18 pages, LaTeX. This version contains new material, proving correct relationship with odd Chern classes (this was wrongly stated earlier). The version without odd Chern classes will appear in the Asian Journal of Mathematics
The complex Lie group GO(2n,C) by definition consists of all complex matrices A of size 2n, such that A times transpose(A) is a non-zero scalar. In this paper we determine explicitly the singular cohomology ring of the classifying space BGO(2n,C) with mod 2 coefficients, in terms of generators and relations. The method consists of analysing a certain derivation on the cohomology ring of BO(2n) (which is a polynomial ring in the Stiefel-Whitney classes) via a Koszul complex, and using this to `solve' the Gysin sequence for the bundle BO(2n) over BGO(2n,C).
We apply our earlier work on the higher-dimensional analogue of the Mumford conjecture to two questions. Inspired by work of Ebert we prove non-triviality of certain characteristic classes of bundles of smooth closed manifolds. Inspired by work of Church-Farb-Thibault and Church-Crossley-Giansiracusa we investigate the dependence of characteristic classes of bundles on characteristic numbers of its fibre, total space and base space.
We establish formulae that explain how the topological Goresky-MacPherson characteristic L-classes as well as the Hodge-theoretic Hirzebruch characteristic classes defined by Brasselet, Schürmann and Yokura transform under Gysin restrictions associated to normally nonsingular embeddings of singular spaces. We find that both types of classes transform in the same manner. We give a first application of these formulae in obtaining algebraic rigidity results for topologically homeomorphic projective varieties.
In this paper, we consider an obstruction-theoretical construction of characteristic classes of fiber bundles by simplicial method. We can get a certain obstruction class for a deformation of $C_\infty$-algebra models of fibers and a characteristic map from the exterior algebra of a vector space of derivations. Applying this construction for a surface bundle, we obtain the Euler class of a sphere bundle and the Morita-Miller-Mumford classes of a bundle with positive genus fiber.
The theory of indices of Morse--Bott vector fields on a manifold is constructed and the famous localization problem for the transfer map is solved on its base in the present paper. As a consequence, we obtained addition theorems for the universal Pontrjagin characteristic classes in cobordisms. These results gave us a possibility to complete the construction, which was begun more than twenty years ago, of the universal characteristic classes' theory.
In this paper, we construct a double chain complex generated by certain graphs and a chain map from that to the Chevalley-Eilenberg double complex of the dgl of symplectic derivations on a free dgl. It is known that the target of the map is related to characteristic classes of fibrations. We can describe some characteristic classes of fibrations whose fiber is a 1-punctured even-dimensional manifold by linear combinations of graphs though the cohomology of the dgl of derivations.
We compute the completed $TMF_0(3)$ cohomology of the 7-connective cover $BString$ of $BO$. We use cubical structures on line bundles over elliptic curves to construct an explicit class which together with the Pontryagin classes freely generates the cohomology ring.
Let $TMF_1(n)$ be the spectrum of topological modular forms equipped with a $Γ_1(n)$-structure. We compute the $K(2)$-local $TMF_1(3)$-cohomology of $B{\mathit String}$ and $B{\mathit Spin}$: both are power series rings freely generated by classes that we explicitly construct and which generalize the classical Pontryagin classes. As a first application of this computation, we show how to construct $TMF(3n)$-cohomology classes from stable positive energy representations of the loop groups $L{\mathit Spin}$.
We give minimal presentations for the $RO(C_2)$-graded Bredon cohomology of the equivariant classifying spaces $B_{C_2}U(n), B_{C_2}SO(n)$ and $B_{C_2}Sp(n)$ with coefficients in the rational Burnside Green functor $A_{\mathbf Q}$. This results in an efficient description of rational $C_2$ equivariant Chern, Pontryagin and symplectic characteristic classes. These classes are then related to each other using the inclusions of maximal tori.
