Skip to main content

All Questions

Filter by
Sorted by
Tagged with
2 votes
1 answer
287 views

How does hyperelliptic involution act on the standard generators of the fundamental group of surfaces of genus g with n punctures?

Let $S_{g,n}$ be the surface of genus $g$ with $n$ punctures. We know that $\pi_1(S_{g,n})$ admits a presentation: $$\left\langle~ \alpha_1,\beta_1,\dots, \alpha_{g},\beta_{g},\gamma_{1},\dots,\gamma_{...
Rajesh Dey's user avatar
6 votes
1 answer
350 views

Reference for a property of Dehn twists

I was reading The symplectic Floer homology of a Dehn twist by P. Seidel, which you can find here. In Lemma 3(ii) the following topological property of Dehn twists is stated without proof: Let $\...
Don's user avatar
  • 293
22 votes
2 answers
978 views

Fixed-point free diffeomorphisms of surfaces fixing no homology classes

One of my graduate students asked me the following question, and I can't seem to answer it. Let $\Sigma_g$ denote a compact oriented genus $g$ surface. For which $g$ does there exist an orientation-...
Robert's user avatar
  • 313
2 votes
1 answer
177 views

Isotopic homeomorphisms of surface induces same map on the space of ends

Let $\Sigma$ be a non-compact orientable connected two-manifold without boundary. Let $f,g\colon \Sigma\to \Sigma$ be two homeomorphisms. Suppose there is a homotopy $H\colon \Sigma\times [0,1]\to \...
Random's user avatar
  • 1,097
5 votes
1 answer
333 views

Proof of homotopic essential simple close curves are isotopic

In the book by Benson Farb and Dan Margalit A primer on mapping class groups, Princeton Mathematical Series 49. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/...
T566y65tt's user avatar
  • 119
4 votes
1 answer
184 views

Inequivalent free $\Bbb Z/n\Bbb Z$-actions on orientable compact bordered surface

Let $S_{g,b}$ denote the orientable connected compact surface of genus $g$ with $b$ boundary components. A group homomorphism $\varphi\colon G\to \text{Homeo}^+(S_{g,b})$ is said to be free $G$-...
Random's user avatar
  • 1,097
1 vote
1 answer
452 views

Homotopy equivalence preserving all geometric intersection numbers

This question again might be silly, like the last post(deleted). Let me know I will delete it. Problem: Let $\Sigma$ be a surface without boundary and $f:\Sigma\to \Sigma$ be a proper homotopy ...
Sumanta's user avatar
  • 632
8 votes
1 answer
957 views

All non-compact simply connected $2$-manifolds with boundary

There are two corresponding posts MSE and MSE by me without any answers. Problem: Let $\Sigma$ be a non-compact simply-connected $2$-dimensional manifold, with boundary. Then, up to homeomorphism $\...
Sumanta's user avatar
  • 632
6 votes
1 answer
658 views

Generalized Birman exact sequence for surfaces with boundaries

Let $S_g^n$ be a surface of genus g with n boundaries and let $Mod(S_g^n)$ be its mapping class group. We will also denote by $S_{g,m}^n$ a surface of genus g with n boundaries and m punctures. The ...
Philippe Tranchida's user avatar
21 votes
0 answers
777 views

Is the mapping class group of $\Bbb{CP}^n$ known?

In his paper "Concordance spaces, higher simple homotopy theory, and applications", Hatcher calcuates the smooth, PL, and topological mapping class groups of the $n$-torus $T^n$. This requires an ...
mme's user avatar
  • 9,580
14 votes
1 answer
449 views

Is there a Morita cocycle for the mapping class group Mod(g,n) when n > 1?

Write Mod(g,n) for the mapping class group of a genus-$g$ surface $\Sigma$ with $n$ boundary components. When $n=0,1$ we define the Torelli group $T$ to be the subgroup of Mod(g,n) which acts ...
JSE's user avatar
  • 19.2k
6 votes
1 answer
1k views

Dehn-Nielsen-Baer Theorem for surfaces with boundary and punctures

Let $S=S_{g,b}$ be a compact orientable surface with genus $g$ and $b$ boundary components, such that $\chi(S)=2-2g-b<0$. Let $Q=\{x_1,\ldots , x_n\}$ be a set of $n$ distinguished points in the ...
Mark Grant's user avatar
  • 35.9k
2 votes
2 answers
151 views

How to prove that $\phi: \;\mathrm Mod(S_g)\to \mathrm Sp(2g, \mathbb{Z})$ is an epimorphism? [duplicate]

How do I prove that homomorphism $\phi : \; \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an ...
G.Tverisovskikh's user avatar
11 votes
0 answers
654 views

What is known about mapping class groups of 4-manifolds?

I am mostly interested in the case when you have a smooth degree $d$ algebraic surface $X$ over $\mathbb C$ and we can define three distinct groups: $\pi_0(\mathrm{Diff}^+(X))$, $\pi_0(\mathrm{Homeo}^+...
Harry Reed's user avatar
7 votes
0 answers
355 views

Making diffeomorphism of submanifolds boring

This is probably very well known in surgery theory... I'm looking for a modern reference on the following questions (the only one I know is Browder's "Diffeomorphism of 1-connected manifolds"...
Nati's user avatar
  • 1,981
14 votes
3 answers
683 views

Compact manifolds with big mapping class group

I was wondering if compact surfaces were the only compact manifolds with a "big" or "complicated" mapping class group. Are there higher dimensional manifolds (which are not in some way reducible to ...
Selim G's user avatar
  • 2,696
0 votes
2 answers
219 views

If $i(x,z)\neq 0$ and if $y$ is conjugate of $x$, then what can we say about $i(x*y,z)$?

