Homological stability for the moduli spaces of products of spheres (Q2790654)

If \(M\) and \(N\) are manifolds of dimension \(n\), and \(M\) has non-empty boundary, there is a stabilisation map \(\mathrm{Diff}^\partial(M)\to \mathrm{Diff}^\partial(M\# N)\) between the topological groups of diffeomorphisms that restrict to the identity near the boundary. The \(N\)-genus of a manifold \(M\), denoted \(r_N(M)\), is the highest \(g\) such that \(N_g:= N\overbrace{\#\ldots\#}^{g}N\setminus D^n\) embeds into \(M\).NEWLINENEWLINEThe main theorem of this article takes \(N=S^p\times S^q\), and establishes that the homomorphism induced by the corresponding stabilisation map in the homology of the classifying spaces NEWLINE\[NEWLINEH_k(B\mathrm{Diff}^\partial(M);\mathbb Z)\mapsto H_k(B\mathrm{Diff}^\partial(M\# (S^{p}\times S^q));\mathbb Z)NEWLINE\]NEWLINE is an isomorphism provided that {\parindent=0.7cm \begin{itemize}\item[(c1)] \(p<q<2p-2\) and \(\pi_i(M) = 0\) for all \(i\leq q-p+2\), \item[(c2)] \(2k\leq r_{N}(M)-3-d\), NEWLINENEWLINE\end{itemize}} where \(d\) is the smallest number of generators of \(\pi_q(S^p)\).NEWLINENEWLINEThis theorem extends the main theorem of [\textit{S. Galatius} and \textit{O. Randal-Williams}, ``Homological stability for moduli spaces of high dimensional manifolds'', Preprint, \url{arXiv:1403.2334}], where the analogous statement for \(N=S^p\times S^p\) is proven (in particular, the dimension \(n\) is even), which in turn extends the homological stability theorem of \textit{J. L. Harer} [Ann. Math. (2) 121, 215--249 (1985; Zbl 0579.57005)], who takes \(N=S^1\times S^1\).NEWLINENEWLINEThis extension offers a twofold insight in the homological stability of diffeomorphism groups: first, that different stabilisation maps may induce isomorphisms in homology in a range. Second, that diffeomorphism groups of manifolds of odd dimension do satisfy homological stability (this was also proven in an earlier preprint of the author [``Homological stability for moduli spaces of odd dimensional manifolds'', Preprint, \url{arXiv:1311.5648}]). This latter result should be compared to [J. Reine Angew. Math. 684, 1--29 (2013; Zbl 1295.55014)], where \textit{J. Ebert} proves that the odd-dimensional analogue of the results of [\textit{I. Madsen} and \textit{M. Weiss}, Ann. Math. (2) 165, No. 3, 843--941 (2007; Zbl 1156.14021)] and [\textit{S. Galatius} and \textit{O. Randal-Williams}, Acta Math. 212, No. 2, 257--377 (2014; Zbl 1377.55012)] regarding the homology of \(\lim_g B\mathrm{Diff}^\partial (M\# N_g)\) cannot hold.NEWLINENEWLINEComments on the proof: Condition (c1) arises from the disjunction results of \textit{A. Haefliger} [Comment. Math. Helv. 36, 47--82 (1961; Zbl 0102.38603)]: under these conditions every element in \(\pi_p(M)\) and \(\pi_q(M)\) can be represented by an embedding, which moreover is unique up to regular homotopy. This is used to study embeddings of \(N_1\) into \(M\).NEWLINENEWLINEFor the proof, the author considers a simplicial complex \(K_N(M)\) whose set of vertices is the set of embeddings of a core of \(N_1\) into \(M\). A set of vertices spans a simplex if they are disjoint embeddings. The key point of the proof consists in proving that this simplicial complex is \(\frac{1}{2}(r_{N}(M)-4-d)\)-connected, after what a line of argument developed in [\textit{S. Galatius} and \textit{O. Randal-Williams}, ``Homological stability for moduli spaces of high dimensional manifolds'', loc. cit.] is used to infer the main theorem.NEWLINENEWLINEUnder condition (c1), the connectivity of this simplicial complex is equivalent to the following cancellation result proven by the author: If \(r_N(M)\geq 4+d\) and \(M\cong P\# N_1\) and \(M\cong Q\# N_1\), then \(P\cong Q\). The number \(d\) appears here in relation to the fact that any such diffeomorphism must preserve the composition operation \(\pi_p(M)\otimes \pi_q(S^p)\to \pi_{q}(M)\).NEWLINENEWLINEAs in [loc. cit.], he proves the high connectivity of this simplicial complex by comparing it to an algebraic version \(L_N(M)\) of it. This comparison uses strongly the Whitney trick and its high dimensional version as developed by [\textit{R. Wells}, Ill. J. Math. 11, 389--403 (1967; Zbl 0146.45103)] and [\textit{A. E. Hatcher} and \textit{F. Quinn}, Trans. Am. Math. Soc. 200, 327--344 (1974; Zbl 0291.57019)], the latter being used to characterise algebraically whether two embeddings are disjoint.