Trigenus

Last updated March 16, 2019

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.

In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, 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. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

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.

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 look like our universe does to a small enough observer. This is made more precise in the definition below.

That is, a decomposition $M=V_{1}\cup V_{2}\cup V_{3}$ with ${\rm {int}}V_{i}\cap {\rm {int}}V_{j}=\varnothing$ for $i,j=1,2,3$ and being $g_{i}$ the genus of $V_{i}$ .

For orientable spaces, ${\rm {trig}}(M)=(0,0,h)$ , where $h$ is $M$ 's Heegaard genus.

For non-orientable spaces the ${\rm {trig}}$ has the form ${\rm {trig}}(M)=(0,g_{2},g_{3})\quad {\mbox{or}}\quad (1,g_{2},g_{3})$ depending on the image of the first Stiefel–Whitney characteristic class $w_{1}$ under a Bockstein homomorphism, respectively for $\beta (w_{1})=0\quad {\mbox{or}}\quad \neq 0.$

In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. Stiefel–Whitney classes are indexed from 0 to n, where n is the dimension of the vector space fiber of the vector bundle. If the Stiefel–Whitney class of index i is nonzero, then there cannot exist (n−i+1) everywhere linearly independent sections of the vector bundle. A nonzero nth Stiefel–Whitney class indicates that every section of the bundle must vanish at some point. A nonzero first Stiefel–Whitney class indicates that the vector bundle is not orientable. For example, the first Stiefel–Whitney class of the Möbius strip, as a line bundle over the circle, is not zero, whereas the first Stiefel–Whitney class of the trivial line bundle over the circle, S¹×R is zero.

In homological algebra, the Bockstein homomorphism, introduced by Meyer Bockstein, is a connecting homomorphism associated with a short exact sequence

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 $D\colon H^{1}(M;{\mathbb {Z} }_{2})\to H_{2}(M;{\mathbb {Z} }_{2}),$ , that is, $Dw_{1}(M)=[G]$ . If $\beta (w_{1})=0\,$ then ${\rm {trig}}(M)=(0,2g,g_{3})\,$ , and if $\beta (w_{1})\neq 0.\,$ then ${\rm {trig}}(M)=(1,2g-1,g_{3})\,$ .

Theorem

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

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References

J.C. Gómez Larrañaga, W. Heil, V.M. Núñez. Stiefel–Whitney surfaces and decompositions of 3-manifolds into handlebodies, Topology Appl. 60 (1994), 267–280.
J.C. Gómez Larrañaga, W. Heil, V.M. Núñez. Stiefel–Whitney surfaces and the trigenus of non-orientable 3-manifolds, Manuscripta Math. 100 (1999), 405–422.
"On the trigenus of surface bundles over $S^{1}$ ", 2005, Soc. Mat. Mex. | pdf

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