Sharper periodicity and stabilization maps for configuration spaces of closed manifolds (Q2827392)

Let \(C_k(M)\) denote the ``configuration space'' of unordered \(k\) points in \(M\) which are pairwise distinct. Here \(M\) is a manifold and the object of this paper, which fits in an expanding body of work on the question, is to analyze the stability range for the homology of these configuration spaces as \(k\) increases. More precisely, and when \(M\) is an open manifold, there is a ``stabilization'' map \(C_k(M)\rightarrow C_{k+1}(M)\) given by ``bringing and adding a point from infinity'', and thus there is an induced map in homology \(\sigma: H_*(C_k(M);R)\rightarrow H_*(C_{k+1}(M);R)\), and this map can be shown to be an isomorphism for homological degrees \(\ast\leq f(k,M,R)\), where \(f(k,M,R)\) is a ``stable range'' computed already in various papers and depending obviously on \(k,M\) and the coefficient ring \(R\). It is known that \(f(k,M,R)\) is at least \(k/2\) and has largest value \(k\) if \(2\) is a unit in \(R\) and \(\dim M\geq 3\) for example. The main problem is to compare the homology of the configuration spaces in the case of a closed manifold.NEWLINENEWLINETheorem A (Part (ii)) of this paper shows that there is an isomorphism \(\sigma: H_i(C_k(M);\mathbb Z)\cong H_{i+1}(C_{k+1}(M);\mathbb Z))\) for any connected \(M\) of finite type, for a range of degrees \(i\) identical to the case of open manifolds, provided that \(M\) is odd. An analogous stabilization result is not true for even manifolds as illustrated by the case \(M=S^2\) and \(H_1(C_k(S^2))\cong \mathbb Z/(2k-2)\mathbb Z\). As pointed out, for \(M\) closed, no ``adding a point at infinity'' map \(C_k(M)\rightarrow C_{k+1}(M)\) exists, but one can always map \(C_k(M)\) into \(\text{Sym}^{\leq 2}_{k+1}(M)\); the space of unordered \(k+1\) points where two points can coincide (but not three) by adding a fixed basepoint. The authors show (Proposition 3.2) that for odd manifolds, this map induces an isomorphism on homology with \(\mathbb Z[1/2]\)-coefficients. They deduce (Theorem B) that in this case there is a natural split injection NEWLINE\[NEWLINE\sigma: H_*(C_k(M);\mathbb Z[1/2])\rightarrow H_*(C_{k+1}(M);\mathbb Z[1/2])NEWLINE\]NEWLINE which is an isomorphism for \(i\leq f(k,M, \mathbb Z[1/2])\).NEWLINENEWLINEPart (i) of Theorem A, and this is the main part, gives a homological isomorphism NEWLINE\[NEWLINE\sigma: H_*(C_k(M);\mathbb Z/d\mathbb Z)\cong H_*(C_{k+m}(M);\mathbb Z/d\mathbb Z)NEWLINE\]NEWLINE for degrees \(\ast\) less than the range \(f(k,M,\mathbb Z/d\mathbb Z)\), for \(M\) even dimensional and coefficients \(\mathbb Z/d\mathbb Z\) as long as \(d\) divides \(2m\). This result improves on most known stability results in the field [\textit{R. Nagpal}, ``FI-modules and the cohomology of modular representations of symmetric groups'', Preprint, \url{arXiv:1505.04294} and \textit{F. Cantero} and \textit{M. Palmer}, Doc. Math., J. DMV 20, 753--805 (2015; Zbl 1352.55008)]. The proof relies on geometric input from [\textit{O. Randal-Williams}, Q. J. Math. 64, No. 1, 303--326 (2013; Zbl 1264.55009)], homological calculations from [\textit{F. R. Cohen} et al., The homology of iterated loop spaces. Lecture Notes in Mathematics. 533. Berlin-Heidelberg-New York: Springer-Verlag (1976; Zbl 0334.55009)] and an algebraic lemma of independent interest that states that if \(C_*\) is a bounded below chain complex of finitely generated free abelian groups and \(p\) a prime, then the isomorphism type of the group \(H_i(C_*\otimes\mathbb Z/p^r\mathbb Z)\) is determined uniquely by the cardinality of the sets \(H_j(C_*\otimes\mathbb Z/p^w\mathbb Z)\) for \(w\leq r \) and \(j\leq i\). A similar statement is given for \(H_i(C_*)\).