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42
Link concordance and higherorder Blanchfield duality
, 2008
"... In 1997, T. Cochran, K. Orr, and P. Teichner [13] 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 topological 4manifolds. Here we introduce new techn ..."
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Cited by 32 (9 self)
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In 1997, T. Cochran, K. Orr, and P. Teichner [13] 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 topological 4manifolds. Here we introduce new techniques for studying C and use them to prove that, for each n ∈ N0, the group Fn/Fn.5 has infinite rank. We establish the same result for the corresponding filtration of the smooth concordance group. We also resolve a longstanding question as to whether certain natural families of knots, first considered by CassonGordon, and Gilmer, contain slice knots.
New topologically slice knots
, 2005
"... In the early 1980’s Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with ..."
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Cited by 13 (1 self)
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In the early 1980’s Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with fundamental group Z ⋉ Z[1/2]. These two fundamental groups are known to be the only solvable ribbon groups. Our homological condition implies that the Alexander polynomial equals (t − 2)(t −1 − 2) but also contains information about the metabelian cover of the knot complement (since there are many nonslice knots with this Alexander polynomial).
Topology and Combinatorics of Partitions of Masses by Hyperplanes
, 2005
"... An old problem in combinatorial geometry is to determine when one or more measurable sets in R d admit an equipartition by a collection of k hyperplanes (B. Grünbaum, Pacific J. Math. 10 (1960) 1257–1261). A related topological problem is the question of (non)existence of a map f: (S d) k → S(U), eq ..."
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Cited by 12 (2 self)
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An old problem in combinatorial geometry is to determine when one or more measurable sets in R d admit an equipartition by a collection of k hyperplanes (B. Grünbaum, Pacific J. Math. 10 (1960) 1257–1261). A related topological problem is the question of (non)existence of a map f: (S d) k → S(U), equivariant with respect to the Weyl group Wk = Bk: = (Z/2) ⊕k ⋊ Sk, where U is a representation of Wk and S(U) ⊂ U the corresponding unit sphere. We develop general methods for computing topological obstructions for the existence of such equivariant maps. Among the new results is the well known open case of 5 measures and 2 hyperplanes in R 8, (E.A. Ramos, Discrete Comput. Geom. 15 (1996) 147–167). The obstruction in this case is identified as the element 2Xab ∈ H1(D8; Z) ∼ = Z/4, where Xab is a generator, which explains why this result cannot be obtained by the parity count formulas of Ramos (loc. cit.) or the methods based on either StiefelWhitney classes or ideal valued cohomological index theory (E. Fadell, S. Husseini, Ergod. Th. and Dynam.
Surgery on closed 4manifolds with free fundamental group
 Math. Proc. Cambridge Philos. Soc
"... The 4dimensional topological surgery conjecture has been established for a class of groups, including the groups of subexponential growth (see [6], [13] for recent developments), however the general case remains open. The full surgery conjecture is known to be equivalent to the question for a class ..."
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Cited by 12 (4 self)
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The 4dimensional topological surgery conjecture has been established for a class of groups, including the groups of subexponential growth (see [6], [13] for recent developments), however the general case remains open. The full surgery conjecture is known to be equivalent to the question for a class of canonical problems with free fundamental group [5, Chapter 12]. The proof of the conjecture for “good ” groups relies on the disk embedding theorem (see [5]), which is not presently known to hold for arbitrary groups. However, in certain cases it may be shown that surgery works even when the disk embedding theorem is not available for a given fundamental group (such results still use the diskembedding theorem in the simplyconnected setting, proved in [3].) For example, this may be done when the surgery kernel is represented by π1null spheres [4], or more generally by a π1null submanifold satisfying a certain condition on Dwyer’s filtration on second homology [7]. Here we state another instance when the surgery conjecture holds for free groups. The following results are stated in the topological category. Theorem 1. Let X be a 4dimensional Poincaré complex with free fundamental group, and assume the intersection form on X is extended from the integers. Let f: M − → X be a degree 1 normal map, where M is a closed 4manifold. Then the vanishing of the Wall obstruction implies that f is normally bordant to a homotopy equivalence f ′ : M ′ − → X. In the canonical surgery problems, X has free fundamental group and trivial π2, however what makes them harder to analyze is the interplay between the homotopy type of X, and the topology of the boundary. Our result sidesteps this by considering closed manifolds. We also prove a related splitting result: Theorem 2. Let M be a closed orientable 4manifold with free fundamental group, and suppose the intersection form on M is extended from the integers. Then M is scobordant to a connected sum of ♯ n S 1 × S 3 with a simplyconnected 4manifold. Note that if the surgery conjecture fails for free groups, then for both theorems above there is, in general, no extension to 4manifolds with boundary. The assumption on the intersection pairing in theorem 2 is necessary, since there are forms
Link concordance and generalized doubling operators
 Algebr. Geom. Topol
"... Abstract. 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 CheegerGromov bound, a deep analyti ..."
