Let \(X\) be an ANR space with a free involution \(T\), \(\dim X = n\) and \(H^n(X,\mathbb{Z}) \cong \mathbb{Z}\), and \(N\) an \(n\)-dimensional homology sphere. Let \(\varphi : X \multimap N\) be an admissible multimap, i.e., \(\varphi\) is upper semicontinuous and there exist a paracompact Hausdorff space \(\Gamma\) and a pair of continuous maps \(p : \Gamma \to X\) and \(q : \Gamma \to N\) such that \(p\) is a Vietoris map and \(q(p^{-1}(x)) \subset \varphi (x),\) \(\forall x \in X.\)
The main results of the present article are:
(1) Suppose that \(c(X,T)^n \neq 0,\) where \(c(X,T)\) is the first Stiefel-Whitney class and \(\varphi (x) \cap \varphi (Tx) = \emptyset,\) \(\forall x \in X.\) Then there exists a unique odd number \(m\) such that \(\deg \varphi = \{m\}\).
(2) Let \(T'\) be a non trivial involution on \(N\). It is supposed that \(T' \varphi (x) \cap \varphi (Tx) = \emptyset,\) \(\forall x \in X.\) Then there exists a unique even number \(m\) such that \(\deg \varphi = \{m\}.\) In particular, if \(T'\) is orientation preserving, then \(\deg \varphi = \{0\}\).