Non Semi-Simple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathfrak {sl}(2)}$\end{document} Quantum Invariants, Spin Case

Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum 𝔰𝔩(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathfrak {sl}(2)}$\end{document} were obtained by the last three authors in Costantino et al. (To appear in J. Topology. arXiv:1202.3553). In their construction, the quantum parameter q is a root of unity of order 2r where r>1 is odd or congruent to 2 modulo 4. In this paper, we consider the remaining cases where r is congruent to zero modulo 4 and produce invariants of 3-manifolds with colored links, equipped with generalized spin structure. For a given 3-manifold M, the relevant generalized spin structures are (non canonically) parametrized by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{1}(M;\mathbb{C}/2\mathbb{Z})$\end{document}.