Integral Cohomology Research Articles

Cohomology of p-adic Stein spaces

We compute p-adic etale and pro-etale cohomologies of Drinfeld half-spaces. In the pro-etale case, the main input is a comparison theorem for p-adic Stein spaces; the cohomology groups involved here are much bigger than in the case of etale cohomology of algebraic varieties or proper analytic spaces considered in all previous works. In the etale case, the classical p-adic comparison theorems allow us to pass to a computation of integral differential forms cohomologies which can be done because the standard formal models of Drinfeld half-spaces are pro-ordinary and their differential forms are acyclic.

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 of the Grassmannian is a gln-module

The integral singular cohomology ring of the Grassmann variety parametrizing r-dimensional subspaces in the n-dimensional complex vector space is naturally an irreducible representation of the Lie algebra of all the n × n matrices with integral entries. The simplest case, r = 1, recovers the well known fact that any vector space is a module over the Lie algebra of its own endomorphisms. The other extremal case, corresponds to the bosonic vertex representation of the Lie algebra on the polynomial ring in infinitely many indeterminates, due to Date, Jimbo, Kashiwara and Miwa. In the present article we provide the structure of this irreducible representation explicitly, by means of a distinguished Hasse-Schmidt derivation on an exterior algebra, borrowed from Schubert Calculus

ALMOST FIXED POINTS OF FINITE GROUP ACTIONS ON MANIFOLDS WITHOUT ODD COHOMOLOGY

If X is a smooth manifold and $$ \mathcal{G} $$ is a subgroup of Diff(X) we say that (X, $$ \mathcal{G} $$ ) has the almost fixed point property if there exists a number C such that for any finite subgroup G ≤ $$ \mathcal{G} $$ there is some x ∈ X whose stabilizer Gx ≤ $$ \mathcal{G} $$ satisfies [G : Gx] ≤ C. We say that X has no odd cohomology if its integral cohomology is torsion free and supported in even degrees. We prove that if X is compact and possibly with boundary and has no odd cohomology then (X, Diff(X)) has the almost fixed point property. Combining this with a result of Petrie and Randall we conclude that if Z is a non-necessarily compact smooth real affine variety, and Z has no odd cohomology, then (Z, Aut(Z)) has the almost fixed point property, where Aut(Z) is the group of algebraic automorphisms of Z lifting the identity on Spec ℝ. The main ingredients in the proof are: (1) the Jordan property for diffeomorphism groups of compact manifolds with nonzero Euler characteristic, and (2) the study of λ-stability, a condition on actions of finite abelian groups on manifolds that we introduce in this paper.

Two examples of toric arrangements

We show that the integral cohomology algebra of the complement of a toric arrangement is not determined by the poset of layers. Moreover, the rational cohomology algebra is not determine by the arithmetic matroid (however it is determine by the poset of layers).

Chow–Witt rings of classifying spaces for symplectic and special linear groups

We compute the Chow–Witt rings of the classifying spaces for the symplectic and special linear groups. In the structural description we give, contributions from real and complex realization are clearly visible. In particular, the computation of cohomology with I j -coefficients is done closely along the lines of Brown's computation of integral cohomology for special orthogonal groups. The computations for the symplectic groups show that Chow–Witt groups are a symplectically oriented ring cohomology theory. Using our computations for special linear groups, we also discuss the question when an oriented vector bundle of odd-rank splits off a trivial summand.

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.

