Gysin Sequence Research Articles

On the cohomology of reciprocity sheaves

AbstractIn this paper, we show the existence of an action of Chow correspondences on the cohomology of reciprocity sheaves. In order to do so, we prove a number of structural results, such as a projective bundle formula, a blow-up formula, a Gysin sequence and the existence of proper pushforward. In this way, we recover and generalise analogous statements for the cohomology of Hodge sheaves and Hodge-Witt sheaves.We give several applications of the general theory to problems which have been classically studied. Among these applications, we construct new birational invariants of smooth projective varieties and obstructions to the existence of zero cycles of degree 1 from the cohomology of reciprocity sheaves.

A monotone Lagrangian casebook

We present an array of new calculations in Lagrangian Floer theory which demonstrate observations relating to symplectic reduction, grading periodicity, and the closed-open map. We also illustrate Perutz's symplectic Gysin sequence and the quilt theory of Wehrheim and Woodward.

Gysin sequences and SU(2)‐symmetries of C∗‐algebras

Motivated by the study of symmetries of C*-algebras, as well as by multivariate operator theory, we introduce the notion of an SU(2)-equivariant subproduct system of Hilbert spaces. We analyse the resulting Toeplitz and Cuntz-Pimsner algebras and provide results about their topological invariants through Kasparov's bivariant K-theory. In particular, starting from an irreducible representation of SU(2), we show that the corresponding Toeplitz algebra is equivariantly KK-equivalent to the algebra of complex numbers. In this way, we obtain a six term exact sequence of K-groups containing a noncommutative analogue of the Euler class.

Mod-two cohomology rings of alternating groups

Abstract We calculate the direct sum of the mod-two cohomology of all alternating groups, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. A range of techniques are developed, including an almost Hopf ring structure associated to the embeddings of products of alternating groups and Fox–Neuwirth resolutions, which are new techniques. We also extend understanding of the Gysin sequence relating the cohomology of alternating groups to that of symmetric groups and calculation of restriction to elementary abelian subgroups.

Hecke operators in $KK$-theory and the $K$-homology of Bianchi groups

Let \Gamma be a torsion-free arithmetic group acting on its associated global symmetric space X . Assume that X is of non-compact type and let \Gamma act on the geodesic boundary \partial X of X . Via general constructions in KK -theory, we endow the K -groups of the arithmetic manifold X / \Gamma , of the reduced group C^* -algebra C^*_r(\Gamma) and of the boundary crossed product algebra C(\partial X) \rtimes\Gamma with Hecke operators. In the case when \Gamma is a group of real hyperbolic isometries, the K -theory and K -homology groups of these C^{*} -algebras are related by a Gysin six-term exact sequence and we prove that this Gysin sequence is Hecke equivariant. Finally, when \Gamma is a Bianchi group, we assign explicit unbounded Fredholm modules (i.e. spectral triples) to (co)homology classes, inducing Hecke-equivariant isomorphisms between the integral cohomology of \Gamma and each of these K -groups. Our methods apply to case \Gamma \subset \mathbf {PSL}(\mathbf Z) as well. In particular we employ the unbounded Kasparov product to push the Dirac operator an embedded surface in the Borel–Serre compactification of \mathbf H/\Gamma to a spectral triple on the purely infinite geodesic boundary crossed product algebra C(\partial \mathbf H) \rtimes\Gamma .

Topological T-duality for stacks using a Gysin sequence

In this paper we study the topological T-dual of spaces with a non-free circle action mainly using the stack theory method of Bunke and co-workers \cite{Bunke1}. We first compare three formalisms for obtaining the Topological T-dual of a semi-free $S^1$-space in a simple example. Then, we calculate the T-dual of general KK-monopole backgrounds using the stack theory method. We define the dyonic coordinate for these backgrounds. We introduce an approach to Topological T-duality using classifying spaces which simultaneously generalizes the methods of Bunke et al \cite{Bunke1} and Mathai and Wu \cite{MaWu}. Then, we define a cohomology Gysin sequence for prinicpal bundles of stacks and describe an application to Topological T-duality for stacks. We apply the above to calculate the Topological T-dual of a general compact three-manifold with an {\em arbitrary} smooth circle action. We point out a possible application of these T-duals to higher-dimensional black holes.

