In
low-dimensional topology, the trigenus of a
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
is an invariant consisting of an ordered triple
. 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
with
for
and being
the genus of
.
For orientable spaces,
,
where
is
's
Heegaard genus
In the mathematical field of geometric topology, a Heegaard splitting () is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
Definitions
Let ''V'' and ''W'' be handlebodies of genus ''g'', and ...
.
For non-orientable spaces the
has the form
depending on the
image of the first
Stiefel–Whitney characteristic class under a
Bockstein homomorphism In homological algebra, the Bockstein homomorphism, introduced by , is a connecting homomorphism associated with a short exact sequence
:0 \to P \to Q \to R \to 0
of abelian groups, when they are introduced as coefficients into a chain complex '' ...
, respectively for
It has been proved that the number
has a relation with the concept of
Stiefel–Whitney surface, that is, an orientable surface
which is embedded in
, has minimal genus and represents the first Stiefel–Whitney class under the duality map
, that is,