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Infinitesimal presentations of the Torelli groups. (English) Zbl 0915.57001

Seien \(S\) eine kompakte orientierbare Fläche vom Geschlecht \(g\), \[ x_1,\dots, x_n;\;y_1,\dots, y_r\tag{*} \] \(n+r\) verschieden Punkte von \(S\) und \(v_1,\dots, v_r\) Tangentialvektoren \(\neq 0\), wobei \(v_j\) tangential an \(S\) in \(y_j\) sei. Die Gruppe \(\Gamma^n_{g,r}\) der Isotopieklassen orientierungserhaltender Diffeomorphismen von \(S\), die jeden der Punkte (*) und der Tangentialvektoren \(v_j\) fix lassen, nennt man Abbildungsklassengruppe. Der Kern des natürlichen Homomorphismus \[ \Gamma^n_{g,r}\to\operatorname{Aut} H_1(S,\mathbb{Z})\tag{**} \] heißt Torelligruppe \(T^n_{g ,r}\). Der Spezialfall \(T_{0,1}^n\) ist die klassische Zopfgruppe \(P_n\).
In der vorliegenden Arbeit wird eine explizite Präsentation der Ma1cev-Lie-Algebra \(t^n_{g,r}\), die sich der Torelligruppe \(T^n_{g,r}\) zuordnen läßt, für alle \(g\geq 6\) und alle \(r\) und \(n\geq 0\) angegeben. Dies verallgemeinert die von Kohno für \(p_n\), der Malcev-Lie-Algebra für \(P_n\), gezeigte Präsentation, die in der Theorie der Vassiliev-Invarianten von Bedeutung ist. Wesentliches technisches Hilfsmittel für den Autor sind gemischte Hodge-Strukturen und ein Resultat von Kabanov, das garantiert, daß bei der Präsentation von \(t^0_{g,0}\) für \(g\geq 6\) nur quadratische Relationen auftreten. Als Anwendung ergibt sich u.a. der Beweis einer Vermutung von S. Morita über die Struktur der Torelligruppen.

MSC:

57M20 Two-dimensional complexes (manifolds) (MSC2010)
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
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