Satellites of infinite rank in the smooth concordance group

Hedden, Matthew; Pinzón-Caicedo, Juanita

doi:10.1007/s00222-020-01026-w

Cited by 14 publications

(11 citation statements)

References 47 publications

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“…This passage to the double cover gives a homomorphism from the concordance group to Θ 3 ℤ/2 . Hedden and Pinz ón-Caicedo [HPC21] used this result on ℤ/2 homology cobordism to provide a criterion for a satellite operation to have infinite rank as a function on smooth concordance. 4.2.…”

Section: α βmentioning

confidence: 99%

Applications of Instanton Floer Homology

Pinzón-Caicedo¹,

Ruberman²

2022

Notices Amer. Math. Soc.

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Some thirty-five years ago, Andreas Floer introduced a new tool into geometry: an infinite-dimensional version of Morse theory that is now referred to as Floer homology. The original version of Morse theory computes the homology of a finite-dimensional manifold in terms of critical points of a smooth function and the flow of its gradient vector field. Floer theory does something similar in infinite dimensions, where the critical points and gradient flow equations are now differential geometric objects.

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Section: α βmentioning

confidence: 99%

Applications of Instanton Floer Homology

Pinzón-Caicedo¹,

Ruberman²

2022

Notices Amer. Math. Soc.

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“…To sample just a few results, the satellite construction features prominently in the first evidence for a fractal structure on concordance [CHL11]; the first examples of non-smoothly concordant knots with homeomorphic 0-surgeries [Yas15]; and the first knots in homology spheres which do not bound PL discs in any contractible 4-manifold [Lev16]. As a result, satellite operations have become an object of study in their own right, with recent work in the area focusing on the existence of interesting bijective satellite maps [GM95,MP18], the behavior of the 4-genera of knots under satelliting [CH18,Pic19,Mil19,FMPC22], and on satellite maps with image of infinite rank [HPC21].…”

Section: Introductionmentioning

confidence: 99%

Homomorphism obstructions for satellite maps

Miller

2023

Trans. Amer. Math. Soc. Ser. B

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A knot in a solid torus defines a map on the set of (smooth or topological) concordance classes of knots in S 3 S^3 . This set admits a group structure, but a conjecture of Hedden suggests that satellite maps never induce interesting homomorphisms: we give new evidence for this conjecture in both categories. First, we use Casson-Gordon signatures to give the first obstruction to a slice pattern inducing a homomorphism on the topological concordance group, constructing examples with every winding number besides ± 1 \pm 1 . We then provide subtle examples of satellite maps which map arbitrarily deep into the n n -solvable filtration of Cochran, Orr, and Teichner [Ann. of Math. (2) 157 (2003), pp. 433–519], act like homomorphisms on arbitrary finite sets of knots, and yet which still do not induce homomorphisms. Finally, we verify Hedden’s conjecture in the smooth category for all small crossing number satellite operators but one.

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“…Remark 1.8. This project was an offshoot of an attempt to approach the problem of concordance homomorphisms using Chern-Simons theory as applied previously by Hedden, Kirk, and Pinzón-Caicedo (see for example [19,20]). The key idea would be to build a certain negative definite cobordism from Σ q (P (T r,s )#P (−T r,s )) to a 3-manifold built out of surgeries on T r,s .…”

Section: Introductionmentioning

confidence: 99%