In this paper, we provide a concrete interpretation of equivariant Reidemeister torsion, and demonstrate that Bismut–Zhang’s equivariant Cheeger–Müller theorem simplifies considerably when applied to locally symmetric spaces. In a companion paper, this allows us to extend recent results on torsion cohomology growth and torsion cohomology comparison for arithmetic locally symmetric spaces to an equivariant setting.

Vincent Lafforgue's bivariant K-theory for Banach algebras is invariant in the second variable under a rather general notion of Morita equivalence. In particular, the ordinary topological K-theory for Banach algebras is invariant under Morita equivalences.

Let $G$ be a finite group. We classify $G$-equivariant flow equivalence of non-trivial irreducible shifts of finite type in terms of

(i) elementary equivalence of matrices over $ZG$ and

(ii) the conjugacy class in $ZG$ of the group of G-weights of cycles based at a fixed vertex.

In the case $G = Z/2$, we have the classification for twistwise flow equivalence. We include some algebraic results and examples related to the determination of $E(ZG)$ equivalence, which involves $K_1(ZG)$.

For any finite group G we define the moduli space of pointed admissible G-covers and the concept of a G-equivariant cohomological field theory (G-CohFT), which, when G is the trivial group, reduces to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that taking the ‘quotient’ by G reduces a G-CohFT to a CohFT. We also prove that a G-CohFT contains a G-Frobenius algebra, a G-equivariant generalization of a Frobenius algebra, and that the ‘quotient’ by G agrees with the obvious Frobenius algebra structure on the space of G-invariants, after rescaling the metric. We then introduce the moduli space of G-stable maps into a smooth, projective variety X with G action. Gromov–Witten-like invariants of these spaces provide the primary source of examples of G-CohFTs. Finally, we explain how these constructions generalize (and unify) the Chen–Ruan orbifold Gromov–Witten invariants of $[X/G]$ as well as the ring $H^{\bullet}(X,G)$ of Fantechi and Göttsche.

Let $G$ be a countable discrete group and let $M$ be a proper free $C^r$ $G$-manifold and $N$ a $C^r$ $G$-manifold, where $1\leq r\leq\omega$. We prove that if $G$ acts properly and freely also on $N$ and if $\dim(N)\geq2\dim(M)$, then equivariant immersions form an open dense subset in the space $C^r_G(M,N)$ of all equivariant $C^r$ maps from $M$ to $N$. The space $C^r_G(M,N)$ is equipped with a topology, which coincides with the Whitney $C^r$ topology if $G$ is finite and is suited to studying equivariant maps. We also prove an equivariant version of Thom’s transversality theorem and show that $C^\omega_G(M,N)$ is dense in $C^r_G(M,N)$, for $1\leq r\leq\infty$.

AMS 2000 Mathematics subject classification: Primary 57S20