Localization and homological stability of configuration spaces (Q2922450)

{ Background.} Let \(M\) be a smooth, connected (and paracompact) manifold and let \(C_k(M)\) denote the space of subsets of \(M\) of cardinality \(k\). This is the \textit{\(k\)th unordered configuration space on \(M\)}, and formally it is topologised as \((M^k \smallsetminus \Delta)/\Sigma_k\), where \(\Delta\) is the subspace of \(k\)-tuples \((x_1,\ldots,x_k)\) where \(x_i=x_j\) for some \(i\neq j\) and the symmetric group \(\Sigma_k\) acts by permuting the entries of a \(k\)-tuple. If \(M\) is non-compact, the sequence \(\{C_k(M)\}\) is \textit{homologically stable}, meaning that in each fixed degree \(i\), there exist isomorphisms \(H_i(C_j(M)) \cong H_i(C_k(M))\) for \(j,k\gg i\). In fact, there are well-defined ``stabilisation maps'' \(C_k(M) \to C_{k+1}(M)\) which induce all the isomorphisms required for homological stability. This is originally due to \textit{D. McDuff} [Topology 14, 91--107 (1975; Zbl 0296.57001)] and \textit{G. Segal} [Acta Math. 143, 39--72 (1979; Zbl 0427.55006)], and the most recent and general version of this result is proved in [\textit{O.\ Randal-Williams}, Q.\ J.\ Math.\ 64, No.\ 1, 303--326 (2013; Zbl 1264.55009)].NEWLINENEWLINEHowever, when the manifold \(M\) is closed, such maps do not exist and the sequence \(\{C_k(M)\}\) is not homologically stable in general: the standard example for this is \(C_k(S^2)\), whose first homology is \(\mathbb{Z}/(2k-2)\), which does not stabilise. Nevertheless, some stabilisation does occur if one takes homology with coefficients in a field \(\mathbb{F}\), namely: for each fixed degree \(i\), the sequence \(\{H_i(C_k(M);\mathbb{F})\}\) is independent of \(k\) for \(k\gg i\) as long as \(M\) is odd-dimensional or the characteristic of \(\mathbb{F}\) is even. The characteristic-\(2\) case was proven in [\textit{R. J. Milgram} and \textit{P. Löffler}, Contemp. Math. 78, 415--424 (1988; Zbl 0667.55005)], and both the characteristic-\(2\) and odd-dimensional cases were proven in [\textit{C. F. Bödigheimer} et al., Topology 28, No. 1, 111--123 (1989; Zbl 0689.55012)]. More recently, the characteristic-\(0\) case is a corollary of the main result of [\textit{T.\ Church}, Invent.\ Math.\ 188, No.\ 2, 465--504 (2012; Zbl 1244.55012)], and all three cases were proven in O.\ Randal-Williams, [op. cit.].NEWLINENEWLINE{ The main results.} The first main theorem of the paper under review recovers the rational homological stability result for the sequence \(\{C_k(M)\}\) of Church and Randal-Williams (mentioned above). In fact, the theorem of Church assumes that \(M\) is orientable and that of Randal-Williams assumes that it has dimension at least \(3\), so the authors' result is new in the case of non-orientable surfaces. There is one very small caveat: if \(M\) is even-dimensional and the Euler characteristic \(\chi\) of \(M\) is even, say \(\chi=2a\), then the authors' theorem holds only for the subsequence \(\{C_k(M) \mid k\neq a\}\).NEWLINENEWLINEThe second main theorem is a new homological stability result for unordered configuration spaces on closed manifolds, concerning coefficients in the \(p\)-local integers \(\mathbb{Z}_{(p)}\) for any prime \(p\geqslant \frac12(\mathrm{dim}(M)+3)\). They show that for odd-dimensional \(M\), the sequence \(\{C_k(M)\}\) is homologically stable with these coefficients, and for even-dimensional \(M\), it splits as a union of subsequences, each of which is (independently) homologically stable with these coefficients. The subsequences in question are NEWLINE\[NEWLINE \mathcal{C}_e \;=\; \{ C_k(M) \mid \nu_p(2k-\chi)=e \} NEWLINE\]NEWLINE for each non-negative integer \(e\), where \(\nu_p(-)\) is \(p\)-adic valuation. As long as the integral homology of \(C_k(M)\) is finitely-generated in each degree, this implies that the \(p\)-torsion subgroup of \(H_i(C_k(M);\mathbb{Z})\) is independent of \(k\) for \(k\gg i\) under the same conditions. Previous stability results for the torsion in the homology of configuration spaces dealt only with mod-\(p\) homology, and so were unable to distinguish between, for example, a \(\mathbb{Z}/p\) summand and a \(\mathbb{Z}/(p^2)\) summand in \(H_i(C_k(M);\mathbb{Z})\). Another advantage of the authors' result is that they do not just obtain abstract isomorphisms of homology groups (as with the results on homology with field coefficients mentioned above) -- their isomorphisms are induced by zig-zags of maps of spaces, and thus preserve additional structure, such as cup products.NEWLINENEWLINEThe authors note that the hypothesis \(p\geqslant \frac12(\mathrm{dim}(M)+3)\) is sometimes unnecessary. Specifically, they show that when \(M\) is either parallelisable or an orientable surface, each of the subsequences \(\mathcal{C}_e\) is homologically stable with \(\mathbb{Z}_{(p)}\) coefficients even without this hypothesis. (In fact their arguments show this equally well for all stably parallelisable manifolds.)NEWLINENEWLINE{ Overview of the proof.} We give a detailed explanation of the authors' proof in six steps below, but we will first summarise the key points.NEWLINENEWLINEFirst, using a ``scanning'' result of McDuff [op.\ cit.], the theorem may be reduced to the problem of constructing sufficiently many bundle endomorphisms of a certain bundle \(\dot{T}M_{(p)}\) over \(M\), whose fibres are copies of the \(p\)-localisation of \(S^n\), where \(n=\mathrm{dim}(M)\). When \(n\) is odd, we can use obstruction theory directly to construct bundle endomorphisms restricting to degree \(1\) maps on each fibre, and taking any given section \(\sigma\) of \(\dot{T}M_{(p)}\) to any other given section \(\tau\). This implies the result when \(M\) is odd-dimensional. However, this does not work when \(n\) is even -- in this case there is an obstruction living in a non-trivial group, arising from the fact that \(\pi_{2n-1}(S^n)\) has non-zero rank. We can still use obstruction theory to construct bundle endomorphisms having fibrewise degree \(1\), and indeed any fibrewise degree which is a unit in the ring \(\mathbb{Z}_{(p)}\), but we cannot directly control how they act on a given section of \(\dot{T}M_{(p)}\). To get around this, we need to see how an arbitrary bundle endomorphism of fibrewise degree \(r\) acts on sections of degree \(k\). The authors prove an explicit formula for the degree of the resulting section, from which the result in the even-dimensional case follows.NEWLINENEWLINEThis degree formula is where the restriction in the even-dimensional case arises --- we only compare configurations of \(j\) points with configurations of \(k\) points, where \(\nu_p(2j-\chi) = \nu_p(2k-\chi)\) --- since we cannot always get from a section of degree \(j\) to one of degree \(k\) through bundle endomorphisms whose fibrewise degree is a unit of \(\mathbb{Z}_{(p)}\). On the other hand, the lower bound on \(p\) is needed in both the even- and the odd-dimensional case to do the obstruction theory.NEWLINENEWLINE{ More details.} In the next six paragraphs we give a more detailed overview of the authors' proof, slightly biased towards the reviewer's point of view. This is in order to facilitate a comparison (see the end of the review) with a related, more recent, result of F.