Sala Seminari (Dip. Matematica).<\/p>\n
For a closed oriented 3-manifold $M$, a discrete group $G$, a 3-cocycle $\\alpha$ of $G$, and a representation $\\rho \\colon \\pi_1(M) \\to G$, the Dijkgraaf–Witten invariant is defined to be $\\rho^\\ast \\alpha [M]$, where $[M]$ is the fundamental class of $M$, and $\\rho^\\ast \\alpha$ is the pull-back of $\\alpha$ by $\\rho$. We consider an equivalent invariant $\\rho_\\ast [M] \\in H_3(G)$, and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the hyperbolic volume and the Chern–Simons invariant of $M$ in terms of the image of the Dijkgraaf–Witten invariant for $G={\\rm SL}_2 \\Bbb{C}$ by the Bloch–Wigner map $H_3(M)\\to \\mathcal{B}(\\Bbb{C})$, where $\\mathcal{B}(\\Bbb{C})$ is the Bloch group of $\\Bbb{C}$. Further, in 2013, Hutchinson gave a construction of the Bloch–Wigner map $H_3({\\rm SL}_2 \\Bbb{F}_p)\\to \\mathcal{B} (\\Bbb{F}_p)$ explicitly, where $p$ is prime, and $\\Bbb{F}_p$ is the finite field of order $p$. In this talk, I calculate the reduced Dijkgraaf–Witten invariant of the complement of knots, especially twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for SL$_2\\Bbb{F}_p$ by the Bloch–Wigner map $H_3({\\rm SL}_2\\Bbb{F}_p) \\to \\mathcal{B}(\\Bbb{F}_p)$.<\/p>\n","protected":false},"excerpt":{"rendered":"
For a closed oriented 3-manifold $M$, a discrete group $G$, a 3-cocycle $\\alpha$ of $G$, and a representation $\\rho \\colon \\pi_1(M) \\to G$, the Dijkgraaf–Witten invariant is defined to be $\\rho^\\ast \\alpha [M]$, where $[M]$ is the fundamental class…<\/p>\n