# Questions tagged [mapping-class-group]

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### Generators of the Torelli group for surfaces with punctures

Are there any references for the generators of the Torelli group of surfaces with punctures? Johnson only assumes that the surface has boundary components. See Lemma 10 and Lemma 11 here (page 253). ...
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### "Simply connected" like condition for isotopy

I'm looking for literature regarding a locally compact Hausdorff topological space $X$ where any two embeddings $f,g:[0,1]\to X$ with fixed boundary points $f(0)=g(0),f(1)=g(1)$ are isotopic. If we ...
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### Nielsen Realization for punctured surfaces

I am trying to find a reference to a proof of the Nielsen Realization theorem for surfaces with punctures (of negative Euler characteristic). In Farb-Margalit's "A Primer on Mapping Class Groups&...
58 views

### What exactly does the "change of coordinates principle" say?

I'm currently reading the "Primer on mapping class groups" by Farb-Magalit. Something that almost always comes up in the proofs is the "change of coordinates principle". The first ...
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### What is meant by "side of a curve"?

I'm currently reading "A primer on mapping class groups" by Farb-Magalit. A notion that often turns up is that of a "side of a curve". For example in the proof of Proposition 3.2 ...
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### Maximal rank abelian subgroups of mapping class group

I'm currently getting involved with mapping class groups of surfaces ($MCG$). So for a genus g surface with b boundary components $S_{g,b}$ it is well-known that $MCG(S_{g,b})$ contains not only a lot ...
1 vote
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### Every homotopy equivalence of closed surface is homotopic to a homeomorphism

I am reading the first proof of $\text{Theorem 8.9:}$ If $g ≥ 2$, then any homotopy equivalence $S_g → S_g$ is homotopic to a homeomorphism from the book A Primer on Mapping Class Groups. The proof ...
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### Conjugation Classes inside the Orbit of close curve in Hyperbolic Surface under $Mod(S)$

I've heard the statement: Let $S$ be a hyperbolic surface with boundary and $\left[ \alpha \right] \in \pi_1 \left( S \right)$ a non-trivial element that is not conjecture to boundary element, then ...
1 vote
160 views

### mapping class group of the real projective plane

In most literature I've read about the mapping class group, I found that many authors have stated without any explanation that any homeomorphism of a real projective 2-space to itself is isotopic to ...
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### Surface homeomorphism transitively permutes the boundary curves

Let $h: S\to S$ be a surface homeomorphism. What does it mean to say that $h$ transitively permutes components of $\partial S$, and where does this terminology come from? Is this somehow related to a ...
193 views

### Mapping class group of a torus

Seems like i have some false understanding that need to be fixed. I have check varies sources saying that the mapping class group of a torus is isomorphic to $GL(2,\mathbb Z)$, however my false ...
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### The monodromy representation $\pi_1(E)\to Mod(F)$

Given a fiber bundle $F\to X\to E$, with $F,E$ are Riemann surfaces. I know the monodromy gives a permutation of the fiber. But how to see we have the monodromy representatio: $\pi_1(E)\to Mod(F)$? ...
1 vote
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### Mapping class group of $S^1 \times S^1 \times I$

I am interested in the computation of the mapping class group of the manifold $M=S^1 \times S^1 \times I$. One can visualize $M$ as a "torus cross $I$", or as an "annulus cross $S^1$". $M$ is ...
It is known that the mapping class group of the torus $\mathbb{T}^2$ is $\text{Mod}(\mathbb{T}^2) \cong \text{SL}_2(\mathbb{Z})$. We also know that for a pair of pants $P$ (a sphere with three ...