Related papers: Higher order intersection numbers of 2-spheres in …

An obstruction theory for representing homotopy classes of surfaces in 4-manifolds by immersions with pairwise disjoint images is developed, using the theory of non-repeating Whitney towers. The accompanying higher-order intersection…

We show how to measure the failure of the Whitney trick in dimension 4 by constructing higher- order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on…

We prove a surface embedding theorem for 4-manifolds with good fundamental group in the presence of dual spheres, with no restriction on the normal bundles. The new obstruction is a Kervaire-Milnor invariant for surfaces and we give a…

Regular homotopy classes of immersions of a 3-sphere in 5-space constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an…

This paper describes the relationship between the first non-vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed disks…

Wall's finiteness obstruction is an algebraic K-theory invariant which decides if a finitely dominated space is homotopy equivalent to a finite CW complex. The object of this survey is to describe the invariant (which was first formulated…

The SU(3)-Casson invariant for integral homology 3-spheres as studied by Boden-Herald possesses a 'spectral flow obstruction' to being an integer valued invariant which depends only on the non-degenerate (perturbed) moduli space of flat…

Let $M$ be a sphere with handles and holes, $f:M\to\mathbb R^3$ an embedding, and $H_1=H_1(M;\mathbb Z)$. We study a simple isotopy invariant of $f$, the Seifert bilinear form $L(f):H_1\times H_1\to\mathbb Z$. Let $\cap:H_1\times…

We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2 \times S^2$. For a fixed fundamental group, there are primary, secondary and tertiary obstructions, which together with the signature…

Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…

We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional…

Given a link map f into a manifold of the form Q = N \times \Bbb R, when can it be deformed to an unlinked position (in some sense, e.g. where its components map to disjoint \Bbb R-levels) ? Using the language of normal bordism theory as…

We construct an obstruction theory for relative Hilbert schemes in the sense of Behrend-Fantechi and compute it explicitly for relative Hilbert schemes of divisors on smooth projective varieties. In the special case of curves on a surface…

We study smooth maps between smooth manifolds with only fold points as their singularities, and clarify the obstructions to the existence of such a map in a given homotopy class for certain dimensions. The obstructions are described in…

Three-component links in the 3-dimensional sphere were classified up to link homotopy by John Milnor in his senior thesis, published in 1954. A complete set of invariants is given by the pairwise linking numbers p, q and r of the…

Let $f_1,..., f_k:X\to N$ be maps from a complex $X$ to a compact manifold $N$, $k\ge 2$. In previous works \cite{BLM,MS}, a Lefschetz type theorem was established so that the non-vanishing of a Lefschetz type coincidence class…

We continue to develop an obstruction theory for embedding 2-spheres into 4-manifolds in terms of Whitney towers. The proposed intersection invariants take values in certain graded abelian groups generated by labelled trivalent trees, and…

Minimum numbers decide e.g. whether a given map f: S^m --> S^n/G from a sphere into a spherical space form can be deformed to a map f' such that f(x) not equal f'(x) for all x in S^m. In this paper we compare minimum numbers to…

We study a type of left-invariant structure on Lie groups, or equivalently on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the 5-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3).…

For Seifert manifold $M=X({p_1}/_{\f{q_1}},{p_2}/_{\f{q_2}}, ...,{p_n}/_ {\f{q_n}}), \tau^{'}_r(M)$ is calculated for all $r$ odd $\geq 3$. If $r$ is coprime to at least $n-2$ of $p_k$ (e.g. when $M$ is the Poincare homology sphere), it is…