A formula is given for the Seiberg–Witten invariants of a 4–manifold that is cut along certain kinds of 3–dimensional tori. The formula involves a Seiberg–Witten invariant for each of the resulting pieces.

A building is a simplicial complex with a covering by Coxeter complexes (called apartments) satisfying certain combinatorial conditions. A building whose apartments are spherical (respectively Euclidean) Coxeter complexes has a natural piecewise spherical (respectively Euclidean) metric with nice geometric properties. We show that spherical and Euclidean buildings are completely characterized by some simple, geometric properties.

We introduce a homology surgery problem in dimension 3 which has the property that the vanishing of its algebraic obstruction leads to a canonical class of $\pi$–algebraically-split links in 3–manifolds with fundamental group $\pi$. Using this class of links, we define a theory of finite type invariants of 3–manifolds in such a way that invariants of degree $0$ are precisely those of conventional algebraic topology and surgery theory. When finite type invariants are reformulated in terms of clovers, we deduce upper bounds for the number of invariants in terms of $\pi$–decorated trivalent graphs. We also consider an associated notion of surgery equivalence of $\pi$–algebraically split links and prove a classification theorem using a generalization of Milnor’s $\bar{\mu }$–invariants to this class of links.

We study Lefschetz pencils on symplectic four-manifolds via the associated spheres in the moduli spaces of curves, and in particular their intersections with certain natural divisors. An invariant defined from such intersection numbers can distinguish manifolds with torsion first Chern class. We prove that pencils of large degree always give spheres which behave ‘homologically’ like rational curves; contrastingly, we give the first constructive example of a symplectic non-holomorphic Lefschetz pencil. We also prove that only finitely many values of signature or Euler characteristic are realised by manifolds admitting Lefschetz pencils of genus two curves.

In this paper, we show that, for each non-trivial two bridge knot $K$ and for each $g\ge 3$, every genus $g$ Heegaard splitting of the exterior $E\left(K\right)$ of $K$ is reducible.

A knot type is exchange reducible if an arbitrary closed $n$–braid representative $K$ of $\mathcal{K}$ can be changed to a closed braid of minimum braid index $n_{min}\left(\mathcal{K}\right)$ by a finite sequence of braid isotopies, exchange moves and $\pm$–destabilizations. In a preprint of Birman and Wrinkle, a transversal knot in the standard contact structure for $S^3$ is defined to be transversally simple if it is characterized up to transversal isotopy by its topological knot type and its self-linking number. Theorem 2 in the preprint establishes that exchange reducibility implies transversally simplicity. The main result in this note, establishes that iterated torus knots are exchange reducible. It then follows as a corollary that iterated torus knots are transversally simple.

We study the question of existence of a Riemannian metric of positive scalar curvature metric on manifolds with the Sullivan–Baas singularities. The manifolds we consider are $Spin$ and simply connected. We prove an analogue of the Gromov–Lawson Conjecture for such manifolds in the case of particular type of singularities. We give an affirmative answer when such manifolds with singularities accept a metric of positive scalar curvature in terms of the index of the Dirac operator valued in the corresponding “$K$–theories with singularities”. The key ideas are based on the construction due to Stolz, some stable homotopy theory, and the index theory for the Dirac operator applied to the manifolds with singularities. As a side-product we compute homotopy types of the corresponding classifying spectra.

It is shown that, in the 1–jet space of the circle, the swapping and the flyping procedures, which produce topologically equivalent links, can produce nonequivalent legendrian links. Each component of the links considered is legendrian isotopic to the 1–jet of the 0–function, and thus cannot be distinguished by the classical rotation number or Thurston–Bennequin invariants. The links are distinguished by calculating invariant polynomials defined via homology groups associated to the links through the theory of generating functions. The many calculations of these generating function polynomials support the belief that these polynomials carry the same information as a refined version of Chekanov’s first order polynomials which are defined via the theory of holomorphic curves.