For any (n-1)-dimensional simplicial complex, we construct a particular n-dimensional complex vector bundle over the associated Davis-Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley Reisner algebra. We show that the isomorphism type of this complex vector bundle as well as of its realification are completely determined by its characteristic classes. This allows us to show that coloring properties of the simplicial complex are reflected by splitting properties of this bundle and vice versa. Similar question are also disc…
The Miller-Morita-Mumford classes associate to an oriented surface bundle $E\to B$ a class $κ_i(E) \in H^{2i}(B;\Z)$. In this note we define for each prime $p$ and each integer $i\geq 1$ a secondary characteristic class $λ_i(E) \in H^{2i(p-1)-2}(B;\Z)/\Zκ_{i(p-1)-1}$. The mod $p$ reduction $λ_i(E) \in H^*(B; \F_p)$ has zero indeterminacy and satisfies $pλ_i(E) = κ_{i(p-1)-1}(E) \in H^*(B;\Z/p^2)$.
We study relations among characteristic classes of smooth manifold bundles with highly-connected fibers. For bundles with fiber the connected sum of $g$ copies of a product of spheres $S^d \times S^d$ and an odd $d$, we find numerous algebraic relations among the so-called "generalized Miller-Morita-Mumford classes". For all $g > 1$, we show that these infinitely many classes are algebraically generated by a finite subset. Our results contrast with the fact that there are no algebraic relations among these classes in a range of cohomological degrees that grows linearly with $g$, according to r…
In the first part of this paper, we propose a uniform interpretation of characteristic classes as obstructions to the reduction of the structure group and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber of the bundle. By plugging in the correct parameters, we recover several classical theorems. Afterwards, we define a family of invariants of principal bundles that detect the number of group reductions that a principal bundle admits. We prove that they fit into a long exact sequence of abelian groups, together with the cohomology of…
Standard combinatorial construction, due to Kontsevich, associates to any $\ai$-algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We propose an alternative version of this construction based on noncommutative geometry and use it to prove that homotopy equivalent algebras give rise to the same cohomology classes. Along the way we re-prove Kontsevich's theorem relating graph homology to the homology of certain infinite dimensional Lie algebras. An application to topological conformal field theories is gi…
There are characteristic classes that are the obstructions to the vanishing of the differentials in the Lyndon-Hochischild-Serre spectral sequence of an extension of an integral lattice L by a group G. These characteristic classes exist in a given page of the spectral sequence provided the differentials in the previous pages are all zero. When L decomposes into a sum of G-sublattices, we show that there are defining relations between the characteristic classes of L and the characteristic classes of its summands.
A theory of characteristic classes of vector bundles and smooth manifolds plays an important role in the theory of smooth manifolds. An investigation of reasonable notions of characteristic classes of singular spaces started since a systematic study of singular spaces such as singular algebraic varieties. We make a quick survey of characteristic classes of singular varieties, mainly focusing on the functorial aspects of some important ones such as the singular versions of the Chern class, the Todd class and the Thom--Hirzebruch's L-class. Then we explain our recent "motivic" characteristic cla…
Motivic characteristic classes of possibly singular algebraic varieties are homology class versions of motivic characteristics, not classes in the so-called motivic (co)homology. This paper is a survey on them with more emphasis on capturing infinitude finitely and on the motivic nature, in other words, the scissor relation or additivity.
We introduce characteristic classes for the spectral sequence associated to a split short exact sequence of Hopf algebras. We show that these characteristic classes can be seen as obstructions for the vanishing of differentials in the spectral sequence and prove a decomposition theorem. We also interpret our results in the settings of group and Lie algebra extensions and prove some interesting corollaries concerning the collapse of the (Lyndon-)Hochschild-Serre spectral sequence and the order of characteristic classes.
This note shows the compatibility of the differential geometric and the topological formulations of equivariant characteristic classes for a compact connected Lie group action.
Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to a cohomology class of a homotopy colimit BSO of classifying spaces BSO_n. Similarly, characteristic classes of smooth oriented manifold bundles with fibers given by oriented closed smooth manifolds of a fixed dimension d\ge 0 can be identified with cohomology classes of the disjoint union of classifying spaces BDiff M of orientation preserving diffeomorphis…
This paper examines a number of related questions about Euler characteristics and characteristic classes with values in Witt cohomology. We establish a motivic version of the Becker-Gottllieb transfer, generalizing a construction of Hoyois. Ananyevskiy's splitting principle reduces questions about characteristic classes of vector bundles in $\text{SL}$-oriented, $η$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\text{SL}_2$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and t…
In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch- and motivic Chern-classes. As an application we get a description of the differences between the corresponding virtual and fun…
Using certain Thom spectra appearing in the study of cobordism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler characteristic vanish in integral cohomology when restricted to the handlebody subgroup. This is a special case of a more general theorem valid in all dimensions: universal characteristic classes made from monomials in the Pontrjagin classes (and even powers of the Euler class) vanish when pulled back from BDiff(\partial W) to BDiff(W).
