We introduce a new technique for showing classical knots and links are not slice. As one applicat... more We introduce a new technique for showing classical knots and links are not slice. As one application we show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the Cheeger-Gromov bound, a deep analytical tool used by Cochran-Teichner. Our main examples are actually boundary links but cannot be detected in the algebraic boundary link concordance group, nor by any ρ invariants associated to solvable representations into finite unitary groups.
Every undergraduate interested in graduate school should consider attending a Research Experience... more Every undergraduate interested in graduate school should consider attending a Research Experience for Undergraduates (REU) program. The National Science Foundation yearly sponsors approximately twenty REU programs in mathematics that are held at various universities around the country during the summer. The programs range from eight to ten weeks in length, admit anywhere from six to twelve students, and include topics ranging from algebraic geometry and computational group theory to population dynamics and topology. These programs allow an undergraduate to closely work with a faculty member on some component of the faculty’s research. These situations allow students to experience graduate study firsthand very early in their careers. The authors attended Cal Poly San Luis Obispo, a predominately teaching university. As a result of our REU experiences, we both applied to graduate programs for the fall of 1997 and each of us is now in her first year of graduate studies. Andrea is worki...
Recall that a group is called large if it has a finite index subgroup which surjects onto a non-a... more Recall that a group is called large if it has a finite index subgroup which surjects onto a non-abelian free group. By work of Agol and Cooper-Long-Reid, most 3-manifold groups are large; in particular, the fundamental groups of hyperbolic 3-manifolds are large. In previous work, the first author gave examples of closed, hyperbolic 3-manifolds with arbitrarily large first homology rank but whose fundamental groups do not surject onto a non-abelian free group. We call a group very large if it surjects onto a non-abelian free group. In this paper, we consider the question of whether the groups of homology handlebodies-which are very close to being free-are very large. We show that the fundamental group of W. Thurston's tripus manifold, is not very large; it is known to be weakly parafree by Stallings' Theorem and large by the work of Cooper-Long-Reid since the tripus is a hyperbolic manifold with totally geodesic boundary. It is still unknown if a 3-manifold group that is weakly parafree of rank at least 3 must be very large. However, we more generally consider the co-rank of the fundamental group, also known as the cut number of the manifold. For each integer g ≥ 1 we construct a homology handlebody Yg of genus g whose group has co-rank equal to r(g), where r(g) = g/2 for g even and r(g) = (g + 1)/2 for g odd. That is, these groups are weakly parafree of rank g and surject onto a free group of rank roughly half of g but no larger.
Most of the 50-years of study of the set of knot concordance classes, C, has focused on its struc... more Most of the 50-years of study of the set of knot concordance classes, C, has focused on its structure as an abelian group. Here we take a different approach, namely we study C as a metric space admitting many natural geometric operators, especially satellite operators. We consider several knot concordance spaces, corresponding to different categories of concordance, and two different metrics. We establish the existence of quasi-n-flats for every n, implying that C admits no quasi-isometric embedding into a finite product of (Gromov) hyperbolic spaces. We show that every satellite operator is a quasi-homomorphism P : C → C. We show that winding number one satellite operators induce quasi-isometries with respect to the metric induced by slice genus. We prove that strong winding number one patterns induce isometric embeddings for certain metrics. By contrast, winding number zero satellite operators are bounded functions and hence quasi-contractions. These results contribute to the suggestion that C is a fractal space. We establish various other results about the large-scale geometry of arbitrary satellite operators. 1 n. Here C top is the (usual) set of topological knot concordance classes wherein two knots are equivalent if there exists a collared proper topological embedding of an annulus into a topological manifold homeomorphic to S 3 × [0, 1] which restricts on its boundary to the given knots. C ex , short for C exotic , is the set of equivalence classes of knots where two are equivalent if they cobound a properly, smoothly embedded annulus in a smooth manifold homeomorphic to S 3 × [0, 1]; that is to say they are concordant in S 3 × [0, 1] equipped with a possibly exotic smooth structure (see [4][25,
We extend the theory of combinatorial link Floer homology to a class of oriented spatial graphs c... more We extend the theory of combinatorial link Floer homology to a class of oriented spatial graphs called transverse spatial graphs. To do this, we define the notion of a grid diagram representing a transverse spatial graph, which we call a graph grid diagram. We prove that two graph grid diagrams representing the same transverse spatial graph are related by a sequence of graph grid moves, generalizing the work of Cromwell for links. For a graph grid diagram representing a transverse spatial graph f W G ! S 3 , we define a relatively bigraded chain complex (which is a module over a multivariable polynomial ring) and show that its homology is preserved under the graph grid moves; hence it is an invariant of the transverse spatial graph. In fact, we define both a minus and hat version. Taking the graded Euler characteristic of the homology of the hat version gives an Alexander type polynomial for the transverse spatial graph. Specifically, for each transverse spatial graph f , we define a balanced sutured manifold .S 3 n f .G/; .f //. We show that the graded Euler characteristic is the same as the torsion of .S 3 n f .
