Let $G$ be a countable discrete group and let $M$ be a smooth proper cocompact $G$-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant $KO$–homology of $M$. The universal equivariant Euler characteristic of $M$, which lives in a group $U^G\left(M\right)$, counts the equivariant cells of $M$, taking the component structure of the various fixed point sets into account. We construct a natural homomorphism from $U^G\left(M\right)$ to the equivariant $KO$-homology of $M$. The main result of this paper says that this map sends the universal equivariant Euler characteristic to the equivariant Euler class. In particular this shows that there are no “higher” equivariant Euler characteristics. We show that, rationally, the equivariant Euler class carries the same information as the collection of the orbifold Euler characteristics of the components of the $L$–fixed point sets $M^L$, where $L$ runs through the finite cyclic subgroups of $G$. However, we give an example of an action of the symmetric group $S_3$ on the 3–sphere for which the equivariant Euler class has order 2, so there is also some torsion information.

We use the knot filtration on the Heegaard Floer complex $\widehat{CF}$ to define an integer invariant $\tau \left(K\right)$ for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to $\mathbb{Z}$. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, $\tau$ gives sharp bounds on the four-ball genera of torus knots. As another illustration, we calculate the invariant for several ten-crossing knots.

As a corollary of work of Ozsváth and Szabó, it is shown that the classical concordance group of algebraically slice knots has an infinite cyclic summand and in particular is not a divisible group.

We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal $n$–excisive approximation, which may be thought of as its $n$–excisive part. Homogeneous functors, meaning $n$–excisive functors with trivial $\left(n-1\right)$–excisive part, can be classified: they correspond to symmetric functors of $n$ variables that are reduced and $1$–excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen’s algebraic $K$–theory.

The main result of this paper is that every non-trivial Hamiltonian diffeomorphism of a closed oriented surface of genus at least one has periodic points of arbitrarily high period. The same result is true for $S^2$ provided the diffeomorphism has at least three fixed points. In addition we show that up to isotopy relative to its fixed point set, every orientation preserving diffeomorphism $F:S\to S$ of a closed orientable surface has a normal form. If the fixed point set is finite this is just the Thurston normal form.

Suppose $\mathcal{G}$ is an almost simple group containing a subgroup isomorphic to the three-dimensional integer Heisenberg group. For example any finite index subgroup of $SL\left(3,\mathbb{Z}\right)$ is such a group. The main result of this paper is that every action of $\mathcal{G}$ on a closed oriented surface by area preserving diffeomorphisms factors through a finite group.

Given an oriented rational homology $3$–sphere $M$, it is known how to associate to any Spin${}^c$–structure $\sigma$ on $M$ two quadratic functions over the linking pairing. One quadratic function is derived from the reduction modulo $1$ of the Reidemeister–Turaev torsion of $\left(M,\sigma \right)$, while the other one can be defined using the intersection pairing of an appropriate compact oriented $4$–manifold with boundary $M$. In this paper, using surgery presentations of the manifold $M$, we prove that those two quadratic functions coincide. Our proof relies on the comparison between two distinct combinatorial descriptions of Spin${}^c$–structures on $M$: Turaev’s charges vs Chern vectors.

We prove that every closed oriented 3–manifold admits a hyperbolic cone–manifold structure with cone–angle arbitrarily close to $2\pi$.

This is one in a series of papers devoted to the foundations of Symplectic Field. We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov’s compactness theorem as well as compactness theorems in Floer homology theory and in contact geometry.

Using Furuta’s idea of finite dimensional approximation in Seiberg–Witten theory, we refine Seiberg–Witten Floer homology to obtain an invariant of homology 3–spheres which lives in the $S^1$–equivariant graded suspension category. In particular, this gives a construction of Seiberg–Witten Floer homology that avoids the delicate transversality problems in the standard approach. We also define a relative invariant of four-manifolds with boundary which generalizes the Bauer–Furuta stable homotopy invariant of closed four-manifolds.

We state and prove a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups. For that, we explain how to contruct a boundary for a group that is an acylindrical amalgamation of relatively hyperbolic groups over a fully quasi-convex subgroup. We apply our result to Sela’s theory on limit groups and prove their relative hyperbolicity. We also get a proof of the Howson property for limit groups.

A new diffeomorphism invariant of integral homology 3–spheres is defined using a non-abelian “quaternionic” version of the Seiberg–Witten equations.

We develop a general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model categories, extending work of Bousfield–Kan and Bendersky–Thompson for ordinary spaces. This is based on a generalized cosimplicial version of the Dwyer–Kan–Stover theory of resolution model categories, and we are able to construct our homotopy spectral sequences and completions using very flexible weak resolutions in the spirit of relative homological algebra. We deduce that our completion functors have triple structures and preserve certain fiber squares up to homotopy. We also deduce that the Bendersky–Thompson completions over connective ring spectra are equivalent to Bousfield–Kan completions over solid rings. The present work allows us to show, in a subsequent paper, that the homotopy spectral sequences over arbitrary ring spectra have well-behaved composition pairings.

We prove that an infinite family of virtually overtwisted tight contact structures discovered by Honda on certain circle bundles over surfaces admit no symplectic semi–fillings. The argument uses results of Mrowka, Ozsváth and Yu on the translation–invariant solutions to the Seiberg–Witten equations on cylinders and the non–triviality of the Kronheimer–Mrowka monopole invariants of symplectic fillings.