부산대학교 도서관

This paper extends S.~V. Matveev's work on the complexity of 3-manifolds which was well described in his book [{\it Algorithmic topology and classification of 3-manifolds}, second edition, Algorithms Comput. 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 linear manifold is calculated by looking at a minimal spine (i.e., one containing the minimal number of vertices). A spine $S$ of a manifold $M$ is essentially a polyhedron in $M$ such that the manifold minus a finite number of disks can be collapsed to $S$, where collapse essentially means that one can remove open faces $f$ and $f^{\prime}$ if $f\subset f^{\prime},$ $f^{\prime}$ is not contained in any other faces, and $f$ is not contained in any faces other than $f^{\prime}.$ \par Summary: ``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). \par ``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).''