Computation of the First Stiefel–Whitney Class for the Variety ℳ0.nℝ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{{\mathrm{\mathcal{M}}}_{0.n}^{\mathbb{R}}} $$\end{document}

We compute the class Wn−4ℳ0.nℝ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{{\mathrm{\mathcal{M}}}_{0.n}^{\mathbb{R}}} $$\end{document}, which is Poincaré dual to the first Stiefel–Whitney class for the variety ℳ0.nℝ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{{\mathrm{\mathcal{M}}}_{0.n}^{\mathbb{R}}} $$\end{document} in terms of the natural cell decomposition of ℳ0.nℝ¯.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{{\mathrm{\mathcal{M}}}_{0.n}^{\mathbb{R}}}. $$\end{document}