Integral Cohomology Groups Research Articles

Cohomology of a real toric variety and shellability of posets arising from a graph

AbstractGiven a graph G without loops, the pseudograph associahedron PG is a smooth polytope, so there is a projective smooth toric variety XG corresponding to PG. Taking the real locus of XG, we have the projective smooth real toric variety $X^{\mathbb{R}}_G$. The integral cohomology groups of $X^{\mathbb{R}}_G$ can be computed by studying the topology of certain posets of even subgraphs of G; such a poset is neither pure nor shellable in general. We completely characterize the graphs whose posets of even subgraphs are always shellable. It follows that we get a family of projective smooth real toric varieties whose integral cohomology groups are torsion-free or have only 2-torsion.

Torsion in cohomology and dimensional reduction

Conventional wisdom dictates that ℤN factors in the integral cohomology group Hp(Xn, ℤ) of a compact manifold Xn cannot be computed via smooth p-forms. We revisit this lore in light of the dimensional reduction of string theory on Xn, endowed with a G-structure metric that leads to a supersymmetric EFT. If massive p-form eigenmodes of the Laplacian enter the EFT, then torsion cycles coupling to them will have a non-trivial smeared delta form, that is an EFT long-wavelength description of p-form currents of the (n − p)-cycles of Xn. We conjecture that, whenever torsion cycles are calibrated, their linking number can be computed via their smeared delta forms. From the EFT viewpoint, a torsion factor in cohomology corresponds to a ℤN gauge symmetry realised by a Stückelberg-like action, and calibrated torsion cycles to BPS objects that source the massive fields involved in it.

Surfaces with $$c_1^2 =9$$ and $$\chi =5$$ whose canonical classes are divisible by 3

We shall study minimal complex surfaces with $c^2 = 9$ and $\chi=5$ whose canonical classes are divisible by $3$ in the integral cohomology groups, where $c_1^2$ and $\chi$ denote the first Chern number of an algebraic surface and the Euler characteristic of the structure sheaf, respectively. The main results are a structure theorem for such surfaces, the unirationality of the moduli space, and a description of the behavior of the canonical map. As a byproduct, we shall also rule out a certain case mentioned in a paper by Ciliberto--Francia--Mendes Lopes. Since the irregularity $q$ vanishes for our surfaces, our surfaces have geometric genus $p_g = 4$.

Nonsymplectic automorphisms of prime order on O’Grady’s sixfolds

We classify nonsymplectic automorphisms of prime order on irreducible holomorphic symplectic manifolds of O’Grady’s 6-dimensional deformation type. More precisely, we give a classification of the invariant and coinvariant sublattices of the second integral cohomology group.

Elliptic genera of level N for complete intersections

We focus on the elliptic genera of level N at the cusps of a congruence subgroup for any complete intersection. Writing the first Chern class of a complete intersection as a product of an integral coefficient c1 and a generator of the 2nd integral cohomology group, we mainly discuss the values of the elliptic genera of level N for the complete intersection in the cases of c1 >, =, or < 0. In particular, the values about the Todd genus, Â-genus, and Ak-genus can be derived from the elliptic genera of level N.

Integral Cohomology Groups of Real Toric Manifolds and Small Covers

For a simplicial complex $K$ with $m$ vertices, there is a canonical $\mathbb Z_2^m$-space known as a real moment angle complex $\mathbb R \mathcal Z_K$. In this paper, we consider the quotient spaces $Y=\mathbb R \mathcal Z_K / \mathbb Z_2^{k}$, where $K$ is a pure shellable complex and $\mathbb Z_2^k \subset \mathbb Z_2^m$ is a maximal free action on $\mathbb R \mathcal Z_K$. A typical example of such spaces is a small cover, where a small cover is known as a topological analog of a real toric manifold. We compute the integral cohomology group of $Y$ by using the PL cell decomposition obtained from a shelling of $K$. In addition, we compute the Bockstein spectral sequence of $Y$ explicitly.

On the Cohomology Groups of Real Lagrangians in Calabi–Yau Threefolds

The quintic threefold X is the most studied Calabi-Yau 3-fold in the mathematics literature. In this article, using Čech-to-derived spectral sequences, we investigate the mod 2 and integral cohomology groups of a real Lagrangian , obtained as the fixed locus of an anti-symplectic involution in the mirror to X. We show that is the disjoint union of a 3-sphere and a rational homology sphere. Analyzing the mod 2 cohomology further, we deduce a correspondence between the mod 2 Betti numbers of and certain counts of integral points on the base of a singular torus fibration on X. By work of Batyrev, this identifies the mod 2 Betti numbers of with certain Hodge numbers of X. Furthermore, we show that the integral cohomology groups of are 2-primary for , we conjecture that this holds in much greater generality.

The First Integral Cohomology of Pure Mapping Class Groups

AbstractIt is a classical result that pure mapping class groups of connected, orientable surfaces of finite type and genus at least 3 are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface’s simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.