In this note we show that the Chern-Simons and the one-loop terms in the M-theory action can be written in terms of new characters involving the M-theory four-form and the string classes. This sheds a new light on the topological structure behind M-theory and suggests the construction of a theory of `higher' characteristic classes.
It is shown that the characteristic classes of foliations that were defined by Losik and that take values in the de~Rham cohomology of the space of infinite order frames over the leaf space may be mapped to the characteristic classes with values in the Čech-de~Rham cohomology of the leaf space studied in details by Crainic and Moerdijk. This map is in general non-injective. All constructions are done using Losik's approach to Gelfand formal geometry. A similar result is obtained for the exotic characteristic classes as well as for the group actions of the diffeomorphisms. As illustrating examp…
Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet-Schuermann-Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and Goresky-MacPherson L-classes, respectively). In this note we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms, and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We…
We obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasi-projective varieties. More concretely, we study equivariant versions of Todd, Chern and Hirzebruch classes for singular spaces, with values in delocalized Borel-Moore homology of external and symmetric products. As a byproduct, we recover our previous characteristic class formulae for symmetric products, and obtain new equivariant generalizations of these results, in particular also in the context of twisting by representations of the symmetric group.
In this note we clarify the relevance of ``connections up to homotopy'' to the theory of characteristic classes. We have already remarked \cite{Crai} that such connections up to homotopy can be used to compute the classical Chern characters. Here we present a slightly different argument for this, and then proceed with the discussion of the flat (secondary) characteristic classes. As an application, we clarify the relation between the two different approaches to characteristic classes of algebroids (and of Poisson manifolds in particular): we explain that the intrinsic characteristic classes ar…
We introduce basic characteristic classes and numbers as new invariants for Riemannian foliations. If the ambient Riemannian manifold M is simply connected (or more generally if the foliation is a transversely orientable Killing foliation), if M is complete and if the space of leaf closures is compact, then the basic characteristic numbers are determined by the infinitesimal dynamical behavior of the foliation at the union of its closed leaves. In fact, they can be computed with an Atiyah-Bott-Berline-Vergne-type integration formula for equivariant basic cohomology.
We construct characteristic classes of smooth (Hamiltonian) fibrations as as fiber integrals of products of Pontriagin (or Chern) classes of vertical vector bundles over the total space of the universal fibration. We give explicit formulae of these fiber integrals for toric manifolds and get estimates of the dimension of the cohomology groups of classifying spaces.
Families of smooth closed oriented 4-manifolds with a complex spin structure are studied by means of a family version of the Bauer--Furuta invariants in the context of parametrised stable homotopy theory, leading to a definition of characteristic cohomotopy classes on Thom spectra associated to the classifying spaces of their complex spin diffeomorphism groups. This is illustrated with mapping tori of such diffeomorphisms and related to the equivariant invariants.
We recall the construction of triality automorphism of so(8) given by E. Cartan and we give a matrix representation for the real form so(4,4). We compute the induced results on the characteristic classes. Paralelly we study the triality automorphism of the singularity D4 (in Arnolds classification of smooth functions) and its miniversal deformation. The similarity with Lie theory leads us to a definition of G2 singularity.
We construct new subtle Stiefel--Whitney classes of quadratic forms. These classes are much more informative than the ones introduced by Milnor. In particular, they see all the powers of the fundamental ideal of the Witt ring, contain the Arason invariant and it's higher analogues. Moreover, the new classes allow to treat the J-invariant of quadrics. This invariant, introduced in 2005, has been so far completely isolated from characteristic classes. In addition, our classes allow to describe explicitly the structure of some motives associated with quadratic forms.
