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A space $X$ is said to be $C$-trivial if the total Chern class $c(\alpha)$ equals $1$ for every complex vector bundle $\alpha$ over $X$. In this note we give a complete homological classification of $C$-trivial closed smooth manifolds of dimension $< 7$. In dimension $7$ we give a complete classification of orientable $C$-trivial manifolds and in the non-orientable case we give necessary homological conditions for the manifold to be $C$-trivial. Our main tool is the Atiyah-Hirzebruch spectral sequence and orders of its differentials., Comment: 13 pages

We compute the upper characteristic rank of the projective Stiefel manifolds over $\mathbb R, \mathbb C$ and $\mathbb H$ and of the flip Stiefel manifolds. We provide bounds for the cup lengths of these spaces. We also provide necessary conditions for existence of $S^3$-map between quaternionic Stiefel manifolds using Fadell-Husseini index., Comment: 10 pages

In 1956 Hirzebruch found an explicit formula for the denominators of the Todd polynomials, which was proved later in his joint work with Atiyah. We present a new formula for the Todd polynomials in terms of the ``forgotten symmetric functions", which follows from our previous work on complex cobordisms. In particular, this leads to a simpler proof of the Hirzebruch formula and provides new interpretations for the Hirzebruch numbers., Comment: Slightly revised version, published in the Proceedings of the Steklov Institute of Mathematics, 2024, Vol. 325

A space $X$ is $W$-trivial if for every real vector bundle $\alpha$ over $X$ the total Stiefel-Whitney class $w(\alpha)$ is 1. It follows from a result of Milnor that if $X$ is an orientable closed smooth manifold of dimension $1,2,4$ or $8$, then $X$ is not $W$-trivial. In this note we completely characterize $W$-trivial orientable connected closed smooth manifolds in dimensions $3,5$ and $6$. In dimension $7$, we describe necessary conditions for an orientable connected closed smooth $7$-manifold to be $W$-trivial., Comment: 12 pages, Accepted in Manuscripta Mathematica

We introduce notions of {\it upper chernrank} and {\it even cup length} of a finite connected CW-complex and prove that {\it upper chernrank} is a homotopy invariant. It turns out that determination of {\it upper chernrank} of a space $X$ sometimes helps to detect whether a generator of the top cohomology group can be realized as Euler class for some real (orientable) vector bundle over $X$ or not. For a closed connected $d$-dimensional complex manifold we obtain an upper bound of its even cup length. For a finite connected even dimensional CW-complex with its {\it upper chernrank} equal to its dimension, we provide a method of computing its even cup length. Finally, we compute {\it upper chernrank} of many interesting spaces.

In the first part, this paper studies the characteristic rank of the canonical oriented $k$-plane bundle over the Grassmann manifold $SO(n)/(SO(k) \times SO(n-k))$ of oriented $k$-planes in Euclidean $n$-space. It presents infinitely many new exact values if $k = 3$ or $k = 4$, as well as new lower bounds for the number in question if $k > 4$. In the second part, these results enable us to improve on the general upper bounds for the $Z/2Z$-cup-length of $SO(n)/(SO(k) \times SO(n-k))$. In particular, for $SO(2^t)/(SO(3) \times SO(2^t-3))$ (with $t > 2$) we prove that the cup-length is equal to $2^t-3$, which verifies the corresponding claim of Tomohiro Fukaya's conjecture from 2008., Comment: 10 pages, to appear in the Osaka Journal of Mathematics

If a characteristic class for two vector bundles over the same base space does not coincide, then the bundles are not isomorphic. We give under rather common assumptions a lower bound on the topological dimension of the set of all points in the base over which a morphism between such bundles is not bijective. Moreover, we show that this set is topologically non-trivial., Comment: 5 pages

We define the notion of characteristic rank, $\mathrm{charrank}_X(\xi)$, of a real vector bundle $\xi$ over a connected finite $CW$-complex $X$. This is a bundle-dependent version of the notion of characteristic rank introduced by J\'{u}lius Korba\v{s} in 2010. We obtain bounds for the cup length of manifolds in terms of the characteristic rank of vector bundles generalizing a theorem of Korba\v{s} and compute the characteristic rank of vector bundles over the Dold manifolds, the Moore spaces and the stunted projective spaces amongst others., Comment: 15 pages, To appear in Math. Slovaca

We prove Milnor-Wood inequalities for local products of manifolds. As a consequence, we establish the generalized Chern Conjecture for products $M\times \Sigma^k$ for any product of a manifold $M$ with a product of $k$ copies of a surface $\Sigma$ for $k$ sufficiently large., Comment: 8 pages; MSC keywords: characteristic classes and numbers

