1,268 research outputs found
On regular genus and G-degree of PL 4-manifolds with boundary
In this article, we introduce two new PL-invariants: weighted regular genus
and weighted G-degree for manifolds with boundary. We first prove two
inequalities involving some PL-invariants which states that for any PL-manifold
with non spherical boundary components, the regular genus
of is at least the weighted regular genus of which is
again at least the generalized regular genus of . Another
inequality states that the weighted G-degree of is always
greater than or equal to the G-degree of . Let be any compact
connected PL -manifold with number of non spherical boundary components.
Then we compute the following: \tilde{G} (M) \geq 2 \chi(M)+3m+2h-4+2 \hat{m}
\mbox{ and } \tilde{D}_G (M) \geq 12(2 \chi(M)+3m+2h-4+2 \hat{m}), where
and are the rank of fundamental groups of and the corresponding
singular manifold (obtained by coning off the boundary components of
) respectively. As a
consequence we prove that the regular genus satisfies the
following inequality:
which improves the previous known lower bounds for the regular genus
of . Then we define two classes of gems for PL -manifold
with boundary: one consists of semi-simple and other consists of weak
semi-simple gems, and prove that the lower bounds for the weighted G-degree and
weighted regular genus are attained in these two classes respectively.Comment: 13 pages, no figur
Lower bounds for regular genus and gem-complexity of PL 4-manifolds
Within crystallization theory, two interesting PL invariants for
-manifolds have been introduced and studied, namely {\it gem-complexity} and
{\it regular genus}. In the present paper we prove that, for any closed
connected PL -manifold , its gem-complexity and its
regular genus satisfy:
where These lower bounds enable to strictly improve
previously known estimations for regular genus and gem-complexity of product
4-manifolds. Moreover, the class of {\it semi-simple crystallizations} is
introduced, so that the represented PL 4-manifolds attain the above lower
bounds. The additivity of both gem-complexity and regular genus with respect to
connected sum is also proved for such a class of PL 4-manifolds, which
comprehends all ones of "standard type", involved in existing crystallization
catalogues, and their connected sums.Comment: 17 pages, 3 figures. To appear in Forum Mathematicu
3-manifolds efficiently bound 4-manifolds
It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for
instance, every 3-manifold has a surgery diagram. There are several proofs of
this fact, including constructive proofs, but there has been little attention
to the complexity of the 4-manifold produced. Given a 3-manifold M of
complexity n, we show how to construct a 4-manifold bounded by M of complexity
O(n^2). Here we measure ``complexity'' of a piecewise-linear manifold by the
minimum number of n-simplices in a triangulation. It is an open question
whether this quadratic bound can be replaced by a linear bound.
The proof goes through the notion of "shadow complexity" of a 3-manifold M. A
shadow of M is a well-behaved 2-dimensional spine of a 4-manifold bounded by M.
We prove that, for a manifold M satisfying the Geometrization Conjecture with
Gromov norm G and shadow complexity S, c_1 G <= S <= c_2 G^2 for suitable
constants c_1, c_2. In particular, the manifolds with shadow complexity 0 are
the graph manifolds.Comment: 39 pages, 21 figures; added proof for spin case as wel
Computing Matveev's complexity via crystallization theory: the boundary case
The notion of Gem-Matveev complexity has been introduced within
crystallization theory, as a combinatorial method to estimate Matveev's
complexity of closed 3-manifolds; it yielded upper bounds for interesting
classes of such manifolds. In this paper we extend the definition to the case
of non-empty boundary and prove that for each compact irreducible and
boundary-irreducible 3-manifold it coincides with the modified Heegaard
complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via
Gem-Matveev complexity, we obtain an estimation of Matveev's complexity for all
Seifert 3-manifolds with base and two exceptional fibers and,
therefore, for all torus knot complements.Comment: 27 pages, 14 figure
Generalized Property R and the Schoenflies Conjecture
There is a relation between the generalized Property R Conjecture and the
Schoenflies Conjecture that suggests a new line of attack on the latter. The
approach gives a quick proof of the genus 2 Schoenflies Conjecture and suffices
to prove the genus 3 case, even in the absence of new progress on the
generalized Property R Conjecture.Comment: 29 pages, 8 figure
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