Shintaro
Kuroki,
OCAMI
and
Visiting
Professor
at
University
of
Toronto
Root
systems
of
torus
graphs
and
extended
actions
of
torus
manifolds
A
torus
manifold
is
a
compact
oriented
2n-dimensional
Tn-manifold
with
fixed
points.
We
can
define
a
labelled
graph
from
given
torus
manifold
as
follows:
vertices
are
fixed
points;
edges
are
invariant
2-spheres;
edges
are
labelled
by
tangential
representations
around
fixed
points.
This
labelled
graph
is
called
a
torus
graph
(this
may
be
regarded
as
the
special
class
of
generalized
GKM
graphs).
It
is
known
that
we
can
compute
the
equivariant
cohomology
of
torus
manifold
by
using
combinatorial
data
of
torus
graphs.
In
this
talk,
we
study
when
torus
actions
of
torus
manifolds
can
be
induced
from
non-abelian
compact
connected
Lie
group
(i.e.
when
torus
actions
can
be
extended
to
non-abelian
group
actions).
To
do
this,
we
introduce
root
systems
of
torus
graphs.
By
using
this
root
system,
we
can
characterize
what
kind
of
compact
connected
non-abelian
Lie
group
(whose
maximal
torus
is
Tn)
acts
on
the
torus
manifold.
This
is
a
joint
work
with
Mikiya
Masuda.