On distribution relations of polylogarithmic Eisenstein classes

This paper is devoted to the complete calculation of the additive structure of the 2-torsion of the integral cohomology of the infinite special linear group SL(ℤ) over the ring of integers ℤ. This enables us to determine the best upper bound for the order of the Chern classes of all integral and rational representations of discrete groups.

There is a mysterious connection between the multiple polylogarithms at N-th roots of unity and modular varieties. In this paper we "explain" it in the simplest case of the double logarithm. We introduce an Euler complex data on modular curves. It includes a length two complex on every modular curve. Their second cohomology groups recover the Beilinson-Kato Euler system in K_2 of modular curves. We show that the above connection in the double logarithm case is provided by the specialization at a cusp of the Euler complex data on the modular curve Y_1(N). Furthermore, specializing the Euler complexes at CM points we find new examples of the connection with geometry of modular varieties, this time hyperbolic 3-folds.

In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a negative answer to Thurston’s conjecture. Here counterexamples have been constructed conditional on the fully marked surface theorem. In the second paper, joint with David Gabai, a proof of the fully marked surface theorem is given.

Analytic cycles on complex manifolds

In this paper we construct a Universal chain complex, counting zeros of closed 1-forms on a manifold. The Universal complex is a refinement of the well known Novikov complex; it relates the homotopy type of the manifold, after a suitable noncommutative localization, with the numbers of zeros of different indices which may have closed 1-forms within a given cohomology class. The main theorem of the paper generalizes the result of a joint paper with A. Ranicki, which treats the special case of closed 1-forms having integral cohomology classes. The present paper also describes a number of new inequalities, giving topological lower bounds on the minimum number of zeros of closed 1-forms. In particular, such estimates are provided by the homology of flat line bundles with monodromy described by complex numbers, which are not Dirichlet units.

We prove that a hyper-Kähler fourfold satisfying a mild topological assumption is of K3 [ 2 ] ^{[2]} deformation type. This proves in particular a conjecture of O’Grady stating that hyper-Kähler fourfolds of K3 [ 2 ] ^{[2]} numerical type are of K3 [ 2 ] ^{[2]} deformation type. Our topological assumption concerns the existence of two integral degree-2 cohomology classes satisfying certain numerical intersection conditions. There are two main ingredients in the proof. We first prove a topological version of the statement, by showing that our topological assumption forces the Betti numbers, the Fujiki constant, and the Huybrechts–Riemann–Roch polynomial of the hyper-Kähler fourfold to be the same as those of K3 [ 2 ] ^{[2]} hyper-Kähler fourfolds. The key part of the article is then to prove the hyper-Kähler SYZ conjecture for hyper-Kähler fourfolds for divisor classes satisfying the numerical condition mentioned above.

We show that any integral second cohomology class of a closed manifold Xn, n ≥ 4, admits, as a Poincare dual, a submanifold N such that X ∖ N has a handle decomposition with no handles of index bigger than (n + 1)∕2. In particular, if X is an almost complex manifold of dimension at least 6, the complement can be given a structure of a Stein manifold.

Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let L → X be a complex line bundle. Using the Cartan complex for equivariant cohomology, we give a new proof of a theorem of Hattori and Yoshida which says that the action of G lifts to L if and only if the first Chern class c1(L) of L can be lifted to an integral equivariant cohomology class in H2G(X; ℤ), and that the different lifts of the action are classified by the lifts of c1(L) to H2G(X; ℤ). As a corollary of our method of proof, we prove that, if the action is Hamiltonian and ∇ is a connection on L which is unitary for some metric on L, and which has a G-invariant curvature, then there is a lift of the action to a certain power Ld (where d is independent of L) which leaves fixed the induced metric on Ld and the connection ∇[otimes ]d. This generalises to symplectic geometry a well-known result in geometric invariant theory.

Bill Thurston proved that taut foliations of hyperbolic 3‐manifolds have Euler classes of norm at most one, and conjectured that any integral second cohomology class of norm equal to one is realized as the Euler class of some taut foliation. Recent work of the second author, joint with David Gabai, has produced counterexamples to this conjecture. Since tight contact structures exist whenever taut foliations do and their Euler classes also have norm at most one, it is natural to ask whether the Euler class one conjecture might still be true for tight contact structures. In this paper, we show that the previously constructed counterexamples for Euler classes of taut foliations in Mehdi Yazdi [Acta Math. 225 (2020) no. 2, 313–368] are in fact realized as Euler classes of tight contact structures. This provides some evidence for the Euler class one conjecture for tight contact structures.

