A crucial ingredient in the theory of theta liftings of Kudla and Millson is the construction of a $q$-form $\varphi_{KM}$ on an orthogonal symmetric space, using Howe's differential operators. This form can be seen as a Thom form of a real oriented vector bundle. We show that the Kudla-Millson form can be recovered from a canonical construction of Mathai and Quillen. A similar result was obtaind by Garcia for signature $(2,q)$ in case the symmetric space is hermitian and we extend it to arbitrary signature.

We prove that the only relation imposed on the Hodge and Chern numbers of a compact Kähler manifold by the existence of a nowhere zero holomorphic one-form is the vanishing of the Hirzebruch genus. We also treat the analogous problem for nowhere zero closed one-forms on smooth manifolds.

We compute the cohomology of the right generalised projective Stiefel manifolds. Following this, we discuss some easy applications of the computations to the ranks of complementary bundles and bounds on the span and immersibility.

We study tautological rings for high-dimensional manifolds, that is, for each smooth manifold $M$ the ring $R^{\ast }(M)$ of those characteristic classes of smooth fibre bundles with fibre $M$ which is generated by generalised Miller–Morita–Mumford classes. We completely describe these rings modulo nilpotent elements, when $M$ is a connected sum of copies of $S^{n}\times S^{n}$ for $n$ odd.

Combings of compact, oriented, 3-dimensional manifolds $M$ are homotopy classes of nowhere vanishing vector fields. The Euler class of the normal bundle is an invariant of the combing, and it only depends on the underlying $\text{Spi}{{\text{n}}^{\text{c}}}$-structure. A combing is called torsion if this Euler class is a torsion element of ${{H}^{2}}(M;\,\mathbb{Z})$. Gompf introduced a $\mathbb{Q}$-valued invariant ${{\theta }_{G}}$ of torsion combings on closed 3-manifolds, and he showed that ${{\theta }_{G}}$ distinguishes all torsion combings with the same $\text{Spi}{{\text{n}}^{\text{c}}}$-structure. We give an alternative definition for ${{\theta }_{G}}$ and we express its variation as a linking number. We define a similar invariant ${{p}_{1}}$ of combings for manifolds bounded by ${{S}^{2}}$. We relate ${{p}_{1}}$ to the $\Theta$-invariant, which is the simplest configuration space integral invariant of rational homology 3-balls, by the formula $\Theta \,=\,\frac{1}{4}{{p}_{1}}\,+\,6\text{ }\!\!\lambda\!\!\text{ }\left( {\hat{M}} \right)$, where $\text{ }\!\!\lambda\!\!\text{ }$ is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for 3-manifolds.

For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern– Euler class and was used by Sha to formulate a relative Poincaré–Hopf theorem under the condition that the metric on the manifold is locally product near the boundary. We show that the secondary Chern–Euler form is exact away from the outward and inward unit normal vectors of the boundary by explicitly constructing a transgression form. Using Stokes’ theorem, this evaluates the boundary term in Sha's relative Poincaré–Hopf theorem in terms of more classical indices of the tangential projection of a vector field. This evaluation in particular shows that Sha's relative Poincaré–Hopf theorem is equivalent to the more classical law of vector fields.

We define and study the secondary Chern–Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with non-isolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern.

We give a classifying theory for LG-bundles, where LG is the loop group of a compact Lie group G, and present a calculation for the string class of the universal LG-bundle. We show that this class is in fact an equivariant cohomology class and give an equivariant differential form representing it. We then use the caloron correspondence to define (higher) characteristic classes for LG-bundles and to prove a result for characteristic classes for based loop groups for the free loop group. These classes have a natural interpretation in equivariant cohomology and we give equivariant differential form representatives for the universal case in all odd dimensions.

We give a general method for expressing various characteristic numbers of the multiple-point manifolds of an immersion.

