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A monoclinic (Formula presented.) monolayer is demonstrated to be a topological semimetal (spinless), or zero-gap semiconductor (spinful, without spin–orbit coupling), or topologically trivial insulator (finite spin–orbit coupling). The latter contradicts a previous prediction that (Formula presented.) exhibits quantum-spin Hall (QSH) effect, and the reported value of (Formula presented.) -invariant, calculated from inversion-parity values at four Brillouin-zone high-symmetry points. Namely, as the electronic states of two of these points are double degenerate and transform according to irreducible representations carrying no parity, the band topology of (Formula presented.) is investigated using elementary band representations (EBRs) for layer groups (LGs). Novel subroutines, which output EBRs and Wilson loop operators for low-dimensional systems, are incorporated into POLSym code. Based on the calculated (Formula presented.) band structure decomposition onto EBRs of the relevant symmetry group and Wilson loop eigen-spectra, it is revealed that valence-band Wannier functions do not break the symmetry, and that Wannier centers are localized within a unit cell. Moreover, the QSH state is proved to be not realizable within (Formula presented.) -phase group-VI transition-metal dichalcogenides, because EBRs for the relevant symmetry groups are topologically trivial. However, it is predicted that among compounds that form other monoclinic structure, with symmorphic LG symmetry, there may exist topologically nontrivial phases, including the QSH state. © 2020 Wiley-VCH GmbH