Homomorphism obstructions for satellite maps

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