Let d be the degree of an algebraic one-dimensional foliation $\mathcal F$ on the complex projective space ${\mathbb P}_n$ (i.e. the degree of the variety of tangencies of the foliation with a generic hyperplane). Let $\Gamma$ be an algebraic solution of degree $\delta$, and geometrical genus g. We prove, in particular, the inequality $(d-1)\delta+2-2g\geq {\mathcal B}(\Gamma)$, where ${\mathcal B}(\Gamma)$ denotes the total number of locally irreducible branches through singular points of $\Gamma$ when $\Gamma$ has singularities, and ${\mathcal B}(\Gamma)=1$ (instead of 0) when $\Gamma$ is smooth. Equivalently, when $\Gamma=\bigcap_{\lambda=1}^{n-1} S_\lambda$ is the complete intersection of n - 1 algebraic hypersurfaces $S_\lambda$, we get $(d+n-\sum_{\lambda=1}^{n-1}\delta_\lambda)\delta \geq {\mathcal B}(\Gamma)-{\mathcal E}(\Gamma)$, where $\delta_\lambda$ denotes the degree of $S_\lambda$ and ${\mathcal E}(\Gamma)=2-2g+(\sum_\lambda\delta_\lambda-(n+1))\delta$ the correction term in the genus formula. These results are also refined when $\Gamma$ is reducible.

In previous joint work with Cappell and Shaneson, we have established an Atiyah–Lusztig–Meyer-type multiplicative characteristic class formula for the twisted signature and, more generally, the twisted $L$-class, of a stratified Witt space. The present paper shows that these formulae hold even when the stratified space does not satisfy the Witt condition. It constitutes one of the first applications of signature homology.

an index of a collection of 1-forms on a complex isolated complete intersection singularity corresponding to a chern number is defined and – in the case when the 1-forms are complex analytic – expressed as the dimension of a certain algebra.

Let $\Cal F$ be a holomorphic foliation (possibly with singularities) on a non-singular manifold $M$, and let $V$ be a complex analytic subset of $M$. Usual residue theorems along $V$ in the theory of complex foliations require that $V$ be tangent to the foliation (that is, a union of leaves and singular points of $V$ and $\Cal F$); this is the case for instance for the blow-up of a non-dicritical isolated singularity. In this paper, residue theorems are introduced along subvarieties that are not necessarily tangent to the foliation, including the blow-up of the dicritical situation.

We study cubical sets without degeneracies, which we call $\square$-sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a $\square$-set $C$ has an infinite family of associated $\square$-sets $J^i(C)$, for $i=1,2,\ldots$, which we call James complexes. There are mock bundle projections $p_i \colon |J^i (C)| \to |C|$ (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of $\Omega (S^2)$. The algebra of these classes mimics the algebra of the cohomotopy of $\Omega (S^2)$ and the reduction to cohomology defines a sequence of natural characteristic classes for a $\square$-set. An associated map to $BO$ leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation.

A connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension are constructed. The construction is based on certain structures on the bundle, that is, connections and bundle splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection and with the curving coincides with the obstruction class of the lifting problem with these structures.

This paper is concerned with mod p Morita-Mumford classes of the mapping class group Γg of a closed oriented surface of genus g ≥ 2, especially triviality and nontriviality of them. It is proved that is nilpotent if n ≡ − 1 (mod p − 1), while the stable mod p Morita-Mumford class is proved to be nontrivial and not nilpotent if n ≢ −1 (mod p − 1). With these results in mind, we conjecture that vanishes whenever n ≡ − 1 (mod p − 1), and obtain a few pieces of supporting evidence.

A geometric invariant is associated to the parabolic moduli space on a marked surface and is related to the symplectic structure of the moduli space.

With the general assumption that the manifold admits two orthogonal complementary foliations, one of which is totally geodesic, we study the components of the curvature tensor field of the characteristic connection.

In the case where the manifold is compact, orientable of dimension 6 or 8 and the dimension of the totally geodesic foliation is 4, we relate the sign of the Euler characteristic of the manifold and that of the sectional curvature of the leaves of both foliations.