Vanishing of degree 3 cohomological invariants

For a complex algebraic variety X, we show that triviality of the degree three unramified cohomology H0(X,H3) (occurring on the second page of the Bloch-Ogus spectral sequence [1]) follows from a condition on the integral Chow group CH2X and the integral cohomology group H3(X,Z). In the case that X is an appropriate approximation to the classifying stack BG of a finite p-group G, this result states that the group G has no degree three cohomological invariants. As a corollary we show that the nonabelian groups of order p3 for odd prime p have no degree three cohomological invariants.

The Cohomology Groups of Real Toric Varieties Associated with Weyl Chambers of TypesCandD

AbstractGiven a root system, the Weyl chambers in the co-weight lattice give rise to a real toric variety, called the real toric variety associated with the Weyl chambers. We compute the integral cohomology groups of real toric varieties associated with the Weyl chambers of typeCnandDn, completing the computation for all classical types.

On the integral cohomology ring of toric orbifolds and singular toric varieties

We examine the integral cohomology rings of certain families of $2n$-dimensional orbifolds $X$ that are equipped with a well-behaved action of the $n$-dimensional real torus. These orbifolds arise from two distinct but closely related combinatorial sources, namely from characteristic pairs $(Q,\lambda)$, where $Q$ is a simple convex $n$-polytope and $\lambda$ a labelling of its facets, and from $n$-dimensional fans $\Sigma$. In the literature, they are referred as toric orbifolds and singular toric varieties respectively. Our first main result provides combinatorial conditions on $(Q,\lambda)$ or on $\Sigma$ which ensure that the integral cohomology groups $H^{\ast}(X)$ of the associated orbifolds are concentrated in even degrees. Our second main result assumes these condition to be true, and expresses the graded ring $H^*(X)$ as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. Also, we compute several examples.

Representation theory and the cycle class map of a classifying space

We compute the Chern subgroup of the 4-th integral cohomology group of a certain classifying space and show that it is a proper subgroup. Such a classifying space gives us new counterexamples for the integral Hodge and Tate conjectures modulo torsion.

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^*.

Homology of some surfaces with isogenous to a product

Bauer and Catanese found four families of surfaces of general type with , each of which is the quotient of a product of curves by the action of a finite Abelian group. We compute the integral cohomology groups of these surfaces.

Integral Monsky-Washnitzer cohomology and the overconvergent de Rham-Witt complex

In their paper which introduced Monsky-Washnitzer cohomology, Monsky and Washnitzer described conditions under which the definition can be adapted to give integral cohomology groups. It seems to be well-known among experts that their construction always gives well-defined integral cohomology groups, but this fact also does not appear to be explicitly written down anywhere. In this paper, we prove that the integral Monsky-Washnitzer cohomology groups are well-defined, for any nonsingular affine variety over a perfect field of characteristic p. We then compare these cohomology groups with overconvergent de RhamWitt cohomology. It was shown earlier that if the affine variety has small dimension relative to the characteristic of the ground field, then the cohomology groups are isomorphic. We extend this result to show that for any nonsingular affine variety, regardless of dimension, we have an isomorphism between integral Monsky-Washnitzer cohomology and overconvergent de Rham-Witt cohomology in degrees which are small relative to the characteristic.

Homology computations for complex braid groups

Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincaré polynomial with coefficients in a finite field for one large series of such groups, and compute the second integral cohomology group for all of them. As a consequence we get non-isomorphism results for these groups.

Universal twist in equivariant K -theory for proper and discrete actions

We define equivariant projective unitary stable bundles as the appropriate twists when defining K-theory as sections of bundles with fibers the space of Fredholm operators over a Hilbert space. We construct universal equivariant projective unitary stable bundles for the orbit types, and we use a specific model for these local universal spaces in order to glue them to obtain a universal equivariant projective unitary stable bundle for discrete and proper actions. We determine the homotopy type of the universal equivariant projective unitary stable bundle, and we show that the isomorphism classes of equivariant projective unitary stable bundles are classified by the third equivariant integral cohomology group. The results contained in this paper extend and generalize results of Atiyah–Segal.

The Picard group of the moduli space of curves with level structures

For $4 \nmid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures. In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level $L$ subgroup of the mapping class group (in a previous paper, the author determined this rationally). This entails calculating the abelianization of the level $L$ subgroup of the mapping class group, generalizing previous results of Perron, Sato, and the author. Finally, along the way we calculate the first homology group of the mod $L$ symplectic group with coefficients in the adjoint representation.

On Leading Terms and Values of Equivariant Motivic L-functions

The ‘Equivariant Tamagawa Number Conjecture’ formulated by Flach and the present author is a natural refinement of the seminal Tamagawa Number Conjecture that was originally formulated by Bloch and Kato and then subsequently extended and refined by Kato and by Fontaine and Perrin-Riou. We explain how the additional information implicit in this refinement entails a variety of explicit predictions concerning both the leading terms and values of equivariant motivic L-functions and the module structure of integral motivic cohomology groups. We also discuss several concrete applications of this approach in the setting of Artin and Hasse-Weil L-functions.

Functional equations of the dilogarithm in motivic cohomology

We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH 2 ( F , 3 ) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generating the integral motivic cohomology groups of some number fields.