To a special type of grope embedded in 4-space, that we call a branchsymmetric grope, we associat... more To a special type of grope embedded in 4-space, that we call a branchsymmetric grope, we associate a length function for each real number q ≥ 1. This gives rise to a family of pseudo-metrics d q , refining the slice genus metric, on the set of concordance classes of knots, as the infimum of the length function taken over all possible grope concordances between two knots. We investigate the properties of these metrics. The main theorem is that the topology induced by this metric on the knot concordance set is not discrete for all q > 1. The analogous statement for links also holds for q = 1. In addition we translate much previous work on knot concordance into distance statements. In particular, we show that winding number zero satellite operators are contractions in many cases, and we give lower bounds on our metrics arising from knot signatures and higher order signatures. This gives further evidence in favor of the conjecture that the knot concordance group has a fractal structure.
We introduce a new technique for showing classical knots and links are not slice. As one applicat... more We introduce a new technique for showing classical knots and links are not slice. As one application we resolve a long-standing question as to whether certain natural families of knots contain topologically slice knots. We also present a simpler proof of the result of Cochran-Teichner that the successive quotients of the integral terms of the Cochran-Orr-Teichner filtration of the knot concordance group have rank 1. For links we have similar results. We show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the Cheeger-Gromov bound, a deep analytical tool used by Cochran-Teichner. Our main examples are actually boundary links but cannot be detected in the algebraic boundary link concordance group, nor by any $\rho$ invariants associated to solvable representations into finite unitary groups.
Most of the 50-year history of the study of the set of knot concordance classes, C, has focused o... more Most of the 50-year history of the study of the set of knot concordance classes, C, has focused on its structure as an abelian group. Here we take a different approach, namely we study C as a metric space admitting many natural geometric operators, especially satellite operators. We consider several knot concordance spaces, corresponding to different categories of concordance, and two different metrics. We establish the existence of quasi-n-flats for every n, implying that C admits no quasi-isometric embedding into a finite product of (Gromov) hyperbolic spaces. We show that every satellite operator is a quasi-homomorphism P: C --- C with respect to the metric given by the slice genus. We show that winding number one satellite operators induce quasi-isometries. We prove that strong winding number one satellite operators induce isometric embeddings for certain metrics. By contrast, winding number zero satellite operators are bounded functions and hence quasi-contractions. These results contribute to the conjecture that C is a fractal space. We establish various other results about the large-scale geometry of arbitrary satellite operators.
