The twisted homology of simplicial set (Q2096932)

The paper gives the \(\Delta\)-twisted homology as a generalization of the \(\delta\)-twisted homology introduced by Jingyan Li, Vladimir Vershinin and Jie Wu [\textit{J. Y. Li} et al., Homology Homotopy Appl. 19, No. 2, 111--130 (2017; Zbl 1384.55016)]. This enriches the theory of \(\delta\)-(co)homology introduced by Alexander Grigoryan, Yuri Muranov, and Shing-Tung Yau [\textit{A. Grigor'yan} et al., J. Homotopy Relat. Struct. 11, No. 2, 209--230 (2016; Zbl 1353.05056)]. The paper proves that the Mayer-Vietoris sequence holds for \(\Delta\)-twisted homology, and the paper introduces the notion of \(\Delta\)-twisted Cartesian product on simplicial sets, which generalizes the classical work of Barratt, Gugenheim and Moore on twisted Cartesian products of simplicial sets [\textit{M. Barratt} et al., Am. J. Math. 81, 639--657 (1959; Zbl 0127.39002)]. Under certain hypotheses, the paper shows that the coordinate projection of \(\Delta\)-twisted Cartesian products admits a fibre bundle structure.