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Homological stability for moduli spaces of high dimensional manifolds. II. (English) Zbl 1412.57026
This paper is a continuation and extension of the authors’ work in [Acta Math. 212, No. 2, 257–377 (2014; Zbl 1377.55012); J. Am. Math. Soc. 31, No. 1, 215–264 (2018; Zbl 1395.57044)] (the reader is cautioned that a bibliographical error in the published version elides the distinction between these two). Generally speaking, the program of these papers is to extend into higher dimensions various celebrated results for surfaces, such as Harer stability, the Mumford conjecture, and the Madsen-Weiss theorem.
The main result, which the authors christen “stable stability