Rational manifold models for duality groups

Abstract

We show that a finite type duality group of dimension d > 2 is the fundamental group of a (d + 3)-manifold with rationally acyclic universal cover. We use this to find closed manifolds with rationally acyclic universal cover and some nonvanishing L 2 -Betti numbers outside the middle dimension, which contradicts a rational analogue of a conjecture of Singer.14 Such a manifold would be homotopy equivalent but not properly homotopy equivalent to (Ṫ 2 ) d . 15 Which, in particular, produces a 2d-manifold rationally equivalent to (Ṫ 2 ) d but with very different peripheral properties: its interior can be homotoped to the end, in sharp contrast to the classical (Ṫ 2 ) d manifold.

Homology growth, hyperbolization, and fibering

“…Theorem D (5). If π 1 pN q is virtually special hyperbolic, then π 1 pMq is virtually special hyperbolic.…”

“…In this case, the van Kampen embedding theory method does not produce a 7-dimensional thickening since b 3 pS 3 ; F 2 q ‰ 0, and we suspect that no such thickening exists. However, since finite index torsion free subgroups Γ of G S 3 are duality groups [18,21], we can use the rational homotopy method from [5] to at least produce a rational thickening, i.e. a rationally aspherical, compact 7-manifold with boundary pN, BN q, non-vanishing b p2q 4 and fundamental group Γ.…”

“…We will first concentrate on the upper limit, the lower limit is obtained by flipping signs. Applying sup to tails (subsets of the form C ěx ) defines an operation on bounded functions f Þ Ñ f sup : f sup pxq " sup Cěx f. 5 The partial order on covers of X is defined by…”

Survey on L2‐invariants and 3‐manifolds

“…For some information about the proof and in particular of references in the literature, we refer to [28, Theorem 3.2] except for assertion (5) which is due to Jaikin-Zapirain and Lopez-Alvarez [46, Proposition 6.5]. A group is called locally indicable if every non-trivial finitely generated subgroup admits an epimorphism onto Z.…”

“…One may wonder what happens if we replace by an aspherical finite Poincaré complex in the Singer Conjecture 3.8. There are counterexamples to the Singer Conjecture 3.8 if one weakens aspherical to rationally aspherical, see [5, Theorem 4].…”