Genus-minimal crystallizations of PL 4-manifolds
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry - Tập 59 - Trang 101-111 - 2017
Tóm tắt
For
$$d\ge 2$$
, the regular genus of a closed connected PL d-manifold M is the least genus (resp., half of genus) of an orientable (resp., a non-orientable) surface into which a crystallization of M imbeds regularly. The regular genus of every orientable surface equals its genus, and the regular genus of every 3-manifold equals its Heegaard genus. For every closed connected PL 4-manifold M, it is known that its regular genus
$${{\mathcal {G}}}(M)$$
is at least
$$2 \chi (M) + 5m -4$$
, where m is the rank of the fundamental group of M. In this article, we introduce the concept of “weak semi-simple crystallization” for every closed connected PL 4-manifold M, and prove that
$${{\mathcal {G}}}(M)= 2 \chi (M) + 5m -4$$
if and only if M admits a weak semi-simple crystallization. We then show that the PL invariant regular genus is additive under the connected sum within the class of all PL 4-manifolds admitting a weak semi-simple crystallization. Also, we note that this property is related to the 4-dimensional Smooth Poincaré Conjecture.
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