In the early 1970's, Gelfand, Kalinin and Fuks found an exotic characteristic class of degree 7 in the Gelfand-Fuks cohomology of the Lie algebra of formal Hamiltonian vector fields on the plane. We prove that this cohomology class can be decomposed as a product of a certain leaf cohomology class of degree 5 and the transverse symplectic class. This is similar to the well known factorization of the Godbillon-Vey class for codimension n foliations. We also interpret the characteristic classes of transversely symplectic foliations introduced by Kontsevich in terms of the known classes and prove …
This paper gives an introduction and overview about recent developments on the interaction of the theories of characteristic classes and mixed Hodge theory for singular spaces in the complex algebraic context. It uses M. Saito's deep theory of mixed Hodge modules as a "black box", thinking about them as "constructible or perverse sheaves of Hodge structures", having the same functorial calculus of Grothendieck functors. For the "constant Hodge sheaf", one gets the "motivic characteristic classes" of Brasselet-Schuermann-Yokura, whereas the classes of the "intersection homology Hodge sheaf" wer…
In this note we introduce a construction which assigns to an arbitrary manifold bundle its fiberwise orientation covering. This is used to show that the zeta classes of unoriented surface bundles are not divisible in the stable range.
We prove a closed formula expressing any multiplicative characteristic class evaluated on the tangent bundle of the Hilbert schemes of points on a non-compact simply-connected surface. As a corollary, we deduce a closed formula for the Chern character of the tangent bundles of these Hilbert schemes. We also give a closed formula for the multiplicative characteristic classes of the tautological bundles associated to a line bundle on the surface. We finally remark which implications the results here have for the Hilbert schemes of points of an arbitrary surface.
Michael Gromov has recently initiated what he calls ``symbolic algebraic geometry", in which objects are proalgebraic varieties: a proalgebraic variety is by definition the projective limit of a projective system of algebraic varieties. In this paper we introduce characteristic classes of proalgebraic varieties, using Grothendieck transformations of Fulton--MacPherson's Bivariant Theory, modeled on the construction of MacPherson's Chern class transformation of proalgebraic varieties. We show that a proalgebraic version of the Euler--Poincaré characteristic with values in the Grothendieck ring …
In this paper we give explicit formulas of differential characteristic classes of principal $G$-bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and differential Euler class. Furthermore, we show that the differential Chern class is the unique natural transformation from (Simons-Sullivan) differential $K$-theory to (Cheeger-Simons) differential characters that is compatible with curvature and characteristic class. We also give the explicit formula for the differential Chern c…
Following [14], we compute the motivic cohomology ring of the Nisnevich classifying space of the unitary group associated to the standard split hermitian form of a quadratic extension. This provides us with subtle characteristic classes which take value in the motivic cohomology of the Čech simplicial scheme associated to a hermitian form. Comparing these new classes with subtle Stiefel-Whitney classes arising in the orthogonal case, we obtain relations among the latter ones holding in the motivic cohomology of the Čech simplicial scheme associated to a quadratic form divisible by a 1-fold Pfi…
Given a closed simply connected manifold $M$ of dimension $2n\ge6$, we compare the ring of characteristic classes of smooth oriented bundles with fibre $M$ to the analogous ring resulting from replacing $M$ by the connected sum $M\sharpΣ$ with an exotic sphere $Σ$. We show that, after inverting the order of $Σ$ in the group of homotopy spheres, the two rings in question are isomorphic in a range of degrees. Furthermore, we construct infinite families of examples witnessing that inverting the order of $Σ$ is necessary.
Based on a pair of cohomology operations on so called $δ-2$-formal spaces, we construct the integral cohomology rings of the classifying spaces of the Lie groups $Spin(n)$ and $Spin^{c}(n)$. As applications, we introduce characteristic classes for the reduced topological $K_{Spin}$ theory, determine the ring of integral Weyl invariants of the group $Spin(n)$, and demonstrate their usages in spin geometry.
