We extend the classical theory of homotopical \(\Sigma\) -sets \(\Sigma ^n\) developed by Bieri, Neumann, Renz and Strebel for abstract groups, to \(\Sigma\) -sets \(\Sigma _\textrm{top}^n\) for locally compact Hausdorff groups. Given such a group G, our \(\Sigma _\textrm{top}^n(G)\) are sets of continuous homomorphisms \(G \rightarrow \mathbb {R}\) (“characters”). They match the classical \(\Sigma\) -sets \(\Sigma ^n(G)\) if G is discrete, and refine the homotopical compactness properties \(\mathrm C_n\) of Abels and Tiemeyer. Moreover, our theory recovers the definition of \(\Sigma _\textrm{top}^1\) and \(\Sigma _\textrm{top}^2\) proposed by Kochloukova. Besides presenting various characterizations of \(\Sigma _\textrm{top}^n\) (particularly for \(n\in \{1,2\}\) ), we show that characters in \(\Sigma _\textrm{top}^n(G)\) are also in \(\Sigma _\textrm{top}^n(H)\) if \(H\le G\) is a closed cocompact subgroup, and we generalize several classical results. Namely, we prove that the set of nonzero elements of \(\Sigma _\textrm{top}^n(G)\) is open, we prove that characters in a group of type \(\mathrm C_n\) that do not vanish on the center always lie in \(\Sigma _\textrm{top}^n(G)\) , and we relate the \(\Sigma\) -sets of a group with those of its quotients by closed subgroups of type \(\mathrm C_n\) . Lastly, we describe how \(\Sigma _\textrm{top}^n(G)\) governs whether a closed normal subgroup with abelian quotient is of type \(\mathrm C_n\) , generalizing one of the highlights of the classical theory.
