Let W be a real vector space and let V be an orthogonal representation of a group G such that $V^{G} = \{0\}$ (for the set of fixed points of G). Let $S(V)$ be the sphere of V and suppose that $f: S(V) \to W$ is a continuous map. We estimate the size of the $(H, G)$ -coincidences set if G is a cyclic group of prime power order $\mathbb {Z}_{p^k}$ or a p-torus $\mathbb {Z}_p^k$ .

We compute the rings ${H}^{\ast } (N; { \mathbb{F} }_{2} )$ for $N$ a closed $ \mathbb{S} {\mathrm{ol} }^{3} $-manifold, and then determine the Borsuk–Ulam indices $\text{BU} (N, \phi )$ with $\phi \not = 0$ in ${H}^{1} (N; { \mathbb{F} }_{2} )$.