The average dual surface of a cohomology class and minimal simplicial decompositions of infinitely many lens spaces

Abstract

Discrete normal surfaces are normal surfaces whose intersection with each tetrahedron of a triangulation has at most one component. They are also natural Poincaré duals to 1-cocycles with $\ZZ/2\ZZ$-coefficients. For a fixed cohomology class in a simplicial poset the average Euler characteristic of the associated discrete normal surfaces only depends on the $f$-vector of the triangulation. As an application we determine the minimum simplicial poset representations, also known as crystallizations, of lens spaces $L(2k,q),$ where $2k=qr+1.$ Higher dimensional analogs of discrete normal surfaces are closely connected to the Charney-Davis conjecture for flag spheres.

arXiv admin note: text overlap with arXiv:0805.2425 by other authors In addition to a variety of minor corrections, this version has a new proof of the main lemma so that it applies to all dimensions. There is also a direct connection to the Charney-Davis conjecture for flag spheres

References in corpus (3)

• A note about complexity of lens spaces
• Normal surfaces as combinatorial slicings
• Minimal crystallizations of 3-manifolds

Cited by in corpus (6)

• A note about complexity of lens spaces
• PL 4-manifolds admitting simple crystallizations: framed links and regular genus
• A necessary condition for the tightness of odd-dimensional combinatorial manifolds
• Cataloguing PL 4-manifolds by gem-complexity
• Minimal crystallizations of 3-manifolds
• A new family of triangulations of $\mathbb{R}{P}^d$