Abstract
We compute the mapping class group of the manifolds sharp ^g(S^{2k+1}times 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.
Highlights
Is known as the Torelli group—the subgroup of isotopy classes acting trivially on homology
All groups of diffeomorphisms are equipped with the smooth topology so that (i) BDiff+(Wg,1) and BDiff+(Wg) classify smooth oriented Wg,1-bundles or Wgbundles, respectively, (ii) BDiff∂ (Wg,1) classifies (Wg,1, S2n−1)-bundles, i.e. smooth Wg,1-bundles with a trivialisation of their S2n−1-bundle of boundaries, and (iii) BDiff∂/2(Wg,1) classifies (Wg,1, D2n−1)-bundles, that is, smooth Wg,1-bundles with a trivialised D2n−1-subbundle of its S2n−1-bundle of boundaries
We can well work with Diff∂ (Wg,1) instead of Diff+(Wg), which is advantageous since there is a stabilisation map Diff∂/2(Wg,1) → Diff∂/2(Wg+1,1) by extending diffeomorphisms over an additional boundary connected summand via the identity, which restricts to a map Diff∂ (Wg,1) → Diff∂ (Wg+1,1) and induces stabilisation maps of the form n g,1/2
Summary
Og,g(Z) in general, the description of which we shall recall later. The kernel Tng ⊂. H2(BDiff+(Wg); Z) represented by an oriented smooth fibre bundle π : E → S over a closed oriented surface S with fibre Wg to an eighth of the signature sgn(E) of its total space, the pullback of the second one assigns such a bundle the Pontryagin number p(2n+1)/4(E) up to a fixed constant, and the pullback of the third class evaluates [π ] to a certain linear combination of sgn(E) and p(2n+1)/4(E) In addition to these three classes, our identification of the extension (3) for n ≥ 3 odd involves two particular homotopy spheres: the first one, P ∈ 2n+1, is the Milnor sphere—the boundary of the E8-plumbing [8, Sect. We derive several consequences from this, beginning with deciding when the more commonly considered extensions (1) and (2) split
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