With the solution of Hilbert’s fifth problem, our understanding of connected locally compact groups has significantly increased. Therefore, the contemporary structure problem on locally compact groups concerns the class of totally disconnected locally compact (= t.d.l.c.) groups. The investigation of the class of t.d.l.c. groups can be made more manageable by dividing the infinity of objects under investigation into classes of types with “similar structure”. To this end we introduce the rational discrete cohomology for t.d.l.c. groups and discuss some of the invariants that it produces. For example, the rational discrete cohomological dimension, the number of ends, finiteness properties FP_n and F_n, and the Euler-Poincaré characteristic.

With the solution of Hilbert’s fifth problem, our understanding of connected locally compact groups has significantly increased. Therefore, the contemporary structure problem on locally compact groups concerns the class of totally disconnected locally compact (= t.d.l.c.) groups. The investigation of the class of t.d.l.c. groups can be made more manageable by dividing the infinity of objects under investigation into classes of types with “similar structure”. To this end we introduce the rational discrete cohomology for t.d.l.c. groups and discuss some of the invariants that it produces. For example, the rational discrete cohomological dimension, the number of ends, finiteness properties FP_n and F_n, and the Euler-Poincaré characteristic.