\ Cantero and the reviewer.NEWLINENEWLINE{ 1.\ Scanning maps.} Let \(\dot{T}M\) denote the fibrewise one-point compactification of the tangent bundle \(TM\) and let \(\dot{T}M_{(p)}\) be its fibrewise \(p\)-localisation (with the convention that when \(p=0\) this means rational localisation). There are well-defined ``scanning maps'' \(C_k(M) \to \Gamma_k(\dot{T}M)\) taking values in degree-\(k\) sections of the bundle \(\dot{T}M\), and McDuff [op.\ cit.] proves that these induce isomorphisms on homology in a stable range of degrees. By results of \textit{J. M. Møller} [Trans. Am. Math. Soc. 303, 733--741 (1987; Zbl 0628.55007)], we see that \(\Gamma_k(\dot{T}M_{(p)})\) is the \(p\)-localisation of \(\Gamma_k(\dot{T}M)\) for each \(k\in\mathbb{Z}\) (although the former is defined for any \(k\in\mathbb{Z}_{(p)}\)). It therefore suffices to prove that \(\Gamma_j(\dot{T}M_{(p)})\) and \(\Gamma_k(\dot{T}M_{(p)})\) are homotopy equivalent for different \(j\) and \(k\), under the appropriate conditions. The strategy is to try to construct a bundle endomorphism of \(\dot{T}M_{(p)}\) such that composition with it takes sections of degree \(j\) to sections of degree \(k\), and which has a fibrewise homotopy inverse.NEWLINENEWLINE{ 2.\ Bundle endomorphisms.} The space of bundle endomorphisms of \(\dot{T}M_{(p)}\) is \(\Gamma(\mathrm{end}(\dot{T}M_{(p)}))\), where \(\mathrm{end}(\dot{T}M_{(p)})\) is the bundle over \(M\) whose fibre over \(x\in M\) consists of endomorphisms of the fibre \(\dot{T}M_{(p)}|_x\). Similarly, the subspace of bundle endomorphisms of \(\dot{T}M_{(p)}\) which take a given section \(\sigma\) to another given section \(\tau\) is \(\Gamma(\mathrm{end}^{\sigma,\tau}(\dot{T}M_{(p)}))\), where \(\mathrm{end}^{\sigma,\tau}(\dot{T}M_{(p)})\) is the bundle over \(M\) whose fibre over \(x\in M\) consists of endomorphisms of the fibre \(\dot{T}M_{(p)}|_x\) taking \(\sigma(x)\) to \(\tau(x)\). These bundles both split as a disjoint union over \(r\in\mathbb{Z}_{(p)}\) of the subbundles \(\mathrm{end}_r(\dot{T}M_{(p)})\), respectively \(\mathrm{end}_r^{\sigma,\tau}(\dot{T}M_{(p)})\), whose fibre over \(x\in M\) is the subspace of endomorphisms of degree \(r\).NEWLINENEWLINE{ 3.\ Obstruction theory.} Theorem 3.3 of \textit{A. Dold} [Ann. Math. (2) 78, 223--255 (1963; Zbl 0203.25402)] implies that, for a bundle over a paracompact Hausdorff manifold, an endomorphism of the bundle admits a fibrewise homotopy inverse as long as it restricts to a homotopy equivalence on each fibre. Thus, if \(r\) is a unit in \(\mathbb{Z}_{(p)}\) then any bundle endomorphism in \(\Gamma(\mathrm{end}_r(\dot{T}M_{(p)}))\) admits a fibrewise homotopy inverse. It therefore suffices to find a section of \(\mathrm{end}_1^{\sigma,\tau}(\dot{T}M_{(p)})\) for any given \(\sigma\) and \(\tau\). The fibre of this bundle is homotopy equivalent to \(\Omega_1^n S^n_{(p)}\), where \(n=\mathrm{dim}(M)\), so by obstruction theory a section exists as long as this is \((n-1)\)-connected. When \(n\) is odd, this follows from the calculations of \textit{J.-P. Serre} [Ann. Math. (2) 54, 425--505 (1951; Zbl 0045.26003)], as long as \(p=0\) or \(p\geqslant \frac12(n+3)\), completing the proof for odd-dimensional \(M\).NEWLINENEWLINE{ 4.\ The even-dimensional case.} However, when \(M\) is even-dimensional, \(\pi_{n-1}(\Omega_1^n S^n_{(p)})\) is non-trivial even when \(p=0\). To deal with this case, the authors instead show that the \textit{unbased} mapping space \(\mathrm{Map}_r(S^n_{(p)},S^n_{(p)})\) is \((n-1)\)-connected for any unit \(r\in\mathbb{Z}_{(p)}\), again using the calculations of Serre [op.