The cyclotomic trace and the K-theoretic analogue of Novikov's conjecture.

A trace construction, the cyclotomic trace, is given. It associates to algebraic K-theory of a group ring, or better to Waldhausen's A-theory, equivariant stable homotopy classes of the free-loop space of its classifying space. The cyclotomic trace detects the Borel classes in algebraic K-theory of the integers. It is used to prove, for a wide class of groups, that the K-theory assembly map is rationally injective. This is the K-theoretic analogue of Novikov's conjecture.

Fake tori, the annulus conjecture, and the conjectures of kirby.

The main result of this note (Theorem A) is that the set of piecewise linear (P.L.) manifolds of the same homotopy type as the n-torus, T(n), n >/= 5, is in one-to-one correspondence with the orbits of A(n-3)(pi(1)T(n)) [unk] Z(2) under the natural action of the automorphism group of pi(1)T(n). Every homotopy torus has a finite cover P.L. homeomorphic to T(n); hence the generalized annulus conjecture holds in dimension >/=5 (Kirby, R. C., "Stable homeomorphisms," manuscript in preparation). The methods of this classification are also used to study some conjectures of R. C. Kirby (manuscript in preparation) related to triangulating manifolds.