Proceedings of the London Mathematical Society, 2011
Cochran-Orr-Teichner introduced in [11] a natural filtration of the smooth knot concordance group... more Cochran-Orr-Teichner introduced in [11] a natural filtration of the smooth knot concordance group C • • • ⊂ Fn+1 ⊂ Fn.5 ⊂ Fn ⊂ • • • ⊂ F1 ⊂ F0.5 ⊂ F0 ⊂ C, called the (n)-solvable filtration. We show that each associated graded abelian group {Gn = Fn/Fn.5 | n ∈ N}, n ≥ 2 contains infinite linearly independent sets of elements of order 2 (this was known previously for n = 0, 1). Each of the representative knots is negative amphichiral, with vanishing s-invariant, τ-invariant, δ-invariants and Casson-Gordon invariants. Moreover each is slice in a rational homology 4-ball. In fact we show that there are many distinct such classes in Gn, one for each "distinct" n-tuple P = (p1(t), ..., pn(t)) of knot polynomials. Such a sequence of polynomials records the orders of certain submodules of a sequence of higher-order Alexander modules of the knot.
We define the stability of a subgroup under a class of maps, and establish the basic properties o... more We define the stability of a subgroup under a class of maps, and establish the basic properties of this notion. Loosely speaking, we will say that a normal subgroup, or more generally a normal series {An} of a group A, is stable under a class of homomorphisms H if whenever f : A → B lies in H, we have that f (a) ∈ Bn if and only if a ∈ An. This translates to saying that each element of H induces a monomorphism A/An ֒→ B/Bn. This contrasts with the usual theories of localization wherein one is concerned with situations where f induces an isomorphism. In the literature, the most commonly considered class of maps are those that induce isomorphisms on (low-dimensional) group homology. The model theorem in this regard is the 1963 result of J. Stallings that (each term of) the lower central series is preserved under any Z-homological equivalence of groups [9]. Various other theorems of this nature have since appeared, involving different series of groups-variations of the lower central series. W. Dwyer generalized Stallings' Z results to larger classes of maps [7], work that was completed in the other cases by the authors. More recently the authors proved analogues of the theorems of Stallings and Dwyer for variations of the derived series [4] [6] [5]. The above theorems are all different but clearly have much in common. We interpret all of these results in the framework of stability.
For each sequence P = (p1(t), p2(t),. . .) of polynomials we define a characteristic series of gr... more For each sequence P = (p1(t), p2(t),. . .) of polynomials we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S 3 , such a sequence of polynomials arises naturally as the orders of certain submodules of a sequence of higher-order Alexander modules of K. These group series yield filtrations of the knot concordance group that refine the (n)-solvable filtration. We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higher-order analogues of the p(t)-primary decomposition of the algebraic concordance group. We use these techniques to give evidence that the set of smooth concordance classes of knots is a fractal set. We also show that no Cochran-Orr-Teichner knot is concordant to any Cochran-Harvey-Leidy knot.
In 1964, John Stallings established an important relationship between the lowdimensional homology... more In 1964, John Stallings established an important relationship between the lowdimensional homology of a group and its lower central series. We establish a similar relationship between the low-dimensional homology of a group and its derived series. We also define a torsion-free-solvable completion of a group that is analogous to the Malcev completion, with the role of the lower central series replaced by the derived series. We prove that the torsion-free-solvable completion is invariant under rational homology equivalence.
We study contact manifolds that arise as cyclic branched covers of transverse knots in the standa... more We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic transverse knots with the same self-linking number. These pairs of knots include most of the non-transversely simple knots of Birman-Menasco and Ng-Ozsváth-Thurston.
We define families of invariants for elements of the mapping class group of Σ, a compact orientab... more We define families of invariants for elements of the mapping class group of Σ, a compact orientable surface. Fix any characteristic subgroup H π1(Σ) and restrict to J(H), any subgroup of mapping classes that induce the identity on π1(Σ)/H. To any unitary representation ψ of π1(Σ)/H we associate a higher-order ρ ψ-invariant and a signature 2-cocycle σ ψ. These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular each ρ ψ is a quasimorphism and each σ ψ is a bounded 2-cocycle on J(H). In one of the simplest non-trivial cases, by varying ψ, we exhibit infinite families of linearly independent quasimorphisms and signature cocycles. We show that the ρ ψ restrict to homomorphisms on certain interesting subgroups. Many of these invariants extend naturally to the full mapping class group and some extend to the monoid of homology cylinders based on Σ.
