Let
be a nonorientable
-dimensional
handlebody without
-
and
-handles. We show
that
admits a Lefschetz
fibration over the
-disk,
whose regular fiber is a nonorientable surface with nonempty boundary. This is an
analogue of a result of Harer obtained in the orientable case. As a corollary, we obtain a
-dimensional
proof of the fact that every nonorientable closed
-manifold
admits an open book decomposition, which was first proved by Berstein and Edmonds
using branched coverings. Moreover, the monodromy of the open book we obtain for a given
-manifold belongs to the twist
subgroup of the mapping class group of the page. In particular, we construct an explicit minimal open book
for the connected sum of arbitrarily many copies of the product of the circle with the real projective plane.
We also obtain a relative trisection diagram for
,
based on the nonorientable Lefschetz fibration we construct, similar to the orientable
case first studied by Castro. As a corollary, we get trisection diagrams for some closed
-manifolds, e.g., the product
of the
-sphere with the real
projective plane, by doubling
.
Moreover, if
is a closed
nonorientable
-manifold
which admits a Lefschetz fibration over the
-sphere, equipped with a section
of square
, then we construct
a trisection diagram of
,
which is determined by the vanishing cycles of the Lefschetz fibration. Finally, we include
some simple observations about low-genus Lefschetz fibrations on closed nonorientable
-manifolds.
