Examples of lack of rigidity in crystallographic groups

Topological rigidity and H1–negative involutions on tori

“…With this terminology, our results can be restated as showing that every proper Γ n -action on a contractible manifold is a cocompact manifold model for E fin Γ, that equivariant and isovariant rigidity for Γ n holds when n ≡ 0, 1 (mod 4) or n = 2, 3, and that equivariant and isovariant rigidity fail for all other n. Previous results on equivariant and isovariant rigidity are found in: [45], [19], [20], [55,Section 14.2], [39], and [38]. In particular, [20] gave the first examples of groups where isovariant rigidity fails; this proceeded via a version of Whitehead torsion. Proposition 1.4 below shows that the relevant Whitehead group vanishes for Γ n .…”

A history and survey of the Novikov conjecture

“…For certain groups Γ it is possible to show that there is enough codimension 1 transversality to prove that the surgery spectrum L(Z[Γ]) is a generalized homology spectrum, verifying the Conjectures by showing that the assembly map [Ya2]) applied these methods to the case when Γ is a crystallographic group. Other results, both positive and negative, on topological rigidity statements for crystallographic groups may be found in the work of Connolly and Koźniewski ([CyK1]- [CyK3]). Bounded topology also gives methods for recognizing generalized homology spectra, using the categorical methods initiated by Pedersen Proof.…”

On the generalized Nielsen realization problem

“…By now, many of these are known, [8], [9], [33], [34], [28]. We shall use two examples: one based on surgery theory (Cappell's UNils) and another based on embedding theory.…”

“…It is not hard to see that the action is not topologically conjugate to the original affine action, although it is equivariantly homotopically equivalent to it. ( [8], [9], [33]). This can be detected by an element of the isovariant (that is stratified) structure set in the sense of [33].…”