Additivity of Higher Rho Invariants and Nonrigidity of Topological Manifolds

Abstract

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.

Codimension 2 transfer of higher index invariants

“…Each of the first equalities are the 'boundary of Dirac is Dirac' or the 'boundary of signature is 2 ǫ times signature' principles, which is proved in [HR00b, Proposition 11.2.15], [HSX18, 2.13], [PS14, Theorem 1.22] (see also [Zei16,Theorem 5.15]) and [WYX20, Appendix D] respectively. Each of second equalities again follows from the 'boundary of Dirac is Dirac' or the 'boundary of signature is 2 ǫ times signature' principles, applied for the manifold N ∼ = N × R. We remark that the proof of (4) requires a further discussion because our construction of the signature higher ρ-invariant is not the same as the one dealt with in [WYX20]. This gap is filled in Appendix A.…”

C*-Algebraic higher Signature on Non-Witt space

“…K-homology Class on the Geometrically Controlled Hilbert Poincare Complex. In the appendix of [35], Weinberger, Xie and Yu construct the analytic K-homology class of signature class on the PL manifold M . In fact, their methods can generalize to any combinatorial geometrically controlled Poincare complex on the PL pseudomanifold in [24].…”

“…It is natural to construct the element of K-homology Definition 2.33. [35] Let σ = [v 0 , ....v n ] be a standard simplex where the vertices v i with given order. Define the standard subdivision Sub(σ) as below :…”

“…Then it is possible to use the uniformly bounded subdivision to control the propagation. Next is the standard construction of K-homology class in [35] Appendix B. There is an outline in section 2.9.…”

Slant products on the Higson–Roe exact sequence

“…The first main entity we study in this paper is the analytic exact sequence of Higson and Roe [41]. Questions around this sequence have generated substantial activity in higher index theory, for a selection of recent works see [23,48,49,50,57,63,68]. Its original motivation as devised by Higson and Roe was to serve as the target of certain analytic index maps defined on the surgery sequence from geometric topology.…”