On regular genus of PL 4-manifold with boundary

Abstract: A crystallization of a PL $d$-manifold is a certain type of edge colored graph that represents the manifold. Extending the notion of genus in dimension 2, the notion of regular genus for a $d$-manifold has been introduced, which is strictly related to the existence of regular embeddings of crystallizations of manifold into surfaces. The regular genus of a closed connected orientable (resp. non-orientable) surface coincides with its genus (resp. half of its genus), while the regular genus of a closed connected 3-manifold coincides with its Heegaard genus. Let $M$ be a compact connected PL 4-manifold with boundary. In this talk, I shall give lower bounds for regular genus of the manifold $M$. In particular, if $M$ is a connected compact PL $4$- manifold with $h$ boundary components then its regular genus $\mathcal{G}(M)$ satisfies the following inequalities:

$\mathcal{G}(M)\geq 2\chi(M)+3m+2h-4$ and $\mathcal{G}(M)\geq \mathcal{G}(\partial M)+2\chi(M)+2m+2h-4,$

where $m$ is the rank of the fundamental group of the manifold $M$.

commutative algebraalgebraic topologycombinatorics

Audience: researchers in the topic

Applications of Combinatorics in Algebra, Topology and Graph Theory

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