Pimsner algebras and Gysin sequences from principal circle actions

A self Morita equivalence over an algebra B , given by a B -bimodule E , is thought of as a line bundle over B . The corresponding Pimsner algebra \mathcal O_E is then the total space algebra of a noncommutative principal circle bundle over B . A natural Gysin-like sequence relates the KK -theories of \mathcal O_E and of B . Interesting examples come from \mathcal O_E a quantum lens space over B a quantum weighted projective line (with arbitrary weights). The KK -theory of these spaces is explicitly computed and natural generators are exhibited.

The Gysin sequence for quantum lens spaces

We define quantum lens spaces as `direct sums of line bundles' and exhibit them as `total spaces' of certain principal bundles over quantum projective spaces. For each of these quantum lens spaces we construct an analogue of the classical Gysin sequence in K-theory. We use the sequence to compute the K-theory of the quantum lens spaces, in particular to give explicit geometric representatives of their K-theory classes. These representatives are interpreted as `line bundles' over quantum lens spaces and generically define `torsion classes'. We work out explicit examples of these classes.

Coincidences of Fibrewise Maps Between Sphere Bundles Over the Circle

AbstractWhen can two fibrewise maps be deformed in a fibrewise fashion until they are coincidence free? In order to get a thorough understanding of this problem (and, more generally, of minimum numbers that are closely related to it) we study the strength of natural geometric obstructions, such as ω-invariants and Nielsen numbers, as well as the related Nielsen theory.In the setting of sphere bundles, a certain degree map degB turns out to play a decisive role. In many explicit cases it also yields good descriptions of the set ℱ of fibrewise homotopy classes of fibrewise maps. We introduce an addition on ℱ, which is not always single valued but still very helpful. Furthermore, normal bordism Gysin sequences and (iterated) Freudenthal suspensions play a crucial role.

The Gysin sequence for $\mathbb S^{3}$-actions on manifolds

We construct a Gysin sequence associated to any smooth ${\mathbb S}^3$-action on a smooth manifold.

Characteristic classes for GO(2n) in étale cohomology

Let $GO(2n)$ be the general orthogonal group (the group of similitudes) over any algebraically closed field of characteristic $\ne 2$ . We determine the smooth-étale cohomology ring with $\mathbb F_2$ coefficients of the algebraic stack $BGO(2n)$ . In the topological category, Holla and Nitsure determined the singular cohomology ring of the classifying space $BGO(2n)$ of the complex Lie group $GO(2n)$ in terms of explicit generators and relations. We extend their results to the algebraic category. The chief ingredients in this are: (i) an extension to étale cohomology of an idea of Totaro, originally used in the context of Chow groups, which allows us to approximate the classifying stack by quasi projective schemes; and (ii) construction of a Gysin sequence for the ${\mathbb G_m}$ -fibration $BO(2n)\to BGO(2n)$ of algebraic stacks.

Euler characteristics and Gysin sequences for group actions on boundaries

Let G be a locally compact group, let X be a universal proper G-space, and let Z be a G-equivariant compactification of X that is H-equivariantly contractible for each compact subgroup H of G. Let W be the resulting boundary. Assuming the Baum-Connes conjecture for G with coefficients C and C(W), we construct an exact sequence that computes the map on K-theory induced by the embedding of the reduced group C*-algebra of G into the crossed product of G by C(W). This exact sequence involves the equivariant Euler characteristic of X, which we study using an abstract notion of Poincare duality in bivariant K-theory. As a consequence, if G is torsion-free and the Euler characteristic of the orbit space X/G is non-zero, then the unit element of the boundary crossed product is a torsion element whose order is equal to the absolute value of the Euler characteristic of X/G. Furthermore, we get a new proof of a theorem of Lueck and Rosenberg concerning the class of the de Rham operator in equivariant K-homology.

