International Congress of Mathematicians

328 Xiaochun RongTheorem 1.4 (Rationality of geometric signature). ([Ro3]) If an open completemanifold, M 4 , of finite volume has bounded covering geometry outside a compactsubset, then the integral of the Hirzebruch signature form over M 4 is a rationalnumber.The main idea is to show that M 4 admits a polarized F-structure T outsidesome compact subset and an exhaustion by compact submanifolds, Mf, such thatthe restriction of T to the boundary of M is injective. The integral over M 4 is thelimit of the integrals over Mf, to which we apply the Atiyah-Patodi-Singer formulato reduce to showing the rationality of the limit of the ^-invariant terms. By makinguse of the special property of T and Theorem 1.5 below, we are able to concludethat the limit of the ^-invariant term is rational.Cheeger-Gromov showed that if a sequence of volume collapsed metrics ona closed manifold N 4 "-^1have bounded covering geometry, then the sequence ofthe associated ^-invariants converges and the limit is independent of the particularsequence of such metrics. They conjectured that the limit is rational.Theorem 1.5 (Rationality of limiting ^-invariants). ([Rol]) If a closed manifoldN 3 admits a sequence of volume collapsed metrics with bounded covering geometry,then N 3 admits an injective F-structure and the limit of the n-invariantsis rational.The idea is to show that N 3 admits an injective F-structure T. For an injectiveF-structure, the collapsing constructed in (1.2.1) has bounded covering geometryand may be used to compute the limit. Results from 3-manifold topology play arole in the proof of the existence of the injective F-structure.2. Collapsed manifolds with nonpositive sectionalcurvatureA classical result of Preismann says that for a closed manifold M n with negativesectional curvature, any abelian subgroup of the fundamental group is cyclic.By bringing in the discrete group technique, Margulis showed that if the metric isnormalized such that — 1 < SCCM» < 0, then there exists at least one point at whichthe injectivity radius is bounded below by a constant e(n) > 0.The study of the subsequent study of collapsed manifolds with — 1 < sec < 0may be viewed as an attempt to describe the special circumstances under whichthe conclusions of the Preismann and Margulis theorem can fail, if the hypothesisis weakened to nonpositive curvature; see [Bul-3], [CCR1,2], [Eb], [GW], [LY], [Sc].A collapsed metric with nonpositive curvature tends to be rigid in a precisesense; see (2.2.1) and (2.2.2). Namely, there exists a canonical Cr-structure whoseorbits are flat totally geodesic submanifolds. Of necessity, the construction of thisCr-structure is global. By contrast, the construction of less precise (but more generallyexisting) F-structure is local; see [CG2].Let M n = M n /Y, where M n denotes the universal covering space of M n withthe pull-back metric. A local splitting structure on a Riemannian manifold is a F-equivariant assignment to each point (of an open dense subset of M n ) a specified