부산대학교 도서관

This paper extends S.~V. Matveev's work on the complexity of3-manifolds which was well described in his book [{\it Algorithmictopology and classification of 3-manifolds}, second edition, AlgorithmsComput. Math., 9, Springer, Berlin, 2007; MR2341532(2008e:57021)] and was originally introduced by him in [Acta Appl. Math. {\bf 19} (1990), no.~2, 101--130; MR1074221 (92e:57029)]. The complexity of a piecewise linearmanifold is calculated by looking at a minimal spine (i.e., onecontaining the minimal number of vertices). A spine $S$ of a manifold$M$ is essentially a polyhedron in $M$ such that the manifold minus afinite number of disks can be collapsed to $S$, where collapseessentially means that one can remove open faces $f$ and $f^{\prime}$if $f\subset f^{\prime},$ $f^{\prime}$ is not contained in any otherfaces, and $f$ is not contained in any faces other than $f^{\prime}.$\parSummary: ``We extend Matveev's complexity of 3-manifolds to PL compactmanifolds of arbitrary dimension, and we study its properties. Thecomplexity of a manifold is the minimum number of vertices in a simplespine. We study how this quantity changes under the most commontopological operations (handle additions, finite coverings, drillingand surgery of spheres, products, connected sums) and its relationswith some geometric invariants (Gromov norm, spherical volume, volumeentropy, systolic constant).\par``Complexity distinguishes some homotopically equivalent manifolds andis positive on all closed aspherical manifolds (in particular, onmanifolds with nonpositive sectional curvature). There are finitelymany closed hyperbolic manifolds of any given complexity. On the otherhand, there are many closed 4-manifolds of complexity zero (manifoldswithout 3-handles, doubles of 2-handlebodies, infinitely many exotic$K3$ surfaces, symplectic manifolds with arbitrary fundamentalgroup).''