We give new information about the relationship between the lowdimensional homology of a space and... more We give new information about the relationship between the lowdimensional 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 non-abelian 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 lowdimensional homology of a group and its lower central series quotients. In 1975 W. Dwyer extended Stallings' theorem by weakening the hypothesis on H 2. In 2003 the second author introduced a new characteristic series, G (n) H , 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.
In 1997, T. Cochran, K. Orr, and P. Teichner [12] defined a filtration of the classical knot conc... more In 1997, T. Cochran, K. Orr, and P. Teichner [12] defined a filtration of the classical knot concordance group C,
The question was raised as to whether the cut number of a 3-manifold X is bounded from below by 1... more The question was raised as to whether the cut number of a 3-manifold X is bounded from below by 1 3 β 1 (X). We show that the answer to this question is "no." For each m ≥ 1, we construct explicit examples of closed 3-manifolds X with β 1 (X) = m and cut number 1. That is, π 1 (X) cannot map onto any non-abelian free group. Moreover, we show that these examples can be assumed to be hyperbolic.
We propose and analyze a structure with which to organize the difference between a knot in S 3 bo... more We propose and analyze a structure with which to organize the difference between a knot in S 3 bounding a topologically embedded 2-disk in B 4 and it bounding a smoothly embedded disk. The n-solvable filtration of the topological knot concordance group, due to Cochran-Orr-Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the n-solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, {Bn}, that is simultaneously a refinement of the n-solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each Bn/Bn+1 has infinite rank. But our primary interest is in the induced filtration, {Tn}, on the subgroup, T , of knots that are topologically slice. We prove that T /T0 is large, detected by gauge-theoretic invariants and the τ , s,-invariants; while the non-triviliality of T0/T1 can be detected by certain d-invariants. All of these concordance obstructions vanish for knots in T1. Nonetheless, going beyond this, our main result is that T1/T2 has positive rank. Moreover under a "weak homotopy-ribbon" condition, we show that each Tn/Tn+1 has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.
We introduce a new technique for showing classical knots and links are not slice. As one applicat... more We introduce a new technique for showing classical knots and links are not slice. As one application we show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the Cheeger-Gromov bound, a deep analytical tool used by Cochran-Teichner. Our main examples are actually boundary links but cannot be detected in the algebraic boundary link concordance group, nor by any ρ invariants associated to solvable representations into finite unitary groups.
Every undergraduate interested in graduate school should consider attending a Research Experience... more Every undergraduate interested in graduate school should consider attending a Research Experience for Undergraduates (REU) program. The National Science Foundation yearly sponsors approximately twenty REU programs in mathematics that are held at various universities around the country during the summer. The programs range from eight to ten weeks in length, admit anywhere from six to twelve students, and include topics ranging from algebraic geometry and computational group theory to population dynamics and topology. These programs allow an undergraduate to closely work with a faculty member on some component of the faculty’s research. These situations allow students to experience graduate study firsthand very early in their careers. The authors attended Cal Poly San Luis Obispo, a predominately teaching university. As a result of our REU experiences, we both applied to graduate programs for the fall of 1997 and each of us is now in her first year of graduate studies. Andrea is worki...
Recall that a group is called large if it has a finite index subgroup which surjects onto a non-a... more Recall that a group is called large if it has a finite index subgroup which surjects onto a non-abelian free group. By work of Agol and Cooper-Long-Reid, most 3-manifold groups are large; in particular, the fundamental groups of hyperbolic 3-manifolds are large. In previous work, the first author gave examples of closed, hyperbolic 3-manifolds with arbitrarily large first homology rank but whose fundamental groups do not surject onto a non-abelian free group. We call a group very large if it surjects onto a non-abelian free group. In this paper, we consider the question of whether the groups of homology handlebodies-which are very close to being free-are very large. We show that the fundamental group of W. Thurston's tripus manifold, is not very large; it is known to be weakly parafree by Stallings' Theorem and large by the work of Cooper-Long-Reid since the tripus is a hyperbolic manifold with totally geodesic boundary. It is still unknown if a 3-manifold group that is weakly parafree of rank at least 3 must be very large. However, we more generally consider the co-rank of the fundamental group, also known as the cut number of the manifold. For each integer g ≥ 1 we construct a homology handlebody Yg of genus g whose group has co-rank equal to r(g), where r(g) = g/2 for g even and r(g) = (g + 1)/2 for g odd. That is, these groups are weakly parafree of rank g and surject onto a free group of rank roughly half of g but no larger.
