Abstract. The paper contains a proof that the mapping class group of the manifold S 3 × S 3 is isomorphic to a central extension of the (full) Jacobi group Γ J by the group of 7-dimensional homotopy spheres. Using a presentation of the group Γ J and the µ-invariant of the homotopy spheres, we give a presentation of this mapping class group with generators and defining relations. We also compute the cohomology of the group Γ J and determine 2-cocycles that correspond to the mapping class group of S 3 × S 3 .
IntroductionThe central theme of this paper is the group of isotopy classes of orientationpreserving diffeomorphisms on S 3 × S 3 . We denote this group by π 0 Diff (S 3 × S 3 ). In general, the group of isotopy classes of orientation-preserving diffeomorphisms on a closed oriented smooth manifold M will be denoted by π 0 Diff (S 3 × S 3 ) and called the mapping class group of M by analogy with the 2-dimensional case.The article consists of two parts. Our goal in the first part will be to give a presentation of the mapping class group of S 3 × S 3 with generators and defining relations. The main step in this direction is Theorem 1, where we prove that π 0 Diff (S 3 × S 3 ) is a central extension of the (full) Jacobi group Γ J by the group of 7-dimensional homotopy spheres Θ 7 . The second part is concerned with the cohomology group H 2 (Γ J , Z 28 ). We show that this group is isomorphic to Z 28 ⊕ Z 4 ⊕ Z 2 and determine a 2-cocycle that corresponds to π 0 Diff (S 3 × S 3 ). The mapping class group of S 3 × S 3 was also studied by D. Fried in [10]. In particular, he proves that the group of the isotopy classes of diffeomorphisms that act trivially on H 3 (S 3 × S 3 ) is isomorphic to H Z /28Z, where H Z is the group of the integral upper unitriangular 3 × 3 matrices and Z is the center of H Z . The proof uses the µ-invariant of J. Eells and N. Kuiper [7] to show that v ⊗ v goes to the generator of Θ 7 under the bilinear pairing π 3 (SO(3)) ⊗ π 3 (SO(3)) −→ Θ 7 defined by J. Milnor [17] (v here is the generator of π 3 (SO(3)), given by the conjugation action of unit quaternions). We give details in section 1.3. Moreover, Fried (see Theorem 2) derives the exact sequence:where Φ 2 is the automorphism group of F 2 , the free group on two generators, and the kernel S ⊂ F 2 = [F 2 , F 2 ] ⊂ F 2 ⊂ Φ 2 is characterized by the commutative