Let $\Sigma$ be a smooth complex curve, and let $S$ be the product ruled surface $\Sigma \times ℂP^1$. We prove a correspondence conjectured by Donaldson between finite energy $U\left(2\right)$–instantons over $\Sigma \times S^1\times ℝ$, and rank 2 holomorphic bundles over $S$ whose restrictions to $\Sigma \times \left\{0\right\},\Sigma \times \left\{\infty \right\}$ are stable.

We use the criteria of Lalonde and McDuff to show that a path that is generated by a generic autonomous Hamiltonian is length minimizing with respect to the Hofer norm among all homotopic paths provided that it induces no non-constant closed trajectories in $M$. This generalizes a result of Hofer for symplectomorphisms of Euclidean space. The proof for general $M$ uses Liu–Tian’s construction of $S^1$–invariant virtual moduli cycles. As a corollary, we find that any semifree action of $S^1$ on $M$ gives rise to a nontrivial element in the fundamental group of the symplectomorphism group of $M$. We also establish a version of the area-capacity inequality for quasicylinders.

We provide a framework for studying the interplay between concordance and positive mutation and identify some of the basic structures relating the two.

The fundamental result in understanding knot concordance is the structure theorem proved by Levine: for $n>1$ there is an isomorphism $\varphi$ from the concordance group ${\mathcal{C}}_n$ of knotted $\left(2n-1\right)$–spheres in $S^{2n+1}$ to an algebraically defined group ${\mathcal{G}}_{\pm }$; furthermore, ${\mathcal{G}}_{\pm }$ is isomorphic to the infinite direct sum ${\mathbb{Z}}^{\infty }\oplus {\mathbb{Z}}_2^{\infty }\oplus {\mathbb{Z}}_4^{\infty }$. It was a startling consequence of the work of Casson and Gordon that in the classical case the kernel of $\varphi$ on ${\mathcal{C}}_1$ is infinitely generated. Beyond this, little has been discovered about the pair $\left({\mathcal{C}}_1,\varphi \right)$.

In this paper we present a new approach to studying ${\mathcal{C}}_1$ by introducing a group, $\mathcal{ℳ}$, defined as the quotient of the set of knots by the equivalence relation generated by concordance and positive mutation, with group operation induced by connected sum. We prove there is a factorization of $\varphi$, ${\mathcal{C}}_1\stackrel{{\varphi }_1}{\to }\mathcal{ℳ}\stackrel{{\varphi }_2}{\to }{\mathcal{G}}_{-}$. Our main result is that both maps have infinitely generated kernels.

Among geometric constructions on classical knots, the most subtle is positive mutation. Positive mutants are indistinguishable using classical abelian knot invariants as well as by such modern invariants as the Jones, Homfly or Kauffman polynomials. Distinguishing positive mutants up to concordance is a far more difficult problem; only one example has been known until now. The results in this paper provide, among other results, the first infinite families of knots that are distinct from their positive mutants, even up to concordance.

From any configuration of finitely many points in Euclidean three-space, Atiyah constructed a determinant and conjectured that it was always non-zero. In this article we prove the conjecture for the case of four points.

We study the space of positive scalar curvature (psc) metrics on a 4–manifold, and give examples of simply connected manifolds for which it is disconnected. These examples imply that concordance of psc metrics does not imply isotopy of such metrics. This is demonstrated using a modification of the 1–parameter Seiberg–Witten invariants which we introduced in earlier work. The invariant shows that the diffeomorphism group of the underlying 4–manifold is disconnected. We also study the moduli space of positive scalar curvature metrics modulo diffeomorphism, and give examples to show that this space can be disconnected. The (non-orientable) 4–manifolds in this case are explicitly described, and the components in the moduli space are distinguished by a ${Pin}^c$ eta invariant.

In this note we seek to remedy errors which appeared in [On the classification of tightcontact structures I, Geom. Topol. 4 (2000) 309–368] and were propagated in subsequent papers.

Sections 3 and 4 of “Lefschetz fibrations on compact Stein surfaces” [Geometry and Topology 5 (2001) 319–334] contained errors due to insufficient care with framing conventions. This note corrects these errors. The following material is a verbatim substitute for half of section 3 and section 4 starting at the heading “General case” on page 330.