Browsing by Author "Manchester, Alex"

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    Action of the Mazur pattern up to topological concordance
    (arXiv, 2024) Manchester, Alex
    In the '80s, Freedman showed that the Whitehead doubling operator acts trivally up to topological concordance. On the other hand, Akbulut showed that the Whitehead doubling operator acts nontrivially up to smooth concordance. The Mazur pattern is a natural candidate for a satellite operator which acts by the identity up to topological concordance but not up to smooth concordance. Recently there has been a resurgence of study of the action of the Mazur pattern up to concordance in the smooth and topological categories. Examples showing that the Mazur pattern does not act by the identity up to smooth concordance have been given by Cochran--Franklin--Hedden--Horn and Collins. In this paper, we give evidence that the Mazur pattern acts by the identity up to topological concordance. In particular, we show that two satellite operators $P_{K_0,\eta_0}$ and $P_{K_1,\eta_1}$ with $\eta_0$ and $\eta_1$ freely homotopic have the same action on the topological concordance group modulo the subgroup of $(1)$-solvable knots, which gives evidence that they act in the same way up to topological concordance. In particular, the Mazur pattern and the identity operator are related in this way, and so this is evidence for the topological side of the analogy to the Whitehead doubling operator. We give additional evidence that they have the same action on the full topological concordance group by showing that up to topological concordance they cannot be distinguished by Casson-Gordon invariants or metabelian $\rho$-invariants.
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    Satellite constructions and topological concordance
    (2024-03-25) Manchester, Alex; Harvey, Shelly
    In [CFT07], Cochran-Friedl-Teicher unified and generalized many existing constructions of topologically slice links using the language of satellite constructions. Many such links have been shown to not be smoothly slice, and links which are topologically but not smoothly slice are some of the most fundamental examples of exotic behavior in 4-dimensional topology. In this thesis, we will give an improvement on the Milnor’s µ¯-invariant condition that appears [CFT07], which will allow us to give some examples of topologically slice links which are not covered by [CFT07]. We will then move on to prove an approximate relativization of this theorem, and then show that a wide class of metabelian invariants, in particular Casson-Gordon invariants and metabelian ρ-invarinats, do not obstruct the honest relativization from holding. If the honest relativization did hold, it would give strong evidence that knots with homology cobordant 0-surgeries are topologically concordant, which is known to be false smooothly (see [CFHH13] and [Col22]). We will also discuss how topological concordance can be interpreted for links in homology spheres other than S3. While moving to the more general setting of homology spheres does produce new knots and links up to concordance smoothly (see [Don83] and [Lev16]), there is some evidence (see [Dav20a] and [Dav23]) that every knot in a homology sphere is topologically concordant to a knot in S3. We will record the fundamental fact that there is a canonical homology cobordism between any two homology spheres characterized by its simple connectedness, which gives a concrete place to look for concordances.