# an invariant: trig of a 3-manifold

H of Heegaard genus

The Heegaard genus is a well known topological invariant for 3-manifolds,

see for example the wiki version of the moment here:

That invariant tells something of the topology from the manifold. Understanding the topological behavior of space allows to peer into some other branches of math such as algebra, geometry, analysis and dynamicla systems.

In fact decompose each 3-manifold into a disjoint union of the interiors of embedded handlebodies of the same genus plus the common intesection closures-boundaries.

For orientable spaces the splitting handlebodies are of the same genus and are orientable,

but for non-orientable spaces the handlebodies are non-orientable ones,

giving less visual 3d-intuition that the former type.

TRIG of tri-genus of Gómez-L and Nuñez around 1993

Years after had been understood how to decompose 3-manifolds into three orientable handlebodies same genus not necessarily and by working backwards i mean: find what spaces are constituted as three orientable handlebodies taking into account the genera.

The tri-genus also is an invariant for 3-spaces decomposing them into 3 disjoint interiors union of orientable handlebodies the space plus the common boundary and no necessarily of the same genus  for little more:

Q questions

what 3-manifolds are those which are union of three 3-balls? (splitting type $(0,0,0)$)

which 3-manifolds are those who are union of two 3-balls and a solid torus? (splitting type $(0,0,1)$)

which 3-manifolds are those who are union of one 3-ball and a two solid tori? (splitting type $(0,1,1)$)

which 3-manifolds are those who are union of three solid tori? (splitting type $(1.1,1)$)

minimal atlases, splitting type, trigenus