It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the ring of intege
rs or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group $G$ the same information about the number of ends of $G$ in the sense of H. Abels can be provided by $mathrm{dH}^1(G,mathrm{Bi}(G))$, where $mathrm{Bi}(G)$ is the rational discrete standard bimodule of $G$, and $mathrm{dH}^bullet(G,_)$ denotes rational discrete cohomology as introduced in [6]. As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B).
Rational discrete cohomology and homology for a totally disconnected locally compact group $G$ is introduced and studied. The $mathrm{Hom}$-$otimes$ identities associated to the rational discrete bimodule $mathrm{Bi}(G)$ allow to introduce the notion
of rational duality groups in analogy to the discrete case. It is shown that semi-simple groups defined over a non-discrete, non-archimedean local field are rational t.d.l.c. duality groups, and the same is true for certain topological Kac-Moody groups. However, Y. Neretins group of spheromorphisms of a locally finite regular tree is not even of finite rational discrete cohomological dimension. For a unimodular t.d.l.c. group $G$ of type $mathrm{FP}$ it is possible to define an Euler-Poincare characteristic $chi(G)$ which is a rational multiple of a Haar measure. This value is calculated explicitly for Chevalley groups defined over a non-discrete, non-archimedean local field $K$ and some other examples.