Intersection cohomology of stratified circle actions

For any stratified pseudomanifold $X$ and any action of the unit circle $\mathbb{S}^1$ on $X$ preserving the stratification and the local structure, the orbit space $X/\mathbb{S}^1$ is also a stratified pseudomanifold. For each perversity $\overline{q}$ in $X$ the orbit map $\pi : X/\mathbb{S}^1$ induces a Gysin sequence relating the $\overline{q}$-intersection cohomologies of $X$ and $X/\mathbb{S}^1$. The third term of this sequence can be given by means of a spectral sequence on $X/\mathbb{S}^1 whose second term is the cohomology of the set of fixed points $X^{S^{1}}$ with values on a constructible sheaf.

T-duality for principal torus bundles and dimensionally reduced Gysin sequences

We reexamine the results on the global properties of T-duality for principal circle bundles in the context of a dimensionally reduced Gysin sequence. We will then construct a Gysin sequence for principal torus bundles and examine the consequences. In particular, we will argue that the T-dual of a principal torus bundle with nontrivial H-flux is, in general, a continuous field of noncommutative, nonassociative tori.

Intersection cohomology of [formula omitted]1-actions on pseudomanifolds

For any smooth free action of the unit circle S 1 in a manifold M; the Gysin sequence of M is a long exact sequence relating the DeRham cohomologies of M and its orbit space M S 1. If the action is not free then M S 1 is not a manifold but a stratified pseudomanifold and there is a Gysin sequence relating the DeRham cohomology of M with the intersection cohomology of M S 1. In this work we extend the above statements for any stratified pseudomanifold X of length 1, whenever the action of S 1 preserves the local structure. We give a Gysin sequence relating the intersection cohomologies of X and X S 1 with a third terni G , the Gysin terni; whose cohomology depends on basic cohomological data of two flavors: global data concerns the Euler class induced by the action, local data relates the Gysin terni and the cohomology of the fixed strata with values on a locally trivial presheaf.

T-duality for principal torus bundles

In this paper we study T-duality for principal torus bundles with H-flux. We identify a subset of fluxes which are T-dualizable, and compute both the dual torus bundle as well as the dual H-flux. We briefly discuss the generalized Gysin sequence behind this construction and provide examples both of non T-dualizable and of T-dualizable H-fluxes.

Gysin sequence and Euler class of spherical Lie algebroids

Gysin sequence and Euler class of spherical Lie algebroids

The Euler class for Riemannian flows

Let F be a Riemannian flow on a closed manifold M. The Gysin sequence relating the basic cohomology of F and the de Rham cohomology of M has been constructed for isometric flows in [6]. In this paper, we extend it to the case of Riemannian flows. We also give a geometric characterization of the vanishing of the Euler class similar to the one given in [9].

Complex and real vector bundle monomorphisms

Given two complex vector bundles over a closed smooth manifold, we compare results concerning the existence and the (homotopy) classification of complex vector bundle monomorphisms on one hand and of real ones on the other hand. The two theories are related by transition homomorphisms which turn out to fit into an exact Gysin sequence of normal bordism groups. A detailed study reveals astonishing phenomena, e.g., situations where no complex but infinitely many real monomorphisms exist. Also all possible combinations of finiteness/infiniteness for the following two numbers occur already over products of spheres: 1. (i) the number of complex monomorphisms which become homotopic as real monomorphisms, and 2. (ii) the number of real monomorphisms which are not homotopic to complex ones.

On the homology of double branched covers

If $\pi :\tilde X \to X$ is a double branched cover, with branching set F, we relate ${H_ \ast }(\tilde X:{\mathbb {Z}_2}),{H_ \ast }(X:{\mathbb {Z}_2}),{H_ \ast }(X,F:{\mathbb {Z}_2})$, and ${H_\ast }(F:{\mathbb {Z}_2})$.