Let
be a compact oriented surface. The Dehn twist along every simple closed curve
induces an automorphism of the fundamental group
of
.
There are two possible ways to generalize such automorphisms if the curve
is allowed
to have self-intersections. One way is to consider the “generalized Dehn twist” along
: an automorphism of the
Maltsev completion of
whose definition involves intersection operations and only depends on the homotopy
class
of . Another way is to
choose in the usual cylinder
a knot projecting onto
, to perform a surgery
along
so as to get a
homology cylinder ,
and let
act on every
nilpotent quotient
of (where
denotes the subgroup of
generated by commutators
of length
). In this
paper, assuming that
is in
for some
, we prove that (whatever
the choice of
is) the
automorphism of induced
by
agrees with the
generalized Dehn twist along
and we explicitly compute this automorphism in terms of
modulo
. As
applications, we obtain new formulas for certain evaluations of the Johnson
homomorphisms showing, in particular, how to realize any element of their targets by
some explicit homology cylinders and/or generalized Dehn twists.
Keywords
Dehn twist, homology cylinder, Dehn–Nielsen representation,
Johnson homomorphism