Most of the 50-years of study of the set of knot concordance classes, C, has focused on its struc... more Most of the 50-years of study of the set of knot concordance classes, C, has focused on its structure as an abelian group. Here we take a different approach, namely we study C as a metric space admitting many natural geometric operators, especially satellite operators. We consider several knot concordance spaces, corresponding to different categories of concordance, and two different metrics. We establish the existence of quasi-n-flats for every n, implying that C admits no quasi-isometric embedding into a finite product of (Gromov) hyperbolic spaces. We show that every satellite operator is a quasi-homomorphism P : C → C. We show that winding number one satellite operators induce quasi-isometries with respect to the metric induced by slice genus. We prove that strong winding number one patterns induce isometric embeddings for certain metrics. By contrast, winding number zero satellite operators are bounded functions and hence quasi-contractions. These results contribute to the suggestion that C is a fractal space. We establish various other results about the large-scale geometry of arbitrary satellite operators. 1 n. Here C top is the (usual) set of topological knot concordance classes wherein two knots are equivalent if there exists a collared proper topological embedding of an annulus into a topological manifold homeomorphic to S 3 × [0, 1] which restricts on its boundary to the given knots. C ex , short for C exotic , is the set of equivalence classes of knots where two are equivalent if they cobound a properly, smoothly embedded annulus in a smooth manifold homeomorphic to S 3 × [0, 1]; that is to say they are concordant in S 3 × [0, 1] equipped with a possibly exotic smooth structure (see [4][25,
We extend the theory of combinatorial link Floer homology to a class of oriented spatial graphs c... more We extend the theory of combinatorial link Floer homology to a class of oriented spatial graphs called transverse spatial graphs. To do this, we define the notion of a grid diagram representing a transverse spatial graph, which we call a graph grid diagram. We prove that two graph grid diagrams representing the same transverse spatial graph are related by a sequence of graph grid moves, generalizing the work of Cromwell for links. For a graph grid diagram representing a transverse spatial graph f W G ! S 3 , we define a relatively bigraded chain complex (which is a module over a multivariable polynomial ring) and show that its homology is preserved under the graph grid moves; hence it is an invariant of the transverse spatial graph. In fact, we define both a minus and hat version. Taking the graded Euler characteristic of the homology of the hat version gives an Alexander type polynomial for the transverse spatial graph. Specifically, for each transverse spatial graph f , we define a balanced sutured manifold .S 3 n f .G/; .f //. We show that the graded Euler characteristic is the same as the torsion of .S 3 n f .
To a special type of grope embedded in 4-space, that we call a branchsymmetric grope, we associat... more To a special type of grope embedded in 4-space, that we call a branchsymmetric grope, we associate a length function for each real number q ≥ 1. This gives rise to a family of pseudo-metrics d q , refining the slice genus metric, on the set of concordance classes of knots, as the infimum of the length function taken over all possible grope concordances between two knots. We investigate the properties of these metrics. The main theorem is that the topology induced by this metric on the knot concordance set is not discrete for all q > 1. The analogous statement for links also holds for q = 1. In addition we translate much previous work on knot concordance into distance statements. In particular, we show that winding number zero satellite operators are contractions in many cases, and we give lower bounds on our metrics arising from knot signatures and higher order signatures. This gives further evidence in favor of the conjecture that the knot concordance group has a fractal structure.
