Universal Minimal Flows of Extensions of and by Compact Groups

The paper’s statements and proofs establish exactly the two conclusions at issue: (i) M(G)/K ≅ M(G/K), and (ii) under a uniformly continuous cross section s: G/K → G and freeness of the K–action (on S(G) for the ambit-level result, and on M(G) for the main theorem), the phase space of M(G) is homeomorphic to M(G/K) × K; moreover, if G is SIN or the sequence splits (so s is a homomorphism), one can recover the G-action via a continuous cocycle ρ, giving a G-flow isomorphism M(G) ≅ M(G/K) × K. This is proved in Lemma 2, Theorem 4, Corollary 2, Theorem 5, and Theorem 9/10 of the uploaded paper 2103.10991 , and summarized in Theorem 3 (main statement) . By contrast, the candidate solution’s central step asserts that j(q,k)=k s(q) is a right-uniform homeomorphism Q×K→G and hence extends to a homeomorphism J:S(Q)×K→S(G). This relies on claims that inversion and multiplication preserve right-uniform continuity on general topological groups; that is false in general (multiplication is not jointly uniformly continuous on the right uniformity unless additional conditions such as SIN hold). The paper instead correctly constructs a compact-level cross-section s′:S(G)/K→S(G) and uses freeness of K to identify S(G) with S(G/K)×K, and then proceeds to M(G) and to equivariance via a cocycle ρ, avoiding the uniform continuity pitfall . Hence, while the candidate’s conclusions match the paper’s, the candidate’s proof has an invalid step; the paper’s arguments are correct.