The generalized Dehn twist along a figure eight. (English) Zbl 1282.57025
The work under review is in some sense a continuation of the work by N. Kawazumi and Y. Kuno [“The logarithms of the Dehn twists”, arXiv:1008.5017]. There the authors consider a compact orientable surface of genus \(g\) with one boundary component, denoted by \(\Sigma\). An invariant \(L^{\theta}\) for unoriented loops on \(\Sigma\) was introduced as a derivation on \(\hat T\) (where \(\hat T\) is the completed tensor algebra generated by the first rational homology group of the surface \(\Sigma\)).
As above \(\Sigma\) denotes a compact orientable surface of genus \(g\) with one boundary component. In the present work, by means of the Magnus expansion \(\theta\), the author obtains an injective group homomorphism, called the total Johnson map, \[ T^{\theta}: M_{g,1} \to Aut(\hat T) \] where \(M_{g,1}\) is the mapping class group of \(\Sigma\). It turns out that if the curve \(C\) is a simple curve then \(T^{\theta}(t_c)=e^{-L^{\theta}(C)}\) holds, where \(t_C\) is the Dehn twist along the simple curve \(C\). As pointed out by the authors, the right-hand side of the equation is an automorphism of \(\hat T\) for any loop \(\gamma\), and it is called the generalized Dehn twist and denoted by \(t_{\gamma}\). When an automorphism is in the image of the injective homomorphism above, then it is called a mapping class. The first main result of the paper is to show:
Theorem 3.9: Suppose that \(t_{\gamma}\) is a mapping class. Then there is a diffeomorphism representing \(T_{\gamma}\) whose support lies in a regular neighborhood of \(\gamma\).
Then the authors completely classify the figure eights on the surface \(\Sigma\), up to homotopy, where roughly speaking a figure eight is an immersed loop with only one double point such that the loop is neither homotopic to a simple curve nor to the the square of a simple curve. Then they show:
Theorem 5.1: Let \(\gamma\) be a figure eight on \(\Sigma\). Then \(t_{\gamma}\) is not a mapping class.
They use as one of the main tools Lie algebras. Most of the background, as well as results from closely related papers are provided.
As above \(\Sigma\) denotes a compact orientable surface of genus \(g\) with one boundary component. In the present work, by means of the Magnus expansion \(\theta\), the author obtains an injective group homomorphism, called the total Johnson map, \[ T^{\theta}: M_{g,1} \to Aut(\hat T) \] where \(M_{g,1}\) is the mapping class group of \(\Sigma\). It turns out that if the curve \(C\) is a simple curve then \(T^{\theta}(t_c)=e^{-L^{\theta}(C)}\) holds, where \(t_C\) is the Dehn twist along the simple curve \(C\). As pointed out by the authors, the right-hand side of the equation is an automorphism of \(\hat T\) for any loop \(\gamma\), and it is called the generalized Dehn twist and denoted by \(t_{\gamma}\). When an automorphism is in the image of the injective homomorphism above, then it is called a mapping class. The first main result of the paper is to show:
Theorem 3.9: Suppose that \(t_{\gamma}\) is a mapping class. Then there is a diffeomorphism representing \(T_{\gamma}\) whose support lies in a regular neighborhood of \(\gamma\).
Then the authors completely classify the figure eights on the surface \(\Sigma\), up to homotopy, where roughly speaking a figure eight is an immersed loop with only one double point such that the loop is neither homotopic to a simple curve nor to the the square of a simple curve. Then they show:
Theorem 5.1: Let \(\gamma\) be a figure eight on \(\Sigma\). Then \(t_{\gamma}\) is not a mapping class.
They use as one of the main tools Lie algebras. Most of the background, as well as results from closely related papers are provided.
Reviewer: Daciberg Lima Gonçalves (São Paulo)
MSC:
| 57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |
| 20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |
| 20F28 | Automorphism groups of groups |
| 57R19 | Algebraic topology on manifolds and differential topology |
Keywords:
surface; mapping class group; Dehn twist; Generalized Dehn twist; automorphism; loops; figure eight; Magnus expansionReferences:
| [1] | DOI: 10.1515/9781400839049 · doi:10.1515/9781400839049 |
| [2] | DOI: 10.1007/BF01389091 · Zbl 0619.58021 · doi:10.1007/BF01389091 |
| [3] | DOI: 10.1090/S0002-9939-2011-10951-1 · Zbl 1241.57025 · doi:10.1090/S0002-9939-2011-10951-1 |
| [4] | Magnus W., Combinatorial Group Theory (1976) |
| [5] | Massuyeau G., Bull. Soc. Math. France 140 pp 101– (2012) · Zbl 1248.57009 · doi:10.24033/bsmf.2625 |
| [6] | S. Morita, Handbook of Teichmüller Theory I (European Math. Soc., 2007) pp. 353–386. · doi:10.4171/029-1/8 |
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