We introduce a new technique for showing classical knots and links are not slice. As one applicat... more We introduce a new technique for showing classical knots and links are not slice. As one application we resolve a long-standing question as to whether certain natural families of knots contain topologically slice knots. We also present a simpler proof of the result of Cochran-Teichner that the successive quotients of the integral terms of the Cochran-Orr-Teichner filtration of the knot concordance group have rank 1. For links we have similar results. We show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the Cheeger-Gromov bound, a deep analytical tool used by Cochran-Teichner. Our main examples are actually boundary links but cannot be detected in the algebraic boundary link concordance group, nor by any $\rho$ invariants associated to solvable representations into finite unitary groups.
Most of the 50-year history of the study of the set of knot concordance classes, C, has focused o... more Most of the 50-year history of the study of the set of knot concordance classes, C, has focused on its structure as an abelian group. Here we take a different approach, namely we study C as a metric space admitting many natural geometric operators, especially satellite operators. We consider several knot concordance spaces, corresponding to different categories of concordance, and two different metrics. We establish the existence of quasi-n-flats for every n, implying that C admits no quasi-isometric embedding into a finite product of (Gromov) hyperbolic spaces. We show that every satellite operator is a quasi-homomorphism P: C --- C with respect to the metric given by the slice genus. We show that winding number one satellite operators induce quasi-isometries. We prove that strong winding number one satellite operators induce isometric embeddings for certain metrics. By contrast, winding number zero satellite operators are bounded functions and hence quasi-contractions. These results contribute to the conjecture that C is a fractal space. We establish various other results about the large-scale geometry of arbitrary satellite operators.
Proceedings of the London Mathematical Society, 2011
Cochran-Orr-Teichner introduced in [11] a natural filtration of the smooth knot concordance group... more Cochran-Orr-Teichner introduced in [11] a natural filtration of the smooth knot concordance group C • • • ⊂ Fn+1 ⊂ Fn.5 ⊂ Fn ⊂ • • • ⊂ F1 ⊂ F0.5 ⊂ F0 ⊂ C, called the (n)-solvable filtration. We show that each associated graded abelian group {Gn = Fn/Fn.5 | n ∈ N}, n ≥ 2 contains infinite linearly independent sets of elements of order 2 (this was known previously for n = 0, 1). Each of the representative knots is negative amphichiral, with vanishing s-invariant, τ-invariant, δ-invariants and Casson-Gordon invariants. Moreover each is slice in a rational homology 4-ball. In fact we show that there are many distinct such classes in Gn, one for each "distinct" n-tuple P = (p1(t), ..., pn(t)) of knot polynomials. Such a sequence of polynomials records the orders of certain submodules of a sequence of higher-order Alexander modules of the knot.
We define the stability of a subgroup under a class of maps, and establish the basic properties o... more We define the stability of a subgroup under a class of maps, and establish the basic properties of this notion. Loosely speaking, we will say that a normal subgroup, or more generally a normal series {An} of a group A, is stable under a class of homomorphisms H if whenever f : A → B lies in H, we have that f (a) ∈ Bn if and only if a ∈ An. This translates to saying that each element of H induces a monomorphism A/An ֒→ B/Bn. This contrasts with the usual theories of localization wherein one is concerned with situations where f induces an isomorphism. In the literature, the most commonly considered class of maps are those that induce isomorphisms on (low-dimensional) group homology. The model theorem in this regard is the 1963 result of J. Stallings that (each term of) the lower central series is preserved under any Z-homological equivalence of groups [9]. Various other theorems of this nature have since appeared, involving different series of groups-variations of the lower central series. W. Dwyer generalized Stallings' Z results to larger classes of maps [7], work that was completed in the other cases by the authors. More recently the authors proved analogues of the theorems of Stallings and Dwyer for variations of the derived series [4] [6] [5]. The above theorems are all different but clearly have much in common. We interpret all of these results in the framework of stability.
