The mapping class group of connect sums of 𝑆²×𝑆¹

Abstract

Let M n M_n be the connect sum of n n copies of S 2 × S 1 S^2 \times S^1 . A classical theorem of Laudenbach says that the mapping class group Mod ⁡ ( M n ) \operatorname {Mod}(M_n) is an extension of Out ⁡ ( F n ) \operatorname {Out}(F_n) by a group ( Z / 2 ) n (\mathbb {Z}/2)^n generated by sphere twists. We prove that this extension splits, so Mod ⁡ ( M n ) \operatorname {Mod}(M_n) is the semidirect product of Out ⁡ ( F n ) \operatorname {Out}(F_n) by ( Z / 2 ) n (\mathbb {Z}/2)^n , which Out ⁡ ( F n ) \operatorname {Out}(F_n) acts on via the dual of the natural surjection Out ⁡ ( F n ) → G L n ( Z / 2 ) \operatorname {Out}(F_n) \rightarrow GL_n(\mathbb {Z}/2) . Our splitting takes Out ⁡ ( F n ) \operatorname {Out}(F_n) to the subgroup of Mod ⁡ ( M n ) \operatorname {Mod}(M_n) consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of M n M_n . Our techniques also simplify various aspects of Laudenbach’s original proof, including the identification of the twist subgroup with ( Z / 2 ) n (\mathbb {Z}/2)^n .