In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac operator on a spin manifold with boundary to the higher rho invariant of the Dirac operator on the boundary, where the boundary is endowed with a positive scalar curvature metric. Our result extends a theorem of Piazza and Schick [27, Theorem 1.17]. *

Let X be a closed oriented connected topological manifold of dimension n ! 5. The structure group S TOP .X/ is the abelian group of equivalence classes of all pairs .f; M / such that M is a closed oriented manifold and f M 3 X is an orientation-preserving homotopy equivalence. The main purpose of this article is to prove that a higher rho invariant map defines a group homomorphism from the topological structure group S TOP .X/ of X to the analytic structure group K n .C £ L;0. z X/ / of X. Here z X is the universal cover of X, h 1 X is the fundamental group of X, and C £ L;0. z X/ is a certain C £-algebra. In fact, we introduce a higher rho invariant map on the homology manifold structure group of a closed oriented connected topological manifold, and prove its additivity. This higher rho invariant map restricts to the higher rho invariant map on the topological structure group. More generally, the same techniques developed in this paper can be applied to define a higher rho invariant map on the homology manifold structure group of a closed oriented connected homology manifold. As an application, we use the additivity of the higher rho invariant map to study nonrigidity of topological manifolds. More precisely, we give a lower bound for the free rank of the algebraically reduced structure group of X by the number of torsion elements in 1 X. Here the algebraically reduced structure group of X is the quotient of S TOP .X/ modulo a certain action of self-homotopy equivalences of X. We also introduce a notion of homological higher rho invariant, which can be used to detect many elements in the structure group of a closed oriented topological manifold, even when the fundamental group of the manifold is torsion free. In particular, we apply this homological higher rho invariant to show that the structure group is not finitely generated for a class of manifolds.

In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the K 0 -group of its group C * -algebra. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the evaluation of this determinant map, the image of a higher rho invariant is precisely the corresponding delocalized eta invariant of Lott. As a consequence, we show that if the Baum-Connes conjecture holds for a group, then Lott's delocalized eta invariants take values in algebraic numbers. We also generalize Lott's delocalized eta invariant to the case where the corresponding conjugacy class does not have polynomial growth, provided that the strong Novikov conjecture holds for the group.

We prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity. We apply this theorem to study Riemannian metrics of positive scalar curvature on manifolds. For every two metrics of positive scalar curvature on a closed manifold and a Galois cover of the manifold, we define a secondary higher index class. Non-vanishing of this higher index class is an obstruction for the two metrics to be in the same connected component of the space of metrics of positive scalar curvature. In the special case where one metric is induced from the other by a diffeomorphism of the manifold, we obtain a formula for computing this higher index class. In particular, it follows that the higher index class lies in the image of the Baum-Connes assembly map.

In this paper, we prove Gromov's dihedral extremality conjecture and dihedral rigidity conjecture regarding comparisons of scalar curvatures, mean curvatures and dihedral angles between two compact manifolds with corners possibly of different dimensions. As a consequence, we answer positively Gromov's dihedral extremality conjecture for polyhedra in all dimensions, and Gromov's dihedral rigidity conjecture for polyhedra in dimension three. Estimates of scalar curvature, mean curvature and dihedral angles on manifolds with cornersIn this section, we establish some estimates for scalar curvature, mean curvature and dihedral angles on manifolds with corners. These estimates constitute a key ingredient of the proof of our main theorem.