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.

Let M be a closed simply connected 2n-dimensional manifold. The paper is concerned with the cohomology of classifying spaces of connected groups of homeomorphisms of M.

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.

The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.

The definition of Thom polynomials for Lagrange, Legendre and critical point function singularities is given. The approach is based on the notion of classifying space of singularities. This approach provides a universal method for computing Thom polynomials. Characteristic classes of complex Lagrange and Legendre singularities of small codimension are computed. Two independent methods for these computations are presented. The first is the direct one based on the resolution of singularities, the Gysin homomorphism, and the notion of adjacency exponents. The second uses the existence theorem for Thom polynomials and is based on Rimányi's idea of using symmetries. The expressions for the resulting polynomials reduced modulo 2 agree with those obtained by Vassiliev for the real case.