Patrick Naylor, Department of Pure Mathematics, University of Waterloo
"Trisections of non-orientable 4—manifolds"
From
the
point
of
view
of
smooth
manifolds,
dimension
four
is
quite
unique—
one
striking
illustration
of
this
is
the
fact
that
R^n
admits
either
one
(if
n
is
not
equal
to
4)
or
uncountably
many
(if
n=4)
smooth
structures.
There
are
many
remaining
fundamental
questions
about
four
dimensional
topology,
but
one
might
hope
to
use
lower
dimensional
tools
to
gain
some
insight.
One
highly
useful
such
tool
is
the
notion
of
a
Heegaard
splitting:
a
decomposition
of
a
3—manifold
into
two
equal
pieces.
In
analogy
with
this
decomposition,
Gay
and
Kirby
recently
defined
the
notion
of
a
trisection
of
a
closed
oriented
4—manifold:
it
is
a
similar
decomposition,
but
into
three
equal
pieces.
Trisections
provide
a
novel
diagrammatic
perspective
on
4—manifolds,
and
have
already
been
used
to
define
new
invariants
and
reprove
fundamental
results
in
gauge
theory.
In
this
talk,
I
will
discuss
my
thesis
(based
on
a
joint
project
with
Maggie
Miller),
which
extends
these
decompositions
to
non-orientable
4—manifolds.
In
particular,
we
prove
a
non-orientable
analogue
of
a
theorem
of
Laudenbach-Poénaru
that
does
not
seem
to
appear
in
the
literature,
and
is
critical
for
descriptions
of
closed
non-orientable
4—manifolds.
We
also
give
a
non-orientable
analogue
of
a
theorem
of
Waldhausen
on
Heegaard
splitting
off
#S^2xS1.
The
talk
will
begin
with
a
brief
introduction
to
the
theory
of
trisections,
and
then
highlight
the
differences
in
the
non-orientable
setting.
Please contact Nancy Maloney (nfmalone@uwaterloo.ca) if you are interested in virtually attending Patrick's talk.