\ cit.] for primes \(p\geqslant \frac12(n+3)\), and using a result of [\textit{J.\ M.\ Møller} and \textit{M.\ Raussen}, Trans.\ Am.\ Math.\ Soc.\ 292, 721-732 (1985; Zbl 0605.55008)] for \(p=0\). This is the fibre of the bundle \(\mathrm{end}_r(\dot{T}M_{(p)})\), so we may find a bundle endomorphism of \(\dot{T}M_{(p)}\) of any fibrewise degree \(r\) which is a unit in \(\mathbb{Z}_{(p)}\). These all admit fibrewise homotopy inverses by the result of Dold cited above, so it remains to show that we have found sufficiently many to deduce the result.NEWLINENEWLINE{ 5.\ The effect of a bundle endomorphism on degree.} The final step (for the even-dimensional case) is to show how a bundle endomorphism of \(\dot{T}M_{(p)}\) of fibrewise degree \(r\) affects the degree of a section of this bundle. Specifically, the authors show that given \(\sigma\in \Gamma(\dot{T}M_{(p)})\) of degree \(k\) and a bundle endomorphism \(\phi\) of \(\dot{T}M_{(p)}\) of fibrewise degree \(r\), the degree of the induced section \(\phi \circ \sigma\) is NEWLINE\[NEWLINE rk + \tfrac12 (1-r)\chi . NEWLINE\]NEWLINE If \(r\) is a unit in \(\mathbb{Z}_{(p)}\) then \(\phi\) has a fibrewise homotopy inverse and therefore induces a homotopy equivalence between \(\Gamma_j(\dot{T}M_{(p)})\) and \(\Gamma_k(\dot{T}M_{(p)})\) for \(j = rk + \tfrac12 (1-r)\chi\). Since such a \(\phi\) exists for any unit \(r\in \mathbb{Z}_{(p)}\), this implies that \(\Gamma_j(\dot{T}M_{(p)})\) and \(\Gamma_k(\dot{T}M_{(p)})\) are homotopy equivalent whenever \(\nu_p(2j-\chi) = \nu_p(2k-\chi)\). (When \(p=0\) we use the convention that \(\nu_0(0)=\infty\) and \(\nu_0(a)=0\) for \(a\neq 0\).) This completes the proof for even-dimensional \(M\).NEWLINENEWLINE{ 6.\ The degree formula.} Finally, the degree formula above is obtained as follows. First assume that \(M\) is orientable. Then the bundle \(\dot{T}M_{(p)}\) -- viewed as just a fibration -- is classified by a map \(M \to BSO(n) \to B\mathrm{Map}_1(S^n_{(p)},S^n_{(p)})\) whose target is \(n\)-connected (see paragraph 4) and is therefore nullhomotopic. The key fact is then that a choice of trivialisation of \(\dot{T}M_{(p)}\) induces a homeomorphism \(\Gamma(\dot{T}M_{(p)}) \cong \mathrm{Map}(M,S^n_{(p)})\) which takes sections of degree \(k\) to maps of degree \(k-\frac12 \chi\). The authors show this by computing the degree of a section as its intersection number with the zero section. The claimed degree formula then follows immediately from this, and can be extended to the case of non-orientable \(M\) by passing to the orientation double cover.NEWLINENEWLINE{ Using the theorem of Dold.} We note that appealing to the result of Dold, as we have done above, slightly simplifies some of the proofs in the paper, since it tells us that certain bundle maps automatically have fibrewise homotopy inverses. Precisely, it simplifies the proofs of Propositions 3.2 and 4.6, and makes Lemma 4.2 unnecessary.NEWLINENEWLINE{ Related subsequent results.} Since the publication of this paper, two other preprints have appeared with related results. The first is [\textit{B.\ Knudsen}, ``Betti numbers and stability for configuration spaces via factorization homology'', \url{arXiv:1405.6696} (2014)], which, as a corollary of its main result, gives an alternative proof of rational homological stability for configuration spaces on arbitrary connected manifolds, and improves the previously known ranges in which stability holds.NEWLINENEWLINEThe second is a preprint of F.