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Heegaard splittings and the CGO invariant of a 3-manifold

#### Joan Birman, Columbia University

Abstract: (Joint with Tara Brendle and Nathan Broaddus)
There is a 1-1 correspondence between equivalence classes of
Heegaard splittings of a closed, oriented 3-manifold $M^3$ and double cosets
in the mapping class group $\mcg_g$ of the Heegaard surface $\Sigma_g$
modulo its handlebody subgroup $\handle_g$. We investigate the images of
these double cosets under the representation $\rho_2$ of $\mcg_g$ which
arises via the action of $\mcg_g$ on the second nilpotent quotient of
$\pi_1(\Sigma_g)$. The 3-manifold invariant that arises is
strongest when $M^3$ is homology-equivalent to the connect sum of $g$ copies
of $S^2\times S^1$. We call it the {\it CGO-invariant} because it was also
discovered in 2001 by Cochran, Gerges and Orr \cite{CGO}, using different
methods.