A singular point of a smooth map F: M -> N of manifolds is a point in M at which the rank of the differential dF is less than the minimum of dimensions of M and N. The classical invariant of the set S of singular points of F of a given type is defined by taking the fundamental class [\bar{S}]\in H_*(M) of the closure of S. We introduce and study new invariants of singular sets for which the classical invariants may not be defined, i.e., for which \bar{S} may not possess the fundamental class. The simplest new invariant is defined by carefully choosing the fundamental class of the intersection of \bar{S} and its slight perturbation in M. Surprisingly, for certain singularity types such an invariant is well-define (and not trivial) despite the fact that \bar{S} does not possess the fundamental class. We determine new invariants for maps with Morin singularities---i.e., singularities of types A_k for k>0 in the ADE-classification of simple singularities by Dynkin diagrams---and, as an application, show that these invariants together with generalized Miller-Morita-Mumford classes form a commutative graded algebra of characteristic classes that completely determine the cobordism groups of maps with at most A_k-singularities for each k>0., Comment: 64 pages

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 diffeomorphism groups of oriented closed manifolds of dimension d. A characteristic class is stable if it extends to a cohomology class of a homotopy colimit of spaces BDiff M. We show that each rational stable characteristic class of oriented manifold bundles of even dimension d is tautological, e.g., if d=2, then each rational stable characteristic class is a polynomial in terms of Miller-Morita-Mumford classes., Comment: 52 pages

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 non-triviality for some of them., Comment: 18 pages

Let M be a compact closed non-orientable surface. We show that the space of representations of the fundamental group of M into PSL(2,R) has exactly two connected components. These two components are the preimages of a certain Stiefel-Whitney characteristic class, computed in a similar way as the Euler class in the orientable case., Comment: 18 pages, 1 figure

A simple trick invoking objective B-fields is employed to refine the concept of characteristic classes for twisted bundles. Then the objective stability and objective Einstein metrics are introduced and a new Hitchin-Kobayashi correspondence is established between them. As an application the SO(3)-instanton moduli space is proved to be always orientable., Comment: 30 pages, to be submitted

We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study index for a vector field with non-isolated singularities on a submanifold. As an application, our studies give conceptual proofs of a classical result of Chern., Comment: Section 2 expanded and fixed. Final version to appear in Canadian Mathematical Bulletin

The cohomology ring of a finite group, with coefficients in a finite field, can be computed by a machine, as Carlson has showed. Here "compute" means to find a presentation in terms of generators and relations, and involves only the underlying (graded) ring. We propose a method to determine some of the extra structure: namely, Stiefel-Whitney classes and Steenrod operations. The calculations are explicitly carried out for about one hundred groups (the results can be consulted on the Internet). Next, we give an application: thanks to the new information gathered, we can in many cases determine which cohomology classes are supported by algebraic varieties., Comment: To appear in Ann. Inst. Fourier

We study geometry on real gerbes in the spirit of Cheeger-Simons theory. The concepts of adaptations and holonomy forms are introduced for flat connections on real gerbes. Their relations to complex gerbes with connections are presented, as well as results in loop and map spaces., Comment: 22 pages

We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old conjecture proved by Milnor in dimension 2. In particular, the Chern conjecture follows in these cases. The proof goes via a new sharp Milnor--Wood inequality for Riemannian manifolds that are locally a product of hyperbolic planes. Furthermore, we analyze the possible flat vector bundles over such manifolds. Over closed Hilbert--Blumenthal modular varieties, we show that there are finitely many flat structures with nonzero Euler number and none of them corresponds to the tangent bundle. Some of the main results were announced in C.R. Acad. Sci. Paris, Ser. I 346 (2008) 661-666., Comment: 36 pages. New title, modified abstract and introduction

We compute the transgressed forms of some modularly invariant characteristic forms,which are related to the twisted elliptic genera. We study the modularity properties of these secondary characteristic forms and relations among them. We also get some twisted anomaly cancellation formulas on some odd dimensional manifolds.

