There are two types of $J$–holomorphic spheres in a symplectic manifold which are invariant under an anti-symplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$–holomorphic disks, which are well studied in the literature. In this paper, we first study moduli spaces describing the latter and then combine the two types of moduli spaces to get a well-defined theory of counting real curves of genus 0. We use equivariant localization to show that these invariants (unlike the disk invariants) are essentially the same for the two (standard) involutions on ${\mathbb{P}}^{4n-1}$.

Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields need not be path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seifert fibred spaces are deformations of taut or Reebless foliations when the genus of the base is positive or the twisting number of the contact structure in the sense of Giroux is non-negative.

We study the problem of finding generators for the fundamental group $G$ of a space of the following sort: one removes a family of complex hyperplanes from ${ℂ}^n$, or complex hyperbolic space $ℂ{ℍ}^n$, or the Hermitian symmetric space for $O\left(2,n\right)$, and then takes the quotient by a discrete group $P\Gamma$. The classical example is the braid group, but there are many similar “braid-like” groups that arise in topology and algebraic geometry. Our main result is that if $P\Gamma$ contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certain property, then $G$ is generated by the analogues of the generators of the classical braid group. We apply this to obtain generators for $G$ in a particular intricate example in $ℂ{ℍ}^{13}$. The interest in this example comes from a conjectured relationship between this braid-like group and the monster simple group $M$, that gives geometric meaning to the generators and relations in the Conway–Simons presentation of $\left(M\times M\right):2$. We also suggest some other applications of our machinery.

We develop techniques for defining and working with virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves which are not necessarily cut out transversally. Such techniques have the potential for applications as foundations for invariants in symplectic topology arising from “counting” pseudo-holomorphic curves.

We introduce the notion of an implicit atlas on a moduli space, which is (roughly) a convenient system of local finite-dimensional reductions. We present a general intrinsic strategy for constructing a canonical implicit atlas on any moduli space of pseudo-holomorphic curves. The main technical step in applying this strategy in any particular setting is to prove appropriate gluing theorems. We require only topological gluing theorems, that is, smoothness of the transition maps between gluing charts need not be addressed. Our approach to virtual fundamental cycles is algebraic rather than geometric (in particular, we do not use perturbation). Sheaf-theoretic tools play an important role in setting up our functorial algebraic “VFC package”.

We illustrate the methods we introduce by giving definitions of Gromov–Witten invariants and Hamiltonian Floer homology over $\mathbb{Q}$ for general symplectic manifolds. Our framework generalizes to the $S^1$–equivariant setting, and we use $S^1$–localization to calculate Hamiltonian Floer homology. The Arnold conjecture (as treated by Floer, by Hofer and Salamon, by Ono, by Liu and Tian, by Ruan, and by Fukaya and Ono) is a well-known corollary of this calculation.

(1) We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. (2) We show that linear growth of mod $p$ Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is “large”.

Normal surface theory, a tool to represent surfaces in a triangulated 3–manifold combinatorially, is ubiquitous in computational 3–manifold theory. In this paper, we investigate a relaxed notion of normal surfaces where we remove the quadrilateral conditions. This yields normal surfaces that are no longer embedded. We prove that it is NP-hard to decide whether such a surface is immersed. Our proof uses a reduction from Boolean constraint satisfaction problems where every variable appears in at most two clauses, using a classification theorem of Feder. We also investigate variants, and provide a polynomial-time algorithm to test for a local version of this problem.

ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain and target are ellipsoids (proved by McDuff), and more generally when the domain is a “concave toric domain” and the target is a “convex toric domain” (proved by Cristofaro-Gardiner). However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk. This paper uses more refined information from ECH to give stronger symplectic embedding obstructions when the domain is a polydisk, or more generally a convex toric domain. We use these new obstructions to reprove a result of Hind and Lisi on symplectic embeddings of a polydisk into a ball, and generalize this to obstruct some symplectic embeddings of a polydisk into an ellipsoid. We also obtain a new obstruction to symplectically embedding one polydisk into another, in particular proving the four-dimensional case of a conjecture of Schlenk.

There are known infinite families of Brieskorn homology $3$–spheres which can be realized as boundaries of smooth contractible $4$–manifolds. In this paper we show that smooth free periodic actions on these Brieskorn spheres do not extend smoothly over a contractible $4$–manifold. We give a new infinite family of examples in which the actions extend locally linearly but not smoothly.

We advance the understanding of $K$–theory of quadratic forms by computing the slices of the motivic spectra representing hermitian $K$–groups and Witt groups. By an explicit computation of the slice spectral sequence for higher Witt theory, we prove Milnor’s conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian $K$–groups in terms of motivic cohomology.

We give counterexamples to a question of Bowditch that asks whether a nonelementary type-preserving representation $\rho :{\pi }_1\left({\Sigma }_{g,n}\right)\to PSL\left(2;ℝ\right)$ of a punctured surface group that sends every nonperipheral simple closed curve to a hyperbolic element must $\rho$ be Fuchsian. The counterexamples come from relative Euler class $\pm 1$ representations of the four-punctured sphere group. We also show that the mapping class group action on each nonextremal component of the character space of type-preserving representations of the four-punctured sphere group is ergodic, which confirms a conjecture of Goldman for this case. The main tool we use are Kashaev and Penner’s lengths coordinates of the decorated character spaces.