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Cited by 10 (2 self)
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Abstract. 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 CheegerGromov bound, a deep analytical tool used by CochranTeichner. Our main examples are boundary links that cannot be detected in the algebraic boundary link concordance group, nor by any ρ invariants associated to solvable representations into finite unitary groups. 1.
The classification of S3bundles over S4
 Differential Geom. Appl
"... We classify the total spaces of bundles over the four sphere with fiber a three sphere up to orientation preserving and reversing homotopy equivalence, homeomorphism and diffeomorphism. These total spaces have been of interest to both topologists and geometers. It has recently been shown by Grove a ..."
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Cited by 10 (5 self)
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We classify the total spaces of bundles over the four sphere with fiber a three sphere up to orientation preserving and reversing homotopy equivalence, homeomorphism and diffeomorphism. These total spaces have been of interest to both topologists and geometers. It has recently been shown by Grove and Ziller [GZ] that each of these total spaces admits metrics with nonnegative sectional curvature. 1
A diffeomorphism classification of manifolds which are like projective planes
, 505
"... We give a complete diffeomorphism classification of 1connected closed manifolds M with integral homology H∗(M) ∼ = Z ⊕ Z ⊕ Z, provided that dim(M) ̸ = 4. The integral homology of a oriented closed manifold 1 M contains at least two copies of Z (in degree 0 resp. dim M). If M is simply connected a ..."
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Cited by 9 (0 self)
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We give a complete diffeomorphism classification of 1connected closed manifolds M with integral homology H∗(M) ∼ = Z ⊕ Z ⊕ Z, provided that dim(M) ̸ = 4. The integral homology of a oriented closed manifold 1 M contains at least two copies of Z (in degree 0 resp. dim M). If M is simply connected and its homology has minimal size (i.e., H∗(M) ∼ = Z ⊕ Z), then M is a homotopy sphere (i.e., M is homotopy equivalent to a sphere). It is wellknown that any homotopy sphere of dimension n ̸ = 3 is homeomorphic to the standard sphere S n of dimension n. By contrast, the cardinality of the set Θn of diffeomorphism classes of homotopy spheres of dimension n can be very large (but finite except possibly for n = 3,4) [7]. In fact, the connected sum of homotopy spheres gives Θn the structure of an abelian group which is closely related to the stable homotopy group πn+k(S k), k ≫ n (currently known approximately in the range n < 100). Somewhat surprisingly, it is easier to obtain an explicit diffeomorphism classification of 1connected closed manifolds whose integral homology consists of three copies of Z. Examples of such manifolds are the 1connected projective planes (i.e., the projective
POLYNOMIAL SPLITTINGS OF METABELIAN VON NEUMANN RHO–INVARIANTS OF KNOTS
, 2008
"... We show that if the connected sum of two knots with coprime Alexander polynomials has vanishing von Neumann ρ–invariants associated with certain metabelian representations, then so do both knots. As an application, we give a new example of an infinite family of knots which are linearly independent ..."
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Cited by 5 (1 self)
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We show that if the connected sum of two knots with coprime Alexander polynomials has vanishing von Neumann ρ–invariants associated with certain metabelian representations, then so do both knots. As an application, we give a new example of an infinite family of knots which are linearly independent in the knot concordance group.