In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple
(
g
1
,
g
2
,
g
3
)
. It is obtained by minimizing the genera of three orientable handle bodies — with no intersection between their interiors— which decompose the manifold as far as the Heegaard genus need only two.
That is, a decomposition
M
=
V
1
∪
V
2
∪
V
3
with
i
n
t
V
i
∩
i
n
t
V
j
=
∅
for
i
,
j
=
1
,
2
,
3
and being
g
i
the genus of
V
i
.
For orientable spaces,
t
r
i
g
(
M
)
=
(
0
,
0
,
h
)
, where
h
is
M
's Heegaard genus.
For non-orientable spaces the
t
r
i
g
has the form
t
r
i
g
(
M
)
=
(
0
,
g
2
,
g
3
)
or
(
1
,
g
2
,
g
3
)
depending on the image of the first Stiefel–Whitney characteristic class
w
1
under a Bockstein homomorphism, respectively for
β
(
w
1
)
=
0
or
≠
0.
It has been proved that the number
g
2
has a relation with the concept of Stiefel–Whitney surface, that is, an orientable surface
G
which is embedded in
M
, has minimal genus and represents the first Stiefel–Whitney class under the duality map
D
:
H
1
(
M
;
Z
2
)
→
H
2
(
M
;
Z
2
)
,
, that is,
D
w
1
(
M
)
=
[
G
]
. If
β
(
w
1
)
=
0
then
t
r
i
g
(
M
)
=
(
0
,
2
g
,
g
3
)
, and if
β
(
w
1
)
≠
0.
then
t
r
i
g
(
M
)
=
(
1
,
2
g
−
1
,
g
3
)
.
