| Title:
|
Finite actions on the Klein four-orbifold and prism manifolds (English) |
| Author:
|
Kalliongis, John |
| Author:
|
Ohashi, Ryo |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
58 |
| Issue:
|
1 |
| Year:
|
2017 |
| Pages:
|
49-68 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We describe the finite group actions, up to equivalence, which can act on the orbifold $\Sigma(2,2,2)$, and their quotient types. This is then used to consider actions on prism manifolds $M(b,d)$ which preserve a longitudinal fibering, but do not leave any Heegaard Klein bottle invariant. If $\varphi\colon G\rightarrow \text{Homeo} (M(b,d))$ is such an action, we show that $M(b,d) = M(b,2)$ and $M(b,2)/\varphi$ fibers over a certain collection of 2-orbifolds with positive Euler characteristic which are covered by $\Sigma(2,2,2)$. For the standard actions, we compute the fundamental group of $M(b,2)/\varphi$ and indicate when it is a Seifert fibered manifold. (English) |
| Keyword:
|
finite group action |
| Keyword:
|
prism 3-manifold |
| Keyword:
|
equivalence of actions |
| Keyword:
|
orbifold |
| Keyword:
|
Klein four-group |
| MSC:
|
57M99 |
| MSC:
|
57S99 |
| idZBL:
|
Zbl 06736743 |
| idMR:
|
MR3631680 |
| DOI:
|
10.14712/1213-7243.2015.193 |
| . |
| Date available:
|
2017-03-12T16:37:40Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146027 |
| . |
| Reference:
|
[1] Dummit D., Foote R.: Abstract Algebra.Wiley, Hoboken, NJ, 2004. Zbl 1037.00003, MR 2286236 |
| Reference:
|
[2] Kalliongis J., Ohashi R.: Finite group actions on prism manifolds which preserve a Heegaard Klein bottle.Kobe J. Math. 28 (2011), no. 1, 69–89. Zbl 1253.57011, MR 2907136 |
| Reference:
|
[3] Kalliongis J., Ohashi R.: Classifying non-splitting fiber preserving actions on prism manifolds.Topology Appl. 178 (2014), 200–218. Zbl 1306.57014, MR 3276737, 10.1016/j.topol.2014.09.010 |
| Reference:
|
[4] Kalliongis J., Ohashi R.: Finite actions on the $2$-sphere and the projective plane.preprint. |
| Reference:
|
[5] McCullough D.: Isometries of elliptic $3$-manifolds.J. London Math. Soc. 65 (2002), no. 1, 167–182. Zbl 1012.57023, MR 1875143, 10.1112/S0024610701002782 |
| Reference:
|
[6] Orlik P.: Seifert Manifolds.Lecture Notes in Mathematics, 291, Springer, Berlin-New York, 1972. Zbl 0263.57001, MR 0426001 |
| Reference:
|
[7] Scott P.: The geometries of $3$-manifolds.Bull. London Math. Soc. 5 (1983), 401–48. Zbl 0662.57001, MR 0705527, 10.1112/blms/15.5.401 |
| . |
