In the 1980s, Freedman showed that the Whitehead doubling operator acts trivially
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 and Horn and by Collins. We
give evidence that the Mazur pattern acts by the identity up to topological
concordance.
In particular, we show that two satellite operators
and
with
and
freely
homotopic have the same action on the topological concordance group modulo the subgroup
of
-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
-invariants.
