Periodicity of the punctured mapping class group
Let \(\Gamma_g^i\) be the punctured mapping class group with \((i\geq 1, g\geq 1)\), and \(\Gamma_g^0\) be the unpunctured mapping class group, where \(g\) is the genus of the closed oriented two-manifold \(S_g\). The work of \textit{H. H. Glover, G. Mislin} and \textit{Y. Xia} [Topology 33, No. 3, 557-574 (1994; Zbl 0828.57011)] shows that \(\Gamma_g^0\) does not have periodic cohomology in general and \(\Gamma_g^0\) is never 2-periodic in group cohomology for \(g > 0\). The aim of paper under review is to prove that \(\Gamma_g^i\) has periodic cohomology with period 2. The result indicates one of the differences between the punctured mapping class group and the unpunctured mapping class group. The method used to prove the main result on periodicity is related to the Yagita invariant which is a generalization of the \(p\)-period of \(\Gamma_g^i\) if \(\Gamma_g^i\) has \(p\)-torsion. The period of the cohomology of a group is the least common multiple of all the \(p\)-periods where \(p\) ranges over the primes such that the group has \(p\)-torsion. Then the author shows that the Yagita invariant (as the \(p\)-period of \(\Gamma_g^i\)) is equal to 2 if \(\Gamma_g^i\) has \(p\)-torsion. The proof follows by induction on \(i\). (1) Suppose that \(\Gamma_g^1\) has \(p\)-torsion. Hence \(\Gamma_g^1\) has \(p\)-periodic cohomology (Theorem 1.2 of the paper) by a short exact sequence relating \(\Gamma_g^0\) and \(\pi_1(S_g)\) and by the relation on (virtual) cohomology dimensions; furthermore, the Yagita invariant \(Y(\Gamma_g^1, p) = 2\) provides the period 2 property by Theorem 1.4 of the paper. The calculation of \(Y(\Gamma_g^1, p)\) is based on the (weakly) homotopy equivalence between the classifying space of \(\Gamma_g^1\) and classifying space of Diff\(^+(S_g, *)\) for \(g > 1\); and the lifting of an arbitrary subgroup of \(\Gamma_g^1\) of order \(p\) to get representations into \(GL_2^+(R)\) and factorization to \(SO(2)\) (\(BGL_2^+(R) = BSO(2)\)) gives the relations on the first Chern class information. (2) The induction on \(i\) is initiated from the short exact sequence \(1 \to \pi_1(S_{g, r}^i) \to \Gamma_{g, r}^{i+1} \to \Gamma_{g, r}^{i} \to 1\) where \(S_{g, r}^i\) is a smooth surface of genus \(g\) with \(r\) boundary components and \(i\) punctures, and \(\Gamma_{g, r}^{i}\) is the corresponding mapping class group. Theorem 1.7 proves that \(\Gamma_g^i\) has 2-period cohomology if \(\Gamma_g^i\) has \(p\)-torsion. The consequences of the main result are (i) all finite subgroups of \(\Gamma_g^i\) are cyclic, and (ii) any cyclic subgroup of \(\Gamma_g^i\) has same normalizer and centralizer in \(\Gamma\). Using the main result, the author studies the \(p\)-torsion of the Farrell cohomology of \(\Gamma_{n(p-1)/2}^i\) for \(i \geq 1\) and \(n=1, 2, 3\) by counting the conjugacy classes of subgroups of \(\Gamma_{n(p-1)/2}^i\) with order \(p\). By the work of Nielsen and Symonds, the elements of order \(p\) are conjugate in \(\Gamma_g\) (or \(\text{Diff}^+(S_g)\)) if and only if they have the same fixed point data. In the author's thesis, the fixed point data notion has been extended for \(\Gamma_g^i\) and a similar result of Nielsen and Symonds holds for characterizing conjugacy relation. The length of section 3 is devoted to the computation of fixed point data for the group \(\Gamma_{n(p-1)/2}^i\) with \(i \geq 1\) and \(n=1, 2, 3\). It would be interesting to know what the virtual cohomology dimensions are for the group \(\Gamma_g^i\) instead of the bounds.
scientific article; zbMATH DE number 1563014
scientific article