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We construct a natural smooth compactification of the space of smooth genus-one curves with distinct points in a projective space. It can be viewed as an analogue of a well-known smooth compactification of the space of smooth genus-zero curves, that is, the space of stable genus-zero maps . In fact, our compactification is obtained from the singular space of stable genus-one maps through a natural sequence of blowups along “bad” subvarieties. While this construction is simple to describe, it requires more work to show that the end result is a smooth space. As a bonus, we obtain desingularizations of certain natural sheaves over the “main” irreducible component of . A number of applications of these desingularizations in enumerative geometry and Gromov–Witten theory are described in the introduction, including the second author’s proof of physicists’ predictions for genus-one Gromov–Witten invariants of a quintic threefold.
Let be the subgroup of the extended mapping class group, , generated by Dehn twists about separating curves. In our earlier paper, we showed that when is a closed, connected, orientable surface of genus . By modifying our original proof, we show that the same result holds for , thus confirming Farb’s conjecture in all cases (the statement is not true for ).
We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The –homotopy theory is “pieced together” from the –homotopy theories for suitable quotient groups of ; a motivation is the way continuous group cohomology of a profinite group is built out of the cohomology of its finite quotient groups. In the model category of equivariant spectra Postnikov towers are studied from a general perspective. We introduce pro––spectra and construct various model structures on them. A key property of the model structures is that pro–spectra are weakly equivalent to their Postnikov towers. We discuss two versions of a model structure with “underlying weak equivalences”. One of the versions only makes sense for pro–spectra. In the end we use the theory to study homotopy fixed points of pro––spectra.
We construct a Teichmüller geodesic which does not have a limit on the Thurston boundary of the Teichmüller space. We also show that for this construction the limit set is contained in a one-dimensional simplex in .
We give new information about the relationship between the low-dimensional homology of a space and the derived series of its fundamental group. Applications are given to detecting when a set of elements of a group generates a subgroup “large enough” to map onto a nonabelian free solvable group, and to concordance and grope cobordism of classical links. We also greatly generalize several key homological results employed in recent work of Cochran–Orr–Teichner in the context of classical knot concordance.
In 1963 J Stallings established a strong relationship between the low-dimensional homology of a group and its lower central series quotients. In 1975 W Dwyer extended Stallings’ theorem by weakening the hypothesis on . In 2003 the second author introduced a new characteristic series, , associated to the derived series, called the torsion-free derived series. The authors previously established a precise analogue, for the torsion-free derived series, of Stallings’ theorem. Here our main result is the analogue of Dwyer’s theorem for the torsion-free derived series. We also prove a version of Dwyer’s theorem for the rational lower central series. We apply these to give new results on the Cochran–Orr–Teichner filtration of the classical link concordance group.
Let be a one-cusped hyperbolic –manifold. A slope on the boundary of the compact core of is called exceptional if the corresponding Dehn filling produces a non-hyperbolic manifold. We give new upper bounds for the distance between two exceptional slopes and in several situations. These include cases where is reducible and where has finite , or is very small, or admits a –injective immersed torus.
Sutured Floer homology, denoted by , is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if is a sutured manifold decomposition then is a direct summand of . To prove the decomposition formula we give an algorithm that computes from a balanced diagram defining that generalizes the algorithm of Sarkar and Wang.
As a corollary we obtain that if is taut then . Other applications include simple proofs of a result of Ozsváth and Szabó that link Floer homology detects the Thurston norm, and a theorem of Ni that knot Floer homology detects fibred knots. Our proofs do not make use of any contact geometry.
Moreover, using these methods we show that if is a genus knot in a rational homology –sphere whose Alexander polynomial has leading coefficient and if then admits a depth taut foliation transversal to .
We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmüller space, complementing a result of Scannell–Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complex-analytic and geometric coordinate systems for the space of complex projective () structures on a surface.
We also study the rays in Teichmüller space associated to the grafting coordinates, obtaining estimates for extremal and hyperbolic length functions and their derivatives along these grafting rays.
For any group G, we define a new characteristic series related to the derived series, that we call the torsion-free derived series of G. Using this series and the Cheeger–Gromov –invariant, we obtain new real-valued homology cobordism invariants for closed –dimensional manifolds. For –dimensional manifolds, we show that is a linearly independent set and for each , the image of is an infinitely generated and dense subset of .
In their seminal work on knot concordance, T Cochran, K Orr and P Teichner define a filtration of the –component (string) link concordance group, called the –solvable filtration. They also define a grope filtration . We show that vanishes for –solvable links. Using this, and the nontriviality of , we show that for each , the successive quotients of the –solvable filtration of the link concordance group contain an infinitely generated subgroup. We also establish a similar result for the grope filtration. We remark that for knots (), the successive quotients of the –solvable filtration are known to be infinite. However, for knots, it is unknown if these quotients have infinite rank when .
We study quakebend deformations in complex hyperbolic quasi-Fuchsian space of a closed surface of genus , that is the space of discrete, faithful, totally loxodromic and geometrically finite representations of the fundamental group of into the group of isometries of complex hyperbolic space. Emanating from an –Fuchsian point , we construct curves associated to complex hyperbolic quakebending of and we prove that we may always find an open neighborhood of in containing pieces of such curves. Moreover, we present generalisations of the well known Wolpert–Kerckhoff formulae for the derivatives of geodesic length function in Teichmüller space.
Pursuing our investigations on the relations between Thompson groups and mapping class groups, we introduce the group (and its companion ) which is an extension of the Ptolemy–Thompson group by the braid group on infinitely many strands. We prove that is a finitely presented group by constructing a complex on which it acts cocompactly with finitely presented stabilizers, and derive from it an explicit presentation. The groups and are in the same relation with respect to each other as the braid groups and , for infinitely many strands . We show that both groups embed as groups of homeomorphisms of the circle and their word problem is solvable.
We develop a method for preserving pseudoholomorphic curves in contact 3–manifolds under surgery along transverse links. This makes use of a geometrically natural boundary value problem for holomorphic curves in a 3–manifold with stable Hamiltonian structure, where the boundary conditions are defined by 1–parameter families of totally real surfaces. The technique is applied here to construct a finite energy foliation for every closed overtwisted contact 3–manifold.
We show that the universal covering of any compact, negatively curved manifold has an exponential growth rate which is strictly greater than the exponential growth rate of any other normal covering . Moreover, we give an explicit formula estimating the difference between and in terms of the systole of and of other elementary geometric parameters of the base space . Then we discuss some applications of this formula to periodic geodesics, to the bottom of the spectrum and to the critical exponent of normal coverings.