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1

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 ...



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