2024/06/20 by Naohiko Kasuya, Kasuya, Naohiko, I. NODA +1 · 1 citation
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2406.14138
Let g be a non-negative integer, Σg a closed orientable surface of genus g, and Mg its mapping class group. We classify all the group homomorphisms π1(Σg)→ G up to the action of Mg on π1(Σg) in the following cases; (1) G=PSL(2;ℤ), (2) G=SL(2;ℤ). As an application of the case (2), we completely classify orientable T2-bundles over closed orientable surfaces up to bundle isomorphisms. In particular, we show that any orientable T2-bundle over Σg with g≥ 1 is isomorphic to the fiber connected sum of g pieces of T2-bundles over T2. Moreover, the classification result in the case (1) can be generalized into the case where G is the free product of finite number of finite cyclic groups. We also apply it to an extension problem of maps from a closed surface to the connected sum of lens spaces.