Hamiltonian circle actions with fixed point set almost minimal

Motivated by recent works on Hamiltonian circle actions satisfying certain minimal conditions, in this paper, we consider Hamiltonian circle actions satisfying an almost minimal condition. More precisely, we consider a compact symplectic manifold $(M, \omega)$ admitting a Hamiltonian circle action with fixed point set consisting of two connected components $X$ and $Y$ satisfying $\dim(X)+\dim(Y)=\dim(M)$. Under certain cohomology conditions, we determine the circle action, the integral cohomology rings of $M$, $X$ and $Y$, and the total Chern classes of $M$, $X$, $Y$, and of the normal bundles of $X$ and $Y$. The results show that these data are unique --- they are exactly the same as those in the standard example $\Gt_2(\R^{2n+2})$, the Grassmannian of oriented $2$-planes in $\R^{2n+2}$, which is of dimension $4n$ with (any) $n\in\N$, equipped with a standard circle action. Moreover, if $M$ is K\"ahler and the action is holomorphic, we can use a few different criteria to claim that $M$ is $S^1$-equivariantly biholomorphic and $S^1$-equivariantly symplectomorphic to $\Gt_2(\R^{2n+2})$.

An n-dimensional Klein bottle

AbstractAn n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.

On Integral Cohomology of Certain Orbifolds

AbstractThe CW-complex structure of certain spaces, such as effective orbifolds, can be too complicated for computational purposes. In this paper we develop the concept of $\mathbf{q}$-CW complex structure on an orbifold, to detect torsion in its integral cohomology. The main result can be applied to well-known classes of orbifolds or algebraic varieties having orbifold singularities, such as toric orbifolds, simplicial toric varieties, torus orbifolds, and weighted Grassmannians.

The Mod Two Cohomology of the Moduli Space of Rank Two Stable Bundles on a Surface and Skew Schur Polynomials

AbstractWe compute cup-product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping-class group action.

The classification of 2‐connected 7‐manifolds

We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first author's thesis. The main additional ingredient is an extension of the Eells-Kuiper invariant for any closed spin 7-manifold, regardless of whether the spin characteristic class p_M in the fourth integral cohomology of M is torsion. In addition we determine the inertia group of 2-connected M - equivalently the number of oriented smooth structures on the underlying topological manifold - in terms of p_M and the torsion linking form.

Hecke modules for arithmetic groups via bivariant K-theory

Let $\Gamma$ be a lattice in a locally compact group $G$. In earlier work, we used $KK$-theory to equip the $K$-groups of any $\Gamma$-$C^{*}$-algebra on which the commensurator of $\Gamma$ acts with Hecke operators. When $\Gamma$ is arithmetic, this gives Hecke operators on the $K$-theory of certain $C^{*}$-algebras that are naturally associated with $\Gamma$. In this paper, we first study the topological $K$-theory of the arithmetic manifold associated to $\Gamma$. We prove that the Chern character commutes with Hecke operators. Afterwards, we show that the Shimura product of double cosets naturally corresponds to the Kasparov product and thus that the $KK$-groups associated to an arithmetic group $\Gamma$ become true Hecke modules. We conclude by discussing Hecke equivariant maps in $KK$-theory in great generality and apply this to the Borel-Serre compactification as well as various noncommutative compactifications associated with $\Gamma$. Along the way we discuss the relation between the $K$-theory and the integral cohomology of low-dimensional manifolds as Hecke modules.

Equivariant cohomology Chern numbers determine equivariant unitary bordism for torus groups

This paper shows that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary $G$-manifolds, which gives an affirmative answer to the conjecture posed by Guillemin--Ginzburg--Karshon in [20, Remark H.5, $\S3$, Appendix H], where $G$ is a torus. As a further application, we also obtain a satisfactory solution of [20, Question (A), $\S1.1$, Appendix H] on unitary Hamiltonian $G$-manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber--Panov--Ray and the Kronecker pairing of bordism and cobordism. Our approach heavily exploits Quillen's geometric interpretation of homotopic unitary cobordism theory. Moreover, this method can also be applied to the study of $({\Bbb Z}_2)^k$-equivariant unoriented bordism and can still derive the classical result of tom Dieck.

