Secondary homological stability for unordered configuration spaces (Q6567161)

For an \(n\)-dimensional manifold \(M\) and \(k\ge 0\), let \(\mathrm{Conf}_k(M)\) be the unordered configuration space of \(k\) particles on \(M\). If \(M\) is open, then there is an embedding \(\mathbb{R}^n\sqcup M\hookrightarrow M\), inducing stabilisation maps \(\mathrm{Conf}_k(M)\to \mathrm{Conf}_{k+1}(M)\) by placing a new particle inside \(\mathbb{R}^n\). A result of McDuff shows that these maps satisfy homological stability of \(M\) is smooth and of finite type; in other words, the relative homology groups \(H_i(\mathrm{Conf}_k(M),\mathrm{Conf}_{k-1}(M))\) vanish for \(i\ll k\).\N\NBy ``placing'' a homology class \(x\in H_a(\mathrm{Conf}_b(\mathbb{R}^n))\) inside \(\mathbb{R}^n\), we get an induced map in relative homology\N\[\NH_i(\mathrm{Conf}_k(M),\mathrm{Conf}_{k-1}(M))\to H_{i+a}(\mathrm{Conf}_{k+b}(M),\mathrm{Conf}_{k-1+b}(M)). \N\]\NUnder certain conditions, these maps are isomorphisms for \(i\ll k\), even outside the range for classical homological stability (where both source and target of the map are trivial, see above); this phenomenon is called \emph{secondary homological stability}.\N\NIn the first part of the paper, the author finds such classes \(x\) and optimal slopes for secondary homological stability (with coefficients in both \(\mathbb{Z}\) and \(\mathbb{F}_p\)). In the second part of the paper, the author studies the case of closed manifolds. Even though there are no obvious stabilisation maps in this setting, the author constructs such maps on the level of chains, inducing morphisms in relative homology as above; this is the main part of the article. Finally, it is shown that these maps satisfy secondary homological stability.