Shelly Harvey
Rice University, Mathematics, Faculty Member
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... 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.
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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... 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...
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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... 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.
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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... 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.
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In the late 90's, Cochran, Orr, and Teichner defined the (n)-solvable filtration, {F_n}, of the smooth knot concordance group, C, which provided a framework for many advances in the study of knot concordance. However it is useless... more
In the late 90's, Cochran, Orr, and Teichner defined the (n)-solvable filtration, {F_n}, of the smooth knot concordance group, C, which provided a framework for many advances in the study of knot concordance. However it is useless for studying the subgroup, T, of topologically slice ...
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Abstract. We define the stability of a subgroup under a class of maps, and establish the basic prop-erties of this notion. Loosely speaking, we will say that a normal subgroup, or more generally a normal series {An} of a group A, is... more
Abstract. We define the stability of a subgroup under a class of maps, and establish the basic prop-erties 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 ...
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Abstract. We define families of invariants for elements of the mapping class group of Σ, a compact ori-entable surface. Fix any characteristic subgroup H ⊲ π1(Σ) and restrict to J(H), any subgroup of mapping classes that induce the... more
Abstract. We define families of invariants for elements of the mapping class group of Σ, a compact ori-entable 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 ...
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Abstract. 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 fami-lies of knots contain topologically slice knots. We also... more
Abstract. 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 fami-lies of knots contain topologically slice knots. We also present a simpler proof of the result of ...
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For each sequence P = (p1(t), p2(t),...) of polynomials we define a characteristic series of groups, called the derived series localized at P. These group series yield filtrations of the knot concordance group that refine the (n)-solvable... 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. These group series yield filtrations of the knot concordance group that refine the (n)-solvable fil-tration. We show that the quotients of ...
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We give homological conditions that ensure that a group homomorphism induces an isomorphism modulo any term of the derived p-series, in analogy to Stallings's 1963 result for the p-lower central series. In fact, we prove a stronger... more
We give homological conditions that ensure that a group homomorphism induces an isomorphism modulo any term of the derived p-series, in analogy to Stallings's 1963 result for the p-lower central series. In fact, we prove a stronger theorem that is analogous to ...
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Abstract. 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 fami-lies of knots contain topologically slice knots. We also... more
Abstract. 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 fami-lies of knots contain topologically slice knots. We also present a simpler proof of the result of ...
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Abstract. 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... more
Abstract. 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 ...
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Abstract. In 1997, T. Cochran, K. Orr, and P. Teichner [12] defined a filtration of the classical knot concordance group C, ···⊆Fn ⊆···⊆F1 ⊆ F0.5 ⊆ F0 ⊆ C. The filtration is important because of its strong connection to the classification... more
Abstract. In 1997, T. Cochran, K. Orr, and P. Teichner [12] defined a filtration of the classical knot concordance group C, ···⊆Fn ⊆···⊆F1 ⊆ F0.5 ⊆ F0 ⊆ C. The filtration is important because of its strong connection to the classification of topo-logical 4-manifolds. Here we ...