# Tag Info

Let me suggest one article, which explicitly describes the simple rotation in $\mathbb{R}^n$ as rotation about an $\mathbb{R}^{n-2}$ subspace: Mortari, Daniele. "On the Rigid Rotation Conept in $n$-Dimensional Spaces." Journal of the Astronautical Sciences 49.3 (2001): 401-420. (PDF download link) This reaches the following expression as the $n \times ... 3 Let$f:\left\{ z\in\mathbb{C}:\left|z\right|=1\right\} \rightarrow\left\{ z\in\mathbb{C}:\left|z\right|=1\right\} $be a homeomorphism. Then$g:\left\{ z\in\mathbb{C}:\left|z\right|\leq1\right\} \rightarrow\left\{ z\in\mathbb{C}:\left|z\right|\leq1\right\} $defined by$0\mapsto0$and$z\mapsto\left|z\right|f\left(\frac{z}{\left|z\right|}\right)$if ... 0 This is my best understanding hopefully will help illustrate, together with other responses: 1) We choose a normal vector field for the core of$T:=S^1 \times D^2$. Then the framing number is the winding number of the normal vector field about the core. 2)Say we select l, a longitude , as a basis element for$H^1 (T;\mathbb Z)$. Then the framing number ... 2 If$M$is a connected closed (i.e. compact without boundary) 3-dimensional manifold, then$\pi_1(M)$cannot be isomorphic to$\pi_1(S)$, where$S$is an orientable surface of genus$\ge 2$. You can see this by first noting that$\pi_2(M)=0$(otherwise, by the sphere theorem,$M$is a nontrivial connected sum which will imply that$\pi_1(M)$is a nontrivial ... 1 Let$f: S\to S$be a homeomorphism of a compact connected surface (possibly with boundary). Theorem. If$\chi(S)<0$then the mapping torus$M=M_f$of$f$is a Seifert manifold if and only if the mapping class of$f$is periodic, i.e.,$f$is isotopic to a periodic homeomorphism. If$\chi(S)\ge 0$then$M$is Seifert unless$S\$ is the torus and ...