In low-dimensional topology

Low-dimensional topology

In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...

, the trigenus of a closed

Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....

 3-manifold

3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...

 is an invariant consisting of an ordered triple . It is obtained by minimizing the genera of three orientable handle bodies

Handlebody

In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds...

 — with no intersection between their interiors— which decompose the manifold as far as the Heegaard

Heegaard splitting

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:...

 genus need only two.

That is, a decomposition with

for and being the genus of .

For orientable spaces, ,
where is 's Heegaard genus.

For non-orientable spaces the has the form
depending on the
image of the first Stiefel–Whitney characteristic class

Stiefel–Whitney class

In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney class is an example of a \mathbb Z_2characteristic class associated to real vector bundles.-General presentation:...

  under a Bockstein homomorphism

Bockstein homomorphism

In homological algebra, the Bockstein homomorphism, introduced by , is a connecting homomorphism associated with a short exact sequenceof abelian groups, when they are introduced as coefficients into a chain complex C, and which appears in the homology groups as a homomorphism reducing degree by...

, 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, . If then , and if
then .

Theorem

A manifold S is a Stiefel–Whitney surface in M, if and only if S and M−int(N(S)) are orientable .