrivial examples, such as the case of flat bundles and symplectic toric manifolds.

Topological entropy of left-invariant magnetic flows on 2-step nilmanifolds

In this paper, we consider magnetic flows on 2-step nilmanifolds , where the Riemannian metric g and the magnetic field σ are left-invariant. Our first result is that when σ represents a rational cohomology class and its restriction to vanishes on the derived algebra, then the associated magnetic flow has zero topological entropy. In particular, this is the case when σ represents a rational cohomology class and is exact. Our second result is the construction of a magnetic field on a 2-step nilmanifold that has positive topological entropy for arbitrarily high energy levels.

Rational cohomologies of classifying spaces for homogeneous spaces of small rank

Let a simply-connected homogeneous space $${X}$$ satisfy the condition of $${{\rm dim} \pi_{\rm even}(X)\otimes {\mathbb{Q}}=2}$$ and $${{\rm dim} \pi_{\rm odd}(X)\otimes {\mathbb{Q}}=3}$$ (then, we say it is of (2, 3) type), which is the smallest rank in non-formal pure spaces. Then, we compute the Sullivan minimal model of the Dold–Lashof classifying space $${{\rm Baut}_1 X}$$ according to Nishinobu and Yamaguchi (Topol Appl 196:290–307, 2015) and observe whether or not its rational cohomology is a polynomial algebra, which is a necessary condition for the Serre spectral sequence of any fibration over a sphere with fibre $${X}$$ to degenerate at term $${E_2}$$ .

Central theorems for cohomologies of certain solvable groups

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 certian results on the cohomology of solvmanifolds and infra-solvmanifolds. Moreover as an application of our results, we give a simple proof of the surprising cohomology vanishing theorem given by Dekimpe-Igodt.

The spaces of non-contractible closed curves in compact space forms

The rational equivariant cohomology of noncontractible loop spaces is calculated for compact space forms. It is also shown how to use these calculations to establish the existence of closed geodesics. Bibliography: 18 titles.

Rational equivariant cohomology theories with toral support

For an arbitrary compact Lie group G, we describe a model for rational G-spectra with toral geometric isotropy and show that there is a convergent Adams spectral sequence based on it. The contribution from geometric isotropy at a subgroup K of the maximal torus of G is captured by a module over H^*(BW_G(K)_e) with an action of \pi_0(W_G(K)), where W_G(K)=N_G(K)/K and the subscript e denotes the identity component.