Vanishing of Symplectic Homology and Obstruction to Flexible Fillability

Abstract For any asymptotically dynamically convex contact manifold $Y$, we show that $SH_{\ast }(W)=0$ is a property independent of the choice of topologically simple (i.e., $c_1(W)=0$ and $\pi _{1}(Y)\rightarrow \pi _1(W)$ is injective) Liouville filling $W$. In particular, if $Y$ is the boundary of a flexible Weinstein domain, then any topologically simple Liouville filling $W$ has vanishing symplectic homology. As a consequence, we answer a question of Lazarev partially: a contact manifold $Y$ admitting flexible fillings determines the integral cohomology of all the topologically simple Liouville fillings of $Y$. The vanishing result provides an obstruction to flexible fillability. As an application, we show that all Brieskorn manifolds of dimension $\ge 5$ cannot be filled by flexible Weinstein manifolds.

Infinite families of equivariantly formal toric orbifolds

Abstract The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and more generally, polyhedral products. In this paper we extend the analysis to include toric orbifolds. Our main results yield infinite families of toric orbifolds, derived from a given one, whose integral cohomology is free of torsion and is concentrated in even degrees, a property which might be termed integrally equivariantly formal. In all cases, it is possible to give a description of the cohomology ring and to relate it to the cohomology of the original orbifold.

H4(Co0; Z) = Z/24

AbstractWe show that the 4th integral cohomology of Conway’s group $\mathrm{Co}_0$ is a cyclic group of order $24$, generated by the 1st fractional Pontryagin class of the $24$-dimensional representation.

Integral cohomology of the generalized Kummer fourfold

Integral cohomology of the generalized Kummer fourfold

A refinement of Betti numbers and homology in the presence of a continuous function, II: The case of an angle-valued map

For a continuous angle-valued map defined on a compact ANR, a fixed field and any degree one proposes a refinement of the Novikov-Betti number and of the Novikov homology of the pair consisting of the ANR and the degree one integral cohomology class represented by the map. For each degree the first refinement is a configuration of points with multiplicity located in the punctured complex plane whose total cardinality is the Novikov-Betti number of the pair. The second refinement is a configuration of submodules of the Novikov homology whose direct sum is isomorphic to the Novikov homology and which has the same support as the first configuration. When the field is a the field of complex numbers the second configuration is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the L2-homology of the infinite cyclic cover of the ANR defined by the angle-valued map. One discusses the properties of these configurations namely, robustness with respect to continuous perturbation of the angle-valued map and the Poincar\'e Duality and one derives some computational applications in topology. The main results parallel the results for the case of real-valued map but with Novikov homology and Novikov-Betti numbers replacing standard homology and standard Betti numbers.

Causality and Legendrian linking for higher dimensional spacetimes

Let ( X m + 1 , g ) be an ( m + 1 ) -dimensional globally hyperbolic spacetime with Cauchy surface M m , and let M ˜ m be the universal cover of the Cauchy surface. Let N X be the contact manifold of all future directed unparameterized light rays in X that we identify with the spherical cotangent bundle S T ∗ M . Jointly with Stefan Nemirovski we showed that when M ˜ m is not a compact manifold, then two points x , y ∈ X are causally related if and only if the Legendrian spheres S x , S y of all light rays through x and y are linked in N X . In this short note we use the contact Bott–Samelson theorem of Frauenfelder, Labrousse and Schlenk to show that the same statement is true for all X for which the integral cohomology ring of a closed M ˜ is not the one of the CROSS (compact rank one symmetric space). If M admits a Riemann metric g ¯ , a point x and a number ℓ > 0 such that all unit speed geodesics starting from x return back to x in time ℓ , then ( M , g ¯ ) is called a Y ℓ x manifold. Jointly with Stefan Nemirovski we observed that causality in ( M × R , g ¯ ⊕ − t 2 ) is not equivalent to Legendrian linking. Every Y ℓ x -Riemann manifold has compact universal cover and its integral cohomology ring is the one of a CROSS. So we conjecture that Legendrian linking is equivalent to causality if and only if one can not put a Y ℓ x Riemann metric on a Cauchy surface M .