Let $S_g$ denote the closed oriented surface of genus $g\geq 2$. Let $x,y$ be two different (upto fixed base point homotopy) but freely homotopic curves, i.e. $y$ is a non-trivial element from a ...
Anubhav Mukherjee's user avatar
9 votes
1 answer
605 views

Mapping Class Groups and torus (JSJ) decomposition of closed 3-manifolds

I am wondering if some intuitive relation exists between Mapping Class Group (MCG) of a 3-manifold (assume "simple" enough manifolds: closed,compact,irreducible, orientable, non-hyperbolic) and its ...
SKShukla's user avatar
  • 231
5 votes
1 answer
437 views

Classifying map for a surface bundle

Let $E\longrightarrow X$ be a surface (with holes) bundle. The structure group is then $M_{g, s}$, the mapping class group of the fiber. It follows from the famous work of Penner that the classifying ...
Denis Gorodkov's user avatar
10 votes
1 answer
557 views

Elements of infinite order in the topological mapping class group

Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ($\operatorname{Homeo}_0(...
John Pardon's user avatar
  • 18.7k
12 votes
1 answer
725 views

Injectivity of the Dehn-Nielsen-Baer map?

If $S$ is a closed hyperbolic surface, is there an easy proof of the injectivity of the Dehn-Nielsen-Baer map from $\mathrm{Mod}(S)$ to $\mathrm{Out}(\pi_1(S))$, taking an element of the mapping class ...
Dylan Thurston's user avatar
3 votes
1 answer
909 views

Isomorphism between a mapping class group and the fundamental group of a moduli space

For some fixed integer $d \geq 3$, let $M(0, d)$ be the mapping class group of self-homeomorphisms of the Riemann sphere which fix each of the $d$ points $0, 1, ... , d-2, \infty$. Let $X$ be the ...
Jeff Yelton's user avatar
  • 1,298
5 votes
1 answer
302 views

In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator?

Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In Meyer, Werner Die Signatur von ...
Patrick's user avatar
  • 51
9 votes
2 answers
1k views

Reference request: Spin structures on surfaces and the spin mapping class group

I am looking for references on the following: Spin structures on surfaces, and particularly the spin mapping class group. What is known about generating the spin mapping class group? Has anybody ...
Victor's user avatar
  • 2,136
5 votes
0 answers
378 views

Eilenberg-Mac Lane spaces for surface group extensions.

(The question has been edited. It was pointed out in the comments that $\Gamma_G$ could be a surface group, thought of as a finite extension of another surface group, in which case $G$ is finite.) ...
Autumn Kent's user avatar
  • 10.6k
22 votes
2 answers
1k views

The image of the point-pushing group in the hyperelliptic representation of the braid group

Let $B_{2g+1}$ be the Artin braid group on $2g+1$ strands. There is a symplectic representation $\rho: B_{2g+1} \rightarrow Sp_{2g}(\mathbf{Z})$ called the "hyperelliptic representation," which ...
JSE's user avatar
  • 19.2k
2 votes
0 answers
430 views

The signature of a mapping torus

Consider a manifold $M$ of dimension $4k + 2$, $k$ an integer. Pick a diffeomorphism $\phi$ of $M$ and construct the mapping torus $T$ of $\phi$. Suppose that there is a $4k+4$ dimensional manifold $B$...
Samuel Monnier's user avatar
3 votes
1 answer
552 views

Is the Action of the mapping class group transitive on embedded arcs?

Let S be a surface of genus g with some parked points (n of them). Assume $n \geq 2$ and fix two of the marked points. Consider the set of embedded arcs going between these two special points. The ...
Chris Schommer-Pries's user avatar
18 votes
2 answers
790 views

The kernel of the map from the handlebody group to Outer automorphisms of a free group

Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the ...
Jeffrey Giansiracusa's user avatar
6 votes
2 answers
3k views

Classification of mapping tori

Assume $M$ is a topological space and $f\in \operatorname{Homeo}(M)$, then $f$ obviously plays a significant role in the torus bundle $$M_f = M\times I/\{(x,0)\sim (f(x),1)\mid x\in M\}.$$ Hence ...
sara's user avatar
  • 257
-1 votes
1 answer
307 views

Are there results about the group of homeomorphisms of $(T^2-\{*,*\})$ up to isotopy?

I am studying a fiber bundle over circle with fiber $T^2-\{*,*\}$. Since this is a mapping torus, the group $Homeo(T^2-\{*,*\})/isotopy$ plays an important role. Are there some existing theorems on ...
student's user avatar
  • 157
3 votes
3 answers
769 views

Reducible 3d torus bundles

Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So, could anyone give me a hint to classify them? In contrast, do you agree ...
janmarqz's user avatar
  • 345
2 votes
2 answers
1k views

Periodic mapping classes of the genus two orientable surface

Please, any information on the periodic mapping classes of the genus two orientable surface, $O_2$, will be greatly thanked. We had been studying the topological structure of 3d surface bundles and ...
janmarqz's user avatar
  • 345