classification of smooth involutions of torus - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T15:38:34Z http://mathoverflow.net/feeds/question/21177 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21177/classification-of-smooth-involutions-of-torus classification of smooth involutions of torus student 2010-04-13T03:27:09Z 2010-04-13T14:48:36Z <p>Let $\mathbb{Z}_2={1,g},T^2={(e^{i\theta_1},e^{i\theta_2})}$ and place $T^2$ in $\mathbb{R}^3$ as the locus of the rotation of $2\pi$ rads of the circle${(y,z)|(y-2)^2+z^2=1}$ around $z$ axis.</p> <p>It is known that there are 5 nonequivalent smooth involutions on torus,and they are:</p> <p>1.$g(e^{i\theta_1},e^{i\theta_2})=(e^{i(\theta_1+\pi)},e^{i\theta_2})$ (rotation$\pi$ rads around $z$ axis) with null fixed point set and orbit space $T^2$</p> <p>2.$g(e^{i\theta_1},e^{i\theta_2})=(e^{-i\theta_1},e^{i\theta_2})$(reflection along $x=0$) with fixed point set $S^1\times S^0$ and orbit space an annulus</p> <p>3.$g(e^{i\theta_1},e^{i\theta_2})=(e^{i\theta_2},e^{i\theta_1})$(switch the two coordinates) with fixed point set the diagonal circle and orbit space Mobius band</p> <p>4.$g(e^{i\theta_1},e^{i\theta_2})=(e^{i(\theta_1 +\pi)},e^{-i\theta_2})$(restriction of the involution $(x,y,z,\mapsto (-x,-y,-z)$ of $\mathbb{R}^3$ to torus)with null fixed point set and orbit space klein bottle</p> <p>5.$g(e^{i\theta_1},e^{i\theta_2})=(e^{-i\theta_1},e^{-i\theta_2})$(reflection along $x=0$ plus reflection along $z=0$) with fixed point set 4 points and orbit space $S^2$</p> <p>i want to know how to derive the result above.for the free case it seems easy.since the action is free,the orbit space must be a manifold also,and has euler char 0,hence must be torus or klein bottle. for the nonfree case,the orbit is not manifold,but "orbifold". and we have Riemann-Hurwitz Formula:</p> <p>$\chi(O)=\chi(X_O)-\sum_{i=1}^n (1-\frac{1}{q_i})-\frac{1}{2}\sum_{j=1}^m (1-\frac{1}{r_j})$</p> <p>here$\chi(O)$ is the orbifold euler char and $\chi(X_o)$ is the euler char of the underlying space associated to the orbifold $O$,and $q_i$and $r_j$ denote the angles for sigular points(cone points and reflector corners can we determine the remaining 3 involutions by using this formula?Thank you!</p> http://mathoverflow.net/questions/21177/classification-of-smooth-involutions-of-torus/21210#21210 Answer by Sam Nead for classification of smooth involutions of torus Sam Nead 2010-04-13T14:48:36Z 2010-04-13T14:48:36Z <p>Here is a sketch -- some of the details are a bit hazy:</p> <p>Suppose that $\iota$ is a smooth involution of $T^2$. Show that the fixed point set of $\iota$ is a submanifold. Show that the orbit space of $\iota$ is an orbifold with orbifold Euler characteristic zero. Using the orbifold Euler characteristic you can enumerate all 17 compact, connected, 2-dimensional orbifolds of orbifold Euler characteristic zero. Now rule out 12 of these for topological reasons. </p> <p>The second to last step is a nice exercise that everybody should do once, after learning about the orbifold Euler characteristic. The non-trivial part in the last step is eliminating $D(2,2;)$ and $P(2,2)$. Getting rid of the others is easy. </p>