Given a finite and connected two-dimensional CW complex K with fundamental group \(\Pi \) and second integer cohomology group \(H^2(K;\mathbb {Z})\) finite of odd order, we prove that: (1) for each local integer coefficient system \(\alpha :\Pi \rightarrow \textrm{Aut}(\mathbb {Z})\) over K, the corresponding twisted cohomology group \(H^2(K;_{\alpha }\!\mathbb {Z})\) is finite of odd order, we say order \(\mathfrak {c}^{*}(\alpha )\), and there exists a natural function—which resemble that one defined by the twisted degree—from the set \([K;\mathbb {R}\textrm{P}^2]_{\alpha }^{*}\) of the based homotopy classes of based maps inducing \(\alpha \) on \(\pi _1\) into \(H^2(K;_{\alpha }\!\mathbb {Z})\), which is a bijection; (2) the set \([K;\mathbb {R}\textrm{P}^2]_{\alpha }\) of the (free) homotopy classes of based maps inducing \(\alpha \) on \(\pi _1\) is finite of order \(\mathfrak {c}(\alpha )=(\mathfrak {c}^{*}(\alpha )+1)/2\); (3) all but one of the homotopy classes \([f]\in [K;\mathbb {R}\textrm{P}^2]_{\alpha }\) are strongly surjective, and they are characterized by the non-nullity of the induced homomorphism \(f^{*}:H^2(\mathbb {R}\textrm{P}^2;_{\varrho }\!\mathbb {Z})\rightarrow H^2(K;_{\alpha }\!\mathbb {Z})\), where \(\varrho \) is the nontrivial local integer coefficient system over the projective plane. Also some calculations of \(H^2(K;_{\alpha }\!\mathbb {Z})\) are provided for several two-complexes K and actions \(\alpha \), allowing to compare \(H^2(K;\mathbb {Z})\) and \(H^2(K;_{\alpha }\!\mathbb {Z})\) for nontrivial \(\alpha \).

给定一个有限且连通的二维CW复数K，其具有奇数阶的基本群\(\Pi \)和第二整数上同调群\(H^2(K;\mathbb {Z})\)有限，​​我们证明： (1) 对于K上的每个局部整数系数系统\(\alpha :\Pi \rightarrow \textrm{Aut}(\mathbb {Z})\)，相应的扭曲上同调群\(H^2(K;_{ \alpha }\!\mathbb {Z})\)是奇数阶的有限，我们说阶\(\mathfrak {c}^{*}(\alpha )\)，并且存在一个自然函数——类似于一个由扭曲度定义——来自集合\([K;\mathbb {R}\textrm{P}^2]_{\alpha }^{*}\)将\(\alpha \) on \(\pi _1\)归纳为\(H^2(K;_{\alpha }\!\mathbb {Z})\) 的基础映射的同伦类，即双射；( 2 )在\ ( \pi _1\)是有限阶的\(\mathfrak {c}(\alpha )=(\mathfrak {c}^{*}(\alpha )+1)/2\)；(3) 除一个同伦类\([f]\in [K;\mathbb {R}\textrm{P}^2]_{\alpha }\) 外，所有同伦类都是强满射，它们的特征是诱导同态的非无效性\(f^{*}:H^2(\mathbb {R}\textrm{P}^2;_{\varrho }\!\mathbb {Z})\rightarrow H^2(K;_{\alpha }\!\mathbb {Z})\)，其中\(\varrho \)是投影平面上的非平凡局部整数系数系统。还为几个双复合体K和动作\( \alpha \) 提供了\(H^2(K;_{\alpha }\!\mathbb {Z})\)的一些计算，允许比较\(H ^2(K;\mathbb {Z})\)和\(H^2(K;_{\alpha }\!\mathbb {Z})\)对于非平凡的\(\alpha \)。