Mapping class groups of highly connected $$(4k+2)$$ ( 4 k + 2 ) -manifolds
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Selecta Mathematica New Series
Mapping class groups of highly connected (4k + 2)-manifolds Manuel Krannich1 Accepted: 13 September 2020 / Published online: 26 November 2020 © The Author(s) 2020
Abstract We compute the mapping class group of the manifolds g (S 2k+1 × S 2k+1 ) for k > 0 in terms of the automorphism group of the middle homology and the group of homotopy (4k + 3)-spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds. Mathematics Subject Classification 57R50 · 55N22 · 57R60
Contents 1 Variations on two extensions of Kreck . . . . . . . . . . . 2 The action on the set of stable framings and Theorem A . 3 Signatures, obstructions, and Theorem B . . . . . . . . . 4 Kreck’s extensions and their abelian quotients . . . . . . 5 Homotopy equivalences . . . . . . . . . . . . . . . . . . Appendix A: Low-degree cohomology of symplectic groups References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 14 16 36 43 44 47
The classical mapping class group g of a genus g surface naturally generalises to all even dimensions 2n as the group of isotopy classes gn = π0 Diff + (Wg ) of orientation-preserving diffeomorphisms of the g-fold connected sum Wg = g (S n × S n ). Its action on the middle cohomology H (g) := Hn (Wg ; Z) ∼ = Z2g provides a n homomorphism g → GL2g (Z) whose image is the symplectic group Sp2g (Z) in the surface case 2n = 2, and a certain arithmetic subgroup G g ⊂ Sp2g (Z) or G g ⊂
B 1
Manuel Krannich [email protected] Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK
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M. Krannich
Og,g (Z) in general, the description of which we shall recall later. The kernel Tng ⊂ gn of the resulting extension 0 −→ Tng −→ gn −→ G g −→ 0
(1)
is known as the Torelli group—the subgroup of isotopy classes acting trivially on homology. In contrast to the surface case, the Torelli group in high dimensions 2n ≥ 6 is comparatively manageable: there is an extension 0 −→ 2n+1 −→ Tng −→ H (g) ⊗ Sπn SO(n) −→ 0
(2)
due to Kreck [35], which relates Tng to the finite abelian group of homotopy spheres 2n+1 and the image of the stabilisation map S : πn SO(n) → πn SO(n + 1) whose isomorphism class is shown in Table 1. The description of gn up to these two extension problems has found a variety of applications [2,5–7,18,23,29,33,38,39], especially in relation to the study of moduli spaces of manifolds [22]. The remaining extensions (1) and (2) have been studied more closely for particular values of g and n [15,19,21,36,37,48] but are generally not well-understood (see e.g. [15, p.1189], [21, p.873], [2, p.425]). In the present work, we res
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