For each sequence P = (p1(t), p2(t),. . .) of polynomials we define a characteristic series of gr... more For each sequence P = (p1(t), p2(t),. . .) of polynomials we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S 3 , such a sequence of polynomials arises naturally as the orders of certain submodules of a sequence of higher-order Alexander modules of K. These group series yield filtrations of the knot concordance group that refine the (n)-solvable filtration. We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higher-order analogues of the p(t)-primary decomposition of the algebraic concordance group. We use these techniques to give evidence that the set of smooth concordance classes of knots is a fractal set. We also show that no Cochran-Orr-Teichner knot is concordant to any Cochran-Harvey-Leidy knot.
In 1964, John Stallings established an important relationship between the lowdimensional homology... more In 1964, John Stallings established an important relationship between the lowdimensional homology of a group and its lower central series. We establish a similar relationship between the low-dimensional homology of a group and its derived series. We also define a torsion-free-solvable completion of a group that is analogous to the Malcev completion, with the role of the lower central series replaced by the derived series. We prove that the torsion-free-solvable completion is invariant under rational homology equivalence.
We study contact manifolds that arise as cyclic branched covers of transverse knots in the standa... more We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic transverse knots with the same self-linking number. These pairs of knots include most of the non-transversely simple knots of Birman-Menasco and Ng-Ozsváth-Thurston.
We define families of invariants for elements of the mapping class group of Σ, a compact orientab... more We define families of invariants for elements of the mapping class group of Σ, a compact orientable surface. Fix any characteristic subgroup H π1(Σ) and restrict to J(H), any subgroup of mapping classes that induce the identity on π1(Σ)/H. To any unitary representation ψ of π1(Σ)/H we associate a higher-order ρ ψ-invariant and a signature 2-cocycle σ ψ. These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular each ρ ψ is a quasimorphism and each σ ψ is a bounded 2-cocycle on J(H). In one of the simplest non-trivial cases, by varying ψ, we exhibit infinite families of linearly independent quasimorphisms and signature cocycles. We show that the ρ ψ restrict to homomorphisms on certain interesting subgroups. Many of these invariants extend naturally to the full mapping class group and some extend to the monoid of homology cylinders based on Σ.
We give new information about the relationship between the lowdimensional homology of a space and... more We give new information about the relationship between the lowdimensional 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 non-abelian 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 lowdimensional homology of a group and its lower central series quotients. In 1975 W. Dwyer extended Stallings' theorem by weakening the hypothesis on H 2. In 2003 the second author introduced a new characteristic series, G (n) H , 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.
In 1997, T. Cochran, K. Orr, and P. Teichner [12] defined a filtration of the classical knot conc... more In 1997, T. Cochran, K. Orr, and P. Teichner [12] defined a filtration of the classical knot concordance group C,
The question was raised as to whether the cut number of a 3-manifold X is bounded from below by 1... more The question was raised as to whether the cut number of a 3-manifold X is bounded from below by 1 3 β 1 (X). We show that the answer to this question is "no." For each m ≥ 1, we construct explicit examples of closed 3-manifolds X with β 1 (X) = m and cut number 1. That is, π 1 (X) cannot map onto any non-abelian free group. Moreover, we show that these examples can be assumed to be hyperbolic.
We propose and analyze a structure with which to organize the difference between a knot in S 3 bo... more We propose and analyze a structure with which to organize the difference between a knot in S 3 bounding a topologically embedded 2-disk in B 4 and it bounding a smoothly embedded disk. The n-solvable filtration of the topological knot concordance group, due to Cochran-Orr-Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the n-solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, {Bn}, that is simultaneously a refinement of the n-solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each Bn/Bn+1 has infinite rank. But our primary interest is in the induced filtration, {Tn}, on the subgroup, T , of knots that are topologically slice. We prove that T /T0 is large, detected by gauge-theoretic invariants and the τ , s,-invariants; while the non-triviliality of T0/T1 can be detected by certain d-invariants. All of these concordance obstructions vanish for knots in T1. Nonetheless, going beyond this, our main result is that T1/T2 has positive rank. Moreover under a "weak homotopy-ribbon" condition, we show that each Tn/Tn+1 has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.
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Papers by Shelly Harvey