On the genus of smooth 4-manifolds

“…Topological classification of closed connected PL 4-manifolds according to the regular genus is a classical problem in combinatorial topology (cf. [13,14]). A PL classification of closed connected PL 4-manifolds according to gem-complexity (up to gem-complexity 8) has been studied in [9].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Regular genus and gem-complexity of some mapping tori

Basak

2019

RACSAM

View full text Add to dashboard Cite

In this article, we construct a crystallization of the mapping torus of some (PL) homeomorphisms f : M → M for a certain class of PL-manifolds M . These yield upper bounds for gem-complexity and regular genus of a large class of PL-manifolds. The bound for the regular genus is sharp for the mapping torus of some (PL) homeomor-In particular, for M = S d−1 × S 1 or S d−1 × − S 1 , our construction gives a crystallization of a mapping torus of a (PL) homeomorphism f : M → M with regular genus d 2 − d. As a consequence, we prove the existence of an orientable mapping torus of a (PL) homeomorphism f : (S 2 × S 1 ) → (S 2 × S 1 ) with regular genus 6. This disproves a conjecture of Spaggiari which states that regular genus six characterizes the topological product RP 3 × S 1 among closed connected prime orientable PL 4-manifolds.MSC 2010 : Primary 57Q15. Secondary 05C15; 57N10; 57N13; 57N15; 57Q05; 55R10.

show abstract

“…S 4 , CP 2 , S 2 ×S 2 , RP 4 , the orientable and non-orientable S 3 -bundles over S 1 and the K3-surface, together with their connected sums, possibly by taking copies with reversed orientation, too). 1 Actually, [16,Proposition 2] yields also a lower bound for closed connected orientable PL 4-manifold, but in the not simply-connected case it is not so significant.…”

Section: Introduction 2 Introductionmentioning

confidence: 99%

“…In particular, for 3 ≤ d ≤ 5, a lot of classifying results in PL-category have been obtained for both closed connected PL d-manifolds and compact connected PL d-manifolds with boundary (cf. [2,3,10,11,15,16]). In [5], we gave a lower bound for the regular genus of a closed connected PL 4-manifold.…”

Section: Introductionmentioning

confidence: 99%

Lower bounds for regular genus and gem-complexity of PL 4-manifolds

Basak

Casali

2016

Forum Mathematicum

View full text Add to dashboard Cite

Within crystallization theory, two interesting PL invariants for d-manifolds have been introduced and studied, namely gem-complexity and regular genus. In the present paper we prove that, for any closed connected PL 4-manifold M , its gem-complexity k (M ) and its regular genus G(M ) satisfy:where rk(π 1 (M )) = m. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of product 4-manifolds. Moreover, the class of 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.MSC 2010 : Primary 57Q15. Secondary 57Q05, 57N13, 05C15.

show abstract

On the genus of smooth 4-manifolds

Cited by 13 publications

References 19 publications

On the genus of real projective spaces

On the genus of real projective spaces

Regular genus and gem-complexity of some mapping tori

Lower bounds for regular genus and gem-complexity of PL 4-manifolds

Contact Info

Product

Resources

About