Reduction of UNil for finite groups with normal abelian Sylow 2-subgroup

Let F be a finite group with a Sylow 2-subgroup S that is normal and abelian. Using hyperelementary induction and cartesian squares, we prove that Cappell's unitary nilpotent groups UNil_*(Z[F];Z[F],Z[F]) have an induced isomorphism to the quotient of UNil_*(Z[S];Z[S],Z[S]) by the action of the group F/S. In particular, any finite group F of odd order has the same UNil-groups as the trivial group. The broader scope is the study of the L-theory of virtually cyclic groups, based on the Farrell--Jones isomorphism conjecture. We obtain partial information on these UNil when S is a finite abelian 2-group and when S is a special 2-group.Comment: 29 pages, revision of decorations, correction of Homological Reductio

On smoothable surgery for 4-manifolds

Under certain homological hypotheses on a compact 4-manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. The main examples are the class of finite connected sums of 4-manifolds with certain product geometries. Most of these compact manifolds have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman-Quinn topological surgery. Necessarily, the *-construction of certain non-smoothable homotopy equivalences requires surgery on topologically embedded 2-spheres and is not attacked here by transversality and cobordism.Comment: 18 pages, separated into two journal submission

Topological rigidity and H_1-negative involutions on tori

We prove there is only one involution (up to conjugacy) on the n-torus which acts as $-\mathrm{Id}$ on the first homology group when $n$ is of the form $4k$, is of the form $4k+1$, or is less than $4$. In all other cases we prove there are infinitely many such involutions up to conjugacy, but each of them has exactly $2^n$ fixed points and is conjugate to a smooth involution. The key technical point is that we completely compute the equivariant structure set for the corresponding crystallographic group action on $\mathbb{R}^n$ in terms of the Cappell $\mathrm{UNil}$-groups arising from its infinite dihedral subgroups. We give a complete analysis of equivariant topological rigidity for this family of groups.Comment: 50 pages, to appear in Geometry & Topolog