The prime decomposition fibre sequence for moduli spaces of reducible 3-manifolds
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2026
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| author | Boyd, Rachael Bregman, Corey Steinebrunner, Jan |
| author_facet | Boyd, Rachael Bregman, Corey Steinebrunner, Jan |
| contents | 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 César de Sá-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 $B\textrm{Diff}^+\!\left((S^1\times S^2)^{\sharp 2}\right)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_02099 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The prime decomposition fibre sequence for moduli spaces of reducible 3-manifolds Boyd, Rachael Bregman, Corey Steinebrunner, Jan Geometric Topology Algebraic Topology 57T20, 58D29 (primary), 57M50, 55R40, 57S05, 58D05 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 César de Sá-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 $B\textrm{Diff}^+\!\left((S^1\times S^2)^{\sharp 2}\right)$. |
| title | The prime decomposition fibre sequence for moduli spaces of reducible 3-manifolds |
| topic | Geometric Topology Algebraic Topology 57T20, 58D29 (primary), 57M50, 55R40, 57S05, 58D05 |
| url | https://arxiv.org/abs/2604.02099 |