Homotopy invariance of higher signatures and $3$-manifold groups

Abstract

Abstract. -For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable 3-manifolds, including the "piecewise geometric" ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable 3-manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients holds. The non-oriented case is also discussed. Nous démontrons que pour des variétés fermées et orientées les signatures qui proviennent des groupes fondamentaux d'une large classe de variétés orientables de dimension 3 sont des invariants homotopiques. Cette classe, que nous décrivons soigneusement, contient en particulier les variétés géométriques par morceaux au sens de Thurston. Si la conjecture de géométrisation de Thurston s'avère vraie cette classe coïncide alors avec celle des groupes fondamentaux de variétés de dimension 3 orientables. Plus précisément nous démontrons que tous les groupes dans cette classe satisfont la conjecture de Baum-Connes avec coefficients. Nous discutons également le cas non-orientable.

K-Theory for Group C∗-algebras

“…Theorem 10 (Matthey, Oyono-Oyono, Pitsch [37]). Let M be a connected orientable 3-dimensional manifold (possibly with boundary).…”

“…• compact groups, • abelian groups, • groups acting simplicially on a tree with all vertex stabilizers satisfying the conjecture with coefficients [42], • amenable groups and, more generally, a-T-menable groups (groups with the Haagerup property) [24], • the Lie group Sp(n, 1) [25], • 3-manifold groups [37].…”

Taming 3-manifolds using scalar curvature

“…Hence H 2 (Bπ 1 ( )) injects into H 2 (B ). By [14], the strong Novikov conjecture holds for . Therefore N and satisfy the conditions of Theorem 4.3, contradicting the result of Theorem 4.2.…”

“…It is not difficult to check the first component of g * (ind(D)) in the above decomposition corresponds to the higher index of the Dirac operator on K in K 1 (C * r (π)), which is nonzero since the strong Novikov conjecture holds for π [14]. This implies that ind(D) is nonzero.…”

3-Manifold group and finite decomposition complexity

“…A surface Σ in M is called 2-sided if it is embedded in ∂M or it admits a tubular neighborhood in M which is a trivial line bundle, see[17].…”