## Algebraic & Geometric Topology

### Complexity of PL manifolds

Bruno Martelli

#### Abstract

We extend Matveev’s complexity of $3$–manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings, drilling and surgery of spheres, products, connected sums) and its relations with some geometric invariants (Gromov norm, spherical volume, volume entropy, systolic constant).

Complexity distinguishes some homotopically equivalent manifolds and is positive on all closed aspherical manifolds (in particular, on manifolds with nonpositive sectional curvature). There are finitely many closed hyperbolic manifolds of any given complexity. On the other hand, there are many closed $4$–manifolds of complexity zero (manifolds without $3$–handles, doubles of $2$–handlebodies, infinitely many exotic K3 surfaces, symplectic manifolds with arbitrary fundamental group).

#### Article information

Source
Algebr. Geom. Topol., Volume 10, Number 2 (2010), 1107-1164.

Dates
Revised: 19 April 2010
Accepted: 21 April 2010
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715128

Digital Object Identifier
doi:10.2140/agt.2010.10.1107

Mathematical Reviews number (MathSciNet)
MR2653058

Zentralblatt MATH identifier
1222.57021

Subjects
Primary: 57Q99: None of the above, but in this section
Secondary: 57M99: None of the above, but in this section

Keywords
complexity spine

#### Citation

Martelli, Bruno. Complexity of PL manifolds. Algebr. Geom. Topol. 10 (2010), no. 2, 1107--1164. doi:10.2140/agt.2010.10.1107. https://projecteuclid.org/euclid.agt/1513715128

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