Harvey, Shelly2024-05-212024-05-212024-052024-03-25May 2024Manchester, Alex. Satellite constructions and topological concordance. (2024). PhD diss., Rice University. https://hdl.handle.net/1911/116103https://hdl.handle.net/1911/116103In [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.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.satelliteconcordance4-manifoldsCochran-Orr-Teichner filtrationhomology spheresThesis2024-05-21