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

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Summary

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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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Different flavours of diffeomorphisms
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Wall’s quadratic form
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Kreck’s extensions
Stabilisation
The action on the set of stable framings and Theorem A
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Obstructions and Pontryagin classes
Highly connected almost closed manifolds
Wall’s classification
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Invariants
The minimal signature
Bundles over surfaces and almost closed manifolds
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Signatures of bundles of symplectic lattices
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Framing obstructions
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The proof of Theorem B
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Kreck’s extensions and their abelian quotients
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A geometric splitting
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Splitting the homology action
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Full Text
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