We define generalized Atiyah-Patodi-Singer boundary conditions of product type for Dirac operators associated to C*-vector bundles on the product of a compact manifold with boundary and a closed manifold. We prove a product formula for the K-theoretic index classes, which we use to generalize the product formula for the topological signature to higher signatures., Comment: 33 pages; revised and expanded, for the odd signature class more general definition (depending on a Lagrangian) used

A version of smooth K-theory is constructed, which is adapted to the total Chern class instead of the Chern character (contrarily to previous theories). Some total Chern class morphism from this K-theory to Cheeger-Simons differential characters is constructed. This answers a question raised by U. Bunke., Comment: Institution, keywords and AMS-classification codes added

This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern--Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary to many other locally symmetric spaces they do admit flat vector bundle of the corresponding dimension. When the manifold is irreducible and of higher rank, it is shown that flat oriented vector bundles are determined completely by the sign of the Euler number., Comment: 6 pages

This note proves that, as K-theory elements, the symbol classes of the de Rham operator and the signature operator on a closed manifold of even dimension are congruent mod 2. An equivariant generalization is given pertaining to the equivariant Euler characteristic and the multi-signature., Comment: 8 pages

This is the final part of the work started in math.DG/0611281 and math.DG/0703916. Here the question of double fibration ois adressed both for relative k-theory and free multiplicative K-theory. In the case of relative and ``nonfree'' multiplicative K-theory, the direct image is proved to be functorial for double submersions.

In this paper, we discuss relations among several invariants of 3-manifolds including Meyer's function, the eta-invariant, the von Neumann rho-invariant and the Casson invariant from the viewpoint of the mapping class group of a surface., Comment: 10 pages

We show how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus g curves with n labeled points. This produces a polynomial with coefficients that include the Euler characteristic of the moduli space, and tautological intersection numbers on the compactified moduli space., Comment: 15 pages, 5figures

A Q-manifold is a graded manifold endowed with a vector field of degree one squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of ``gauge fields'' (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. Any principal bundle yielding canonically a Q-bundle, this construction generalizes Chern-Weil classes. Novel examples include cohomology classes that are locally the de Rham differential of the integrands of topological sigma models obtained by the AKSZ-formalism in arbitrary dimensions. For Hamiltonian Poisson fibrations one obtains a characteristic 3-class in this manner. We also relate to equivariant cohomology and Lecomte's characteristic classes of exact sequences of Lie algebras., Comment: 23 pages, LaTeX, uses diagrams.sty

In 1954 Hirzebruch asked which linear combinations of Chern numbers are topological invariants of smooth complex projective varieties. We give a complete answer to this question in small dimensions, and also prove partial results without restrictions on the dimension., Comment: minor edits; to appear in the Journal of Topology

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., Comment: 6 pages, to appear in Topology and its Applications. v2: changes in exposition

We consider classical potential scattering. If no orbit is trapped at energy E, the Hamiltonian dynamics defines an integer-valued topological degree. This can be calculated explicitly and be used for symbolic dynamics of multi-obstacle scattering. If the potential is bounded, then in the non-trapping case the boundary of Hill's Region is empty or homeomorphic to a sphere. We consider classical potential scattering. If at energy E no orbit is trapped, the Hamiltonian dynamics defines an integer-valued topological degree deg(E) < 2. This is calculated explicitly for all potentials, and exactly the integers < 2 are shown to occur for suitable potentials. The non-trapping condition is restrictive in the sense that for a bounded potential it is shown to imply that the boundary of Hill's Region in configuration space is either empty or homeomorphic to a sphere. However, in many situations one can decompose a potential into a sum of non-trapping potentials with non-trivial degree and embed symbolic dynamics of multi-obstacle scattering. This comprises a large number of earlier results, obtained by different authors on multi-obstacle scattering., Comment: 25 pages, 1 figure Revised and enlarged version, containing more detailed proofs and remarks

This is the sequel of the first part math.DG/0611281. Here, the procedure of transgressing the families index theorem (the so-called $\eta$-form) is adapted to take in account the case of Dirac type operators with kernels of varying dimension. The constructed form is then used to define the direct image under proper submersions of the ``free multiplicative'' $K$-theory which was defined in the first part, the behaviour of the characteristic classes on free multiplicative $K$-theory under submersion is studied. Some universal caracterisation of the forms is provided. Finally, combining our result with Bismut and Lott's results on direct images of flat vector bundles yields a ``Grothendieck-Riemann-Roch'' theorem for Nadel-Chern-Simons classes on relative $K$-theory for flat vector bundles which was defined in the first part.

This paper contains the constructions of a real manifold version of relative K-theory, and of an extension of Karoubi's multiplicative K-theory suggested by U. Bunke (which I call ``free multiplicative K-theory'' in the sequel). Chern-Simons-Nadel type classes on relative K-theory are constructed, while it is proved that on free multiplicative K-theory, there is a notion of Chern-Weil character form, and of a Borel-type characteristic class (which is a differential form modulo exact forms) which recovers the classes $c_k$ of flat vector bundles studied by Bismut and Lott. Finally, a direct image for relative K-theory under proper submersion of compact orientable real manifolds is constructed., Comment: Construction of the topological K-theoretic direct image representatives shortened, considerations added about fibral Hodge symmetry and name of ``transgressive K-theory'' changed to ''free multiplicative K-theory''