A Heegaard splitting of a closed 3-dimensional manifold
is a centuries-old
[e1]
description of as a
union of two handlebodies with their boundaries (surfaces of
genus ) glued together by a diffeomorphism. The 3-sphere
is the union of two 3-balls, or, more interestingly, the
union of two donuts (neighborhoods of the unit circle in the
xy-plane and the z-axis union infinity). The latter
splitting is used to stabilize a Heegaard splitting of
, increasing its genus to . One can ask if two
Heegaard splitting of the same manifold are equivalent; this
was proved by
Reidemeister
[e3]
and
Singer
[e2]
to be true after stabilizing.
A Heegaard splitting of also arises from a Morse-like
function where critical points of index
are mapped to . Then the Heegaard surface is
. Cerf theory then gives another proof of
stabilization because stabilizing amounts to a birth of a
cancelling 1–2-pair.
A key question arose:
Is one stabilization sufficient or
might more stabilizations be necessary to transform one
Heegaard genus splitting to another? All examples known
needed only one stabilization until the
Hass–Thompson–Thurston paper
[1],
which gave examples where
stabilizations are necessary. These
are
obtained by switching the two handlebodies, or equivalently
turning the Morse function upside down by changing to
.
The main theorem is:
For each there is a 3-manifold with two genus Heegaard
splittings that require stabilizations to become equivalent.
It is always true that stabilizations are enough to turn any Heegaard splitting of any
closed 3-manifold upside down. A sketch of the argument is given
in a remark at the
end of the Introduction (Section 2)
of
[1].
The proof that stabilizations are needed
embodies a beautiful topological idea which
requires subtle geometry involving curvature, volume, area and harmonic maps.
Let be the original genus Heegaard surface. Then we stabilize
times to obtain a Heegaard surface of genus , and we want to show that must
at least equal .
To turn the Heegaard surface upside down, we need an isotopy which takes to itself but with the opposite orientation.
can be expressed
as union two spines which are the wedges of
circles. As flips upside down, we have a two parameter family of surfaces,
, where the surface is the image of the
slice under the isotopy at time . These surfaces divide into
two parts, and we color one red and one blue, giving the surface a red and a blue side.