# Stable moduli space of high-dimensional handlebodies

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Boris Botvinnik , University of Oregon
Fine Hall 314

This is a joint Topology / Algebraic Topology seminar.  We study the moduli space of handlebodies diffeomorphic to $(D^{n+1}\times S^n)^{\natural g}$, i.e. the classifying space $\mathrm{BDiff} (D^{n+1}\times S^n)^{\natural g}, D^{2n})$ of the group of diffeomorphisms that restrict to the identity near an embedded disk $D^{2n} \subset \partial (D^{n+1}\times S^n)^{\natural g}$.  We prove that there is a natural map  $$\colim_{g\to\infty}\mathrm{BDiff}((D^{n+1}\times S^n)^{\natural g}, D^{2n}) \;\longrightarrow \; Q_{0}BO(2n+1)\langle n \rangle_{+}$$ which induces an isomorphism in integral homology when $n\geq 4$. Above, $BO(2n+1)\langle n \rangle$ denotes the $n$-connective cover of $BO(2n+1)$. This work is joint with N.Perlmutter. I also plan to discuss some recent results related to this work.