The generalized Miller-Morita-Mumford classes of a manifold bundle with fiber $M$ depend only on the underlying $τ_M$-fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for $τ_M$-fibrations, $Baut(τ_M)$, and its cohomology ring, i.e., the ring of characteristic classes of $τ_M$-fibrations. For a bundle $ξ$ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal orientable $ξ$-fibration together with explicit cocycle representatives for the characteri…
In this paper we identify many striking elements in Leibniz (co)homology which arise from characteristic classes and K-theory. For a group G a field k of characteristic zero, it is shown that all primary characteristic classes, i.e. H^*(BG; k), naturally inject into certain Leibniz cohomology groups via an explicit chain map. Moreover, if f: A \to B is a homomorphism of algebras or rigns, the relative Leibniz homology groups HL_*(f) are defined, and if in addition f is surjective with nilpotent kernel, A and B algebras over the rationals, then there is a natural surjection HL_{*+1}(gl(f)) \to …
In this article we study the construction of characteristic classes for principal $G$-bundles equipped with an additional structure called transitionally commutative structure (TC structure). These structures classify, up to homotopy, possible trivializations of a principal $G$-bundle, such that the induced cocycle have functions that commute in the intersections of their domains. We focus mainly on the cases where the structural group $G$ equals $SU(n)$, $U(n)$ or $\mathrm{Sp}(n)$. Our approach is an algebraic-geometric construction that relies on the so called power maps defined on the space…
In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the complete case fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of …
Making use of the extended flux homomorphism on the group of symplectomorphisms of a closed oriented surface of genus at least 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently area-preserving, total holonomy. These characteristic classes are stable with respect to the genus and we show that they are highly non-trivial. We also prove that the second homology of the group of Hamiltonian symplectomorphisms equipped with the discrete topology is very large.
Using the concept of a cohesive module defined by Block, we use the theory of superconnections in the sense of Quillen to construct natural superconnections on Hermitian cohesive modules. By the Chern-Weil construction, we obtain characteristic classes with values in Bott-Chern cohomology which refines the usual deRham cohomology. The main result of this paper shows that such characteristic classes are independent of the Hermitian metric and only dependent on the homotopy class of the cohesive module in the homotopy category. This extends the Bott-Chern cohomology to the bounded derived catego…
Let G be a connected Lie group, G^d the underlying discrete group, and BG, BG^d their classifying spaces. Let R denote the radical of G. We show that all classes in the image of the canonical map in cohomology H^*(BG,R)->H^*(BG^d,R) are bounded if and only if the derived group [R,R] is simply connected. We also give equivalent conditions in terms of stable commutator length and distortion.
The cohomology of the degree-$n$ general linear group over a finite field of characteristic $p$, with coefficients also in characteristic $p$, remains poorly understood. For example, the lowest degree previously known to contain nontrivial elements is exponential in $n$. In this paper, we introduce a new system of characteristic classes for representations over finite fields, and use it to construct a wealth of explicit nontrivial elements in these cohomology groups. In particular we obtain nontrivial elements in degrees linear in $n$. We also construct nontrivial elements in the mod $p$ homol…
We construct characteristic classes for singular algebraic varieties in motivic Borel-Moore homology, extending the motivic Euler class of the tangent bundle defined for smooth varieties. The two classes we define refine, in the setting of motivic homotopy theory, the top degree of the class constructed by Brasselet-Schuermann-Yokura in G-theory, and the top degree of the pro-CSM class constructed by Aluffi in pro-Chow groups. We deduce an extension of the motivic Gauss-Bonnet formula to non-smooth proper varieties.
We show that every bad orbifold vector bundle can be realized as the restriction of a good orbifold vector bundle to a suborbifold of the base space. We give an explicit construction of this result in which the Chen-Ruan orbifold cohomology of the two base spaces are isomorphic (as additive groups). This construction is used to indicate an extension of the Chern-Weil construction of characteristic classes to bad orbifold vector bundles. In particular, we apply this construction to the orbifold Euler class and demonstrate that it acts as an obstruction to the existence of nonvanishing sections.
In this paper we study the variability and rigidity of secondary characteristic classes which arise from flat connections on a manifold. Considering the connection as a Lie-algebra valued one-form, we study the characteristic map from Lie algebra cohomology to de Rham cohomology of the manifold, and prove that if the Leibniz cohomology of the Lie algebra vanishes, then all secondary characteristic classes are rigid. In the case of codimension one foliations (with trivial normal bundle), we show that a characteristic map from the Leibniz cohomology of formal vector fields to the de Rham cohomol…
By analyzing how the Borel regulator classes vanish on various groups related to $\mathrm{GL}(n,\mathrm{Z})$, we define three series of secondary characteristic classes for subgroups of automorphism groups of free groups. The first case is the $\mathrm{IA}$-automorphism groups and we show that our classes coincide with higher $\mathrm{FR}$ torsions due to Igusa. The second case is the mapping class groups and our classes also turn out to be his higher torsions which are non-zero multiples of the Mumford-Morita-Miller classes of even indices. Our construction gives new group cocycles for these …
