The prime decomposition fibre sequence for moduli spaces of reducible 3-manifolds

Abstract

We study the moduli space $B\textrm{Diff}^+(M)$, for $M$ a reducible, oriented 3-manifold with irreducible prime factors $P_1,\ldots,P_n$. A programme of C\'esar de S\'a-Rourke, Hendriks-Laudenbach, and Hendriks-McCullough studies the homotopy type of $\textrm{Diff}^+(M)$ in terms of the $\textrm{Diff}^+(P_i)$. Inspired by a delooping proposed by Hatcher, we construct a map from $B\textrm{Diff}^+(M)$ to $B\textrm{Diff}^+(P_1 \sqcup \dots \sqcup P_n)$, called the splitting map, that yields a prime decomposition fibre sequence. The fibre $H_g(P_1, \dots, P_n)$ is a space of $1$-handle attachments which we describe geometrically as a homotopy colimit of certain configuration spaces on the $P_i$. Firstly, this allows us to show that for $n>0$ the fibre is equivalent to a finite, connected cell complex. Secondly, this makes the fibre sequence an effective tool for computations, which we illustrate by computing the rational cohomology ring of $B\textrm{Diff}^+\!\left((S^1\times S^2)^{\sharp 2}\right)$.