International Congress of Mathematicians

Collapsed Riemannian Manifolds with Bounded Sectional Curvature 327Theorem 1.2 has been the starting point for many subsequent investigationsof collapsing in various situations. The guiding principle is that additional geometricalproperties of a collapsing should be mirrored in properties of its associatedF-structure, which in turn, puts constraints on the topology. For instance, if a collapsingsatisfies additional geometrical conditions such as: i) volume small, ii) uniformlybounded diameter, iii) nonpositive curvature, iv) positive pinched curvature,v) bounded covering geometry i.e. the injectivity radii of the Riemannian universalcovering has a uniform positive lower bound, then one may expect correspondingtopological properties of the F-structure such as: i) existence of a polarization,ii) pureness, iii) existence of a Cr-structure, iv) the existence of a circle orbit, v)injective F-structure. Results on such correspondences and their applications willoccupy the rest of this paper.d. Topological invariants associated to a volume collapseThe existence of a sufficiently (injectivity radius) collapsed metric as in (1.2.2)imposes constraints on the underlying topology. For instance, the simplicial volumeof M vanishes; see [Gr3]. As mentioned earlier, for a closed M 2n , the Eulercharacteristic of M 2n also vanishes; see [CFG].In this subsection, we focus on some topological invariants associated to certain(partially) volume collapsed metrics: the minimal volume, the L 2 -signature and thelimiting n-invariant; see below.The minimal volume, MinVol(AT), of M, is the infimum of the volumes over allcomplete metrics with |SCCM| < 1- Clearly, MinVol(M) is a topological invariant.Gromov conjectured that there exists a constant e(n) > 0 such that Min Vol (M n )