Introduction. T he KIiiiieth formula enables one to determine the integral cohomology group of the product XX Y of the spaces X and Y in terms of the integral cohomology groups H(X) anid H(Y). However, this formula does not enable one to determine the multiplicative structure of the cohomology rinig II(XX Y) in terms of the initegral cohomology rings(2) 1l(X) and II(Y). It is niatural to ask the question: Is the integral cohomology ring H1(XX Y) determinied by the integral cohomology rings 11(X) and H(Y)? This questioni is aniswered in the niegative by the following example: Let X1 = Y1 be the union of the real projective plane aind a one-sphere (circle) with onie point in commoni. Let X2 Y2 be a Klein bottle. It is easy to check that the rings 11(X1) and 1I(X2) are isomorphic. However, the rings II(X1 X Y1) and II(X2 X Y2) are not isomorphic. In particular, there is a onedimenisionial cohomology class aiid a three-dimensional cohomology class in Ii(X2X Y2) whose product is a nionizero four-dimenisionial cohomology class. Oii the other hanid, all products of onie-dimenisionial aiid three-dimensional cohomology classes of II(X1 X Y1) are zero. Hence these two cohomology rings caninot be isomorplhic. These assertions will follow readily from the theorems provecl below. Sinice the aniswerto the above questioni is niegative, it is niatural to inquire: What iniformation about the colhomology rinigs of X aiid Y is needed to determinie the integral cohomology rinig of XX Y? Let II(X, n) denote the cohomology rinig of X with the integers modulo n as coefficienits. Following J. H. C. Whitehlead [8], and Bocksteini [2], we define the spectrum of coho. mology rings of X, or simiiply the cohomology spectrum of X, to be the set of cohomology rinigs IH(X, n), n_()0, together with the coefficient homomorphisms: II(X, n)-*II(X, m), m>O and the Bocksteini homomorphisms of degree +1: II(X, m)--IH(X, 0), m>O. (These homomorphisms are defined in sectioni (1).) Bocksteiii [2] stated but did not prove that the cohomology spectra of X aiid Y determinie the cohomnology rinig IH(XX Y). From these spectra he constructed a group isomorphic to the group HI(X X Y), however, he did not iintroduce aniy products in this construction; thus the question of

There are many situations in algebraic topology when a geometric construction is possible if, and only if, a certain integral cohomology class, an obstruction is zero. When attempts are made to compute the obstruction, it often happens that it is relatively easy to show that m times the obstruction is zero, where m is an integer, and consequently the geometric construction is possible if the cohomology group in question has no elements of order m.

In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation F evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation F′ such that S is homologous to a union of leaves and such that the plane field of F′ is homotopic to that of F. In particular, F and F′ have the same Euler class. In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper.

A bstract . We give a precise formulation of the M -theory 3-form potential C in a fashion applicable to topologically nontrivial situations. In our model the 3-form is related to the Chern-Simons form of an E 8 gauge field. This leads to a precise version of the Chern-Simons interaction of 11-dimensional supergravity on manifolds with and without boundary. As an application of the formalism we give a formula for the electric C -field charge, as an integral cohomology class, induced by self-interactions of the 3-form and by gravity. As further applications, we identify the M theory Chern-Simons term as a cubic refinement of a trilinear form, we clarify the physical nature of Witten's global anomaly for 5-brane partition functions, we clarify the relation of M -theory ux quantization to K -theoretic quantization of RR charge, and we indicate how the formalism can be applied to heterotic M -theory. INTRODUCTION This paper summarizes a talk given at the conference on Elliptic Cohomology at the Isaac Newton Institute, in December, 2002 In this paper we will discuss the relation of M -theory to E 8 gauge theory in 10, 11, and 12 dimensions. Our basic philosophy is that formulating M -theory in a mathematically precise way, in the presence of nontrivial topology, challenges our understanding of the fundamental formulation of the theory, and therefore might lead to a deeper understanding of how one should express the unified theory of which 11-dimensional supergravity and the five 10-dimensional string theories are distinct limits. To be more specific, let us formulate three motivating problems for the formalism we will develop.

Associated to a differential character is an integral cohomology class, referred to as the characteristic class, and a closed differential form, referred to as the curvature. The characteristic class and curvature are equal in de Rham cohomology, and this is encoded in a commutative square. In the Hopkins–Singer model, where differential characters are equivalence classes of differential cocycles, there is a natural notion of trivializing a differential cocycle. In this paper, we extend the notion of characteristic class, curvature, and de Rham class to trivializations of differential cocycles. These structures fit into a commutative square, and this square is a torsor for the commutative square associated to characters with degree one less. Under the correspondence between degree two differential cocycles and principal circle bundles with connection, we recover familiar structures associated to global sections.

Let p be a prime number and let G be a p-group which is not elementary abelian. For every integral cohomology class \( \xi \) of G which restricts trivially to all proper subgroups, we show that \( \xi^{p} = 0 \) if p > 2 or \( \textrm{deg}(\xi) \) is even, and \( \xi^{3} = 0 \) if p = 2 and \( \textrm{deg}(\xi) \) is odd. This result is applied to get an upper bound, which is \( \frac{|G|}{p} \), for the nilpotence degrees of nilpotent integral cohomology classes of G.