Topological rigidity and H1–negative involutions on tori

Abstract

We show, for n ≡ 0, 1 (mod 4) or n = 2, 3, there is precisely one equivariant homeomorphism class of C 2 -manifolds (N n , C 2 ) for which N n is homotopy equivalent to the n-torus and C 2 = {1, σ} acts so that σ * (x) = −x for all x ∈ H 1 (N).If n ≡ 2, 3 (mod 4) and n > 3, we show there are infinitely many such C 2manifolds. Each is smoothable with exactly 2 n fixed points.The key technical point is that we compute, for all n ≥ 4, the equivariant structure set S TOP (R n , Γ n ) for the corresponding crystallographic group Γ n in terms of the Cappell UNil-groups arising from its infinite dihedral subgroups. 57S17; 57R67

Topological rigidity and actions on contractible manifolds with discrete singular set

“…where L 1 is the 1-connective algebraic L-theory spectrum of the trivial group. Exactly as in [4], for computations we shall use the non-connective, periodic analogue:…”

“…Lastly, we recall the identification of [4,Lemma 4.6], done for L-theory. Lemma 6.4 (Connolly-Davis-Khan).…”

On the stable Cannon Conjecture

“…The upper arrow sends the class of to the difference . This follows from the work of Ranicki [, Proof of Theorem 17.4, p. 191ff] using [, Theorem B1]. A detailed and careful exposition of the proof of the existence of the diagram above can be found in [, Proposition 14.18].…”

Closed aspherical manifolds with center

“…The groups in the theorem are infinitely generated exactly when the UNil groups are nontrivial. In case the fixed set is discrete, Connolly, Davis and Kahn [14] further showed that S(M) is a sum of such UNil groups. Moreover, they also showed that S(M) is the same as the isovariant structure set S iso (T ) of exotic involutions on the torus that are isovariantly homotopic to the standard involution.…”

“…The simplest counterexamples are involutions on tori, which we will use in our construction. Very recently, Connolly, Davis and Kahn [14] gave a very detailed and complete analysis of the equivariant structure set of such involutions.…”