\ Cantero and the reviewer [\textit{F.\ Cantero} and \textit{M.\ Palmer}, ``On homological stability for configuration spaces on closed background manifolds'', \url{arXiv:1406.4916} (2014)]. Here it is shown that the lower bound on \(p\) is not needed for homological stability with coefficients in \(\mathbb{Z}_{(p)}\) to hold (with some caveats when \(p=2\)). This is proved in the paper under review when \(M\) is parallelisable or an orientable surface, but in fact it is true in general. The essential difference is the method by which the necessary bundle endomorphisms are constructed. In the paper under review, this uses obstruction theory for a fibration over an \(n\)-dimensional manifold whose fibre is either \(\Omega_1^n S^n_{(p)}\) or \(\mathrm{Map}_r(S^n_{(p)},S^n_{(p)})\). These spaces only become \((n-1)\)-connected once \(p\) is sufficiently large compared to \(n\) (and assuming that \(n\) is odd in the first case). In the preprint of F.\ Cantero and the reviewer, bundle endomorphisms of fibrewise degree \(r\), equivalently sections of the bundle \(\mathrm{end}_r(\dot{T}M_{(p)})\), are provided by geometrically constructing a certain bundle map NEWLINE\[NEWLINE V_2(TM\oplus\epsilon)_{(p)} \longrightarrow \mathrm{end}_r(\dot{T}M_{(p)}) NEWLINE\]NEWLINE together with sections of the left-hand bundle, where \(\epsilon\) is the trivial one-dimensional bundle over \(M\) and \(V_k(E)\) denotes the bundle of orthonormal \(k\)-frames in the vector bundle \(E\). The idea is that constructing sections of the left-hand side is easier, and in particular does not depend on imposing a lower bound on \(p\). First, there is a section \(\sigma\) of the bundle \(V_1(TM\oplus\epsilon)_{(p)} = S(TM\oplus\epsilon)_{(p)} = \dot{T}M_{(p)}\) of any degree. To see this, reduce to the case \(M=S^n\) by pulling back along a degree-one map \(M\to S^n\), then note that when \(M=S^n\) this bundle is trivial, and there exist maps \(S^n \to S^n_{(p)}\) of any degree in \(\mathbb{Z}_{(p)}\). It then remains to see for which values of \(\mathrm{deg}(\sigma)\) one can solve the lifting problem along the forgetful map NEWLINE\[NEWLINE V_2(TM\oplus\epsilon)_{(p)} \longrightarrow V_1(TM\oplus\epsilon)_{(p)}. NEWLINE\]NEWLINE This has a single obstruction living in the (twisted) cohomology group \(H^n(M;\pi_{n-1}(S^{n-1}_{(p)}))\), which may be analysed as the (twisted) Euler class of an \(S^{n-1}_{(p)}\)-bundle over \(M\). The upshot is that the result in the paper under review concerning coefficients in \(\mathbb{Z}_{(p)}\) holds for any odd prime \(p\) (and also for \(p=2\) under extra conditions), not just for \(p\geqslant \frac12(\mathrm{dim}(M)+3)\).NEWLINENEWLINEAnother related result (not published at the time of writing this review) was obtained in the thesis of R.\ Nagpal, concerning the homology of \(\{C_k(M)\}\) with coefficients in a field \(\mathbb{F}\) of positive characteristic \(p\), assuming that \(M\) is orientable. His result is that in each fixed degree \(i\) the sequence \(\{\mathrm{dim}\, H_i(C_k(M);\mathbb{F})\}\) is periodic in \(k\) for \(k\gg i\). Moreover, the period is \(p^{a(M)}\) for some number \(a(M)\), and he gives an algorithm to compute an upper bound on this number. This result is also recovered in the preprint of F.\ Cantero and the reviewer [op.\ cit.] (using in particular the result on \(\mathbb{Z}_{(p)}\) coefficients), with a much simpler upper bound for \(a(M)\), namely \(a(M)\leqslant 1+\nu_p(\chi)\), where \(\nu_p(-)\) is \(p\)-adic valuation and \(\chi\) is the Euler characteristic of \(M\).