Given a finite group G which acts freely on \(\mathbb {S}^{n}\), H a normal cyclic subgroup of prime order, in de Mattos and dos Santos (Topol. Methods Nonlinear Anal. 33:105–120, 2009) have defined and estimate the cohomological dimension of the set \(A_{\varphi }(f, H, G)\) of (H, G)-coincidence points of a continuous map \(f: X \rightarrow Y\) relative to an essential map \(\varphi : X \rightarrow \mathbb {S}^{n}\), where X is a compact Hausdorff space and Y is a k-dimensional CW-complex. The first aim of this work is to extend this result to a multi-valued map \(F: X \multimap Y\). The second aim of this work is to estimate the cohomological dimension of the set

where \(F: X \multimap M \) is an acyclic multi-valued map and \(h: X \rightarrow N\) is a continuous map such that \(h^{*}: H^{n}(N) \rightarrow H^{n}(X)\) is a non trivial homomorphism, where X is a Hausdorff compact space and N is a connected closed manifold homology \(n-\)sphere and equipped with a free action of the cyclic group \(\mathbb {Z}_{p}\) generated by a periodic homeomorphism \(T: N \rightarrow N\) of prime period p and M is a connected manifold( which M and N suppose orientable if \(p > 2\)).

给定一个自由作用于\(\mathbb {S}^{n}\) 的有限群G，H是素数阶的正常循环子群，在 de Mattos 和 dos Santos 中（Topol.Methods Nonlinear Anal. 33:105–120） ，2009）定义并估计了连续映射\(f: X \rightarrow Y的 ( H , G ) 重合点的集合\(A_{\varphi }(f, H ,  G)\) 的上同调维数\)相对于基本映射\(\varphi : X \rightarrow \mathbb {S}^{n}\)，其中X是紧致豪斯多夫空间，Y是k维 CW 复合体。这项工作的第一个目标是将这个结果扩展到多值映射\(F: X \multimap Y\)。这项工作的第二个目标是估计集合的上同调维数

其中\(F: X \multimap M \)是非循环多值映射，而\(h: X \rightarrow N\)是连续映射，使得\(h^{*}: H^{n}(N ) \rightarrow H^{n}(X)\)是一个非平凡同态，其中X是豪斯多夫紧空间，N是连通闭流形同调\(n-\)球体，并配备循环的自由作用由素数周期p的周期同胚\(T: N \rightarrow N\) 生成的群 \ (\mathbb {Z}_{p}\) 且 M是一个连通流形（如果\ (p > 2\)）。