Brendle, T.E. and Farb, B.
(2007)
The Birman–Craggs–Johnson homomorphism and abelian cycles in the Torelli group.
*Mathematische Annalen*, 338(1),
pp. 33-53.
(doi: 10.1007/s00208-006-0066-y)

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## Abstract

In the 1970s, Birman–Craggs–Johnson (BCJ) (Trans AMS 237: 283–309, 1978; Trans AMS 261(1):423–422, 1980) used Rochlin’s invariant for homology 3-spheres to construct a remarkable surjective homomorphism $${\sigma:\mathcal{I}_{g,1}\to B_3}$$ , where $${\mathcal{I}_{g,1}}$$ is the Torelli group and B 3 is a certain $${{\bf F}_2}$$ -vector space of Boolean (square-free) polynomials. By pulling back cohomology classes and evaluating them on abelian cycles, we construct $${2g^4 + O(g^3)}$$ dimensions worth of nontrivial elements of $${H^2(\mathcal{I}_{g,1}, {\bf F}_2)}$$ which cannot be detected rationally. These classes in fact restrict to nontrivial classes in the cohomology of the subgroup $${\mathcal{K}_{g,1} < \mathcal{I}_{g,1}}$$ generated by Dehn twists about separating curves. We also use the “Casson–Morita algebra” and Morita’s integral lift of the BCJ map restricted to $${\mathcal{K}_{g,1}}$$ to give the same lower bound on $${H^2(\mathcal{K}_{g,1}, {\bf Z})}$$.

Item Type: | Articles |
---|---|

Status: | Published |

Refereed: | Yes |

Glasgow Author(s) Enlighten ID: | Brendle, Professor Tara |

Authors: | Brendle, T.E., and Farb, B. |

Subjects: | Q Science > QA Mathematics |

College/School: | College of Science and Engineering > School of Mathematics and Statistics > Mathematics |

Journal Name: | Mathematische Annalen |

ISSN: | 0025-5831 |

ISSN (Online): | 1432-1807 |

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