English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The Torelli group of a genus    g  $g$    oriented surfaceΣ  g   $\Sigma _g$    is the subgroupI  g   $\mathcal {I}_g$    of the mapping class groupMod  (   Σ  g   )   ${\rm Mod}(\Sigma _g)$    consisting of all mapping classes that act trivially onH  1    (   Σ  g   ,  Z  )    ${\rm H}_1(\Sigma _g, \mathbb {Z})$   . The quotient groupMod   (   Σ  g   )   /   I  g    ${\rm Mod}(\Sigma _g) / \mathcal {I}_g$    is isomorphic to the symplectic groupSp  (  2  g  ,  Z  )   ${\rm Sp}(2g, \mathbb {Z})$   . The cohomological dimension of the groupI  g   $\mathcal {I}_g$    equals to3  g  −  5   $3g-5$   . The main goal of the present paper is to compute the top homology group of the Torelli group in the caseg  =  3   $g = 3$    asSp  (  6  ,  Z  )   ${\rm Sp}(6, \mathbb {Z})$   ‐module. We prove an isomorphismH  4    (   I  3   ,  Z  )   ≅   Ind    S  3   ⋉  SL    (  2  ,  Z  )    ×  3Sp  (  6  ,  Z  )    Z  ,   $$\begin{equation*} \hspace*{4pc}{\rm H}_4(\mathcal {I}_3, \mathbb {Z}) \cong {\rm Ind}^{{\rm Sp}(6, \mathbb {Z})}_{S_3 \ltimes {\rm SL}(2, \mathbb {Z})^{\times 3}} \mathcal {Z}, \end{equation*}$$    where    Z  $\mathcal {Z}$    is the quotient ofZ  3   $\mathbb {Z}^3$    by its diagonal subgroup    Z  $\mathbb {Z}$    with the natural action of the permutation groupS  3   $S_3$    (the action ofSL    (  2  ,  Z  )    ×  3${\rm SL}(2, \mathbb {Z})^{\times 3}$    is trivial). We also construct an explicit set of generators and relations for the groupH  4    (   I  3   ,  Z  )    ${\rm H}_4(\mathcal {I}_3, \mathbb {Z})$   .

The top homology group of the genus 3 Torelli group