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Results tagged with mapping-class-groups
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user 23571
Topology of groups of automorphisms of surfaces, and high dimensional analogues.
16
votes
Mapping Class Group (MCG) of connected sum of 3-torus and $S^2\times S^1$
The mapping class groups of all compact orientable 3-manifolds are essentially known. A fairly detailed summary of the results, focusing on the nonprime case and with references to proofs in the lite …
- 20k
15
votes
Accepted
Mapping class groups of small Seifert-fibred 3-manifolds
The determination of mapping class groups of small Seifert manifolds was completed by M. Boileau and J.-P. Otal in a paper in Invent. Math. 106 (1991), 85-107. They give references for cases previous …
- 20k
14
votes
Periodic mapping classes of the genus two orientable surface
In the paper listed below there is a calculation of all the finite group actions on a genus 2 surface. There are 20 of them, with the groups ranging from order 2 to order 48. Nine of the actions are …
- 20k
14
votes
Accepted
Mapping class group of certain 3-manifolds
Since you write ${\rm Diff}_+(M)$ you are probably assuming $M$ is orientable and diffeomorphisms of $M$ are orientation-preserving. Every diffeomorphism of $M$ can be isotoped to take fibers to fiber …
- 20k
13
votes
Accepted
About the proof of Wajnryb's finite presentation of Mod(S)
It is still an open problem to find a short and simple way to derive a finite presentation for the mapping class group. The book by Farb and Margalit (in the recent preliminary version 4.00) gives a c …
- 20k
10
votes
Accepted
A query about Hatcher flow on arc complex
The flow consists of a sequence of surgeries using one fixed oriented arc $\alpha$ to cut (and isotope) all other arcs $\beta$ to remove one point of $\alpha\cap\beta$ at a time. Each surgery cuts on …
- 20k
8
votes
Reducible 3d torus bundles
OK, time to give some references for this classical material. Orientable 3-manifolds that are torus bundles are classified up to diffeomorphism by the conjugacy class of their monodromy map in SL(2,Z) …
- 20k
6
votes
Accepted
generators for the handlebody group of genus two
As stated in Ian Agol's answer, the mapping class group of a handlebody $B$ maps onto $Out(\pi_1B)$. This is easy to show by lifting known generators for the automorphism group of a free group. Twist …
- 20k
4
votes
Accepted
Action of Mapping Class Group on Arc complex
The quotient complex was studied in a paper by Penner, "The structure and singularities of quotient arc complexes", Journal of Topology 1 (2008), 527-550. An earlier version of the paper is available …
- 20k
3
votes
Mapping Class Group action on triangulated $S^2\times S^1$?
(This is a long comment rather than a complete answer.)
As Igor Rivin points out, the mapping class group is not ${\mathbb Z}_2$. There is another ${\mathbb Z}_2$ direct summand coming from a homeomor …
- 20k
2
votes
Accepted
Dehn-Nielsen-Baer Theorem for surfaces with boundary and punctures
The issue here is Dehn twists along curves parallel to the circles of $\partial S$. These usually generate infinite cyclic subgroups of ${\rm Mod}(S,Q)$, the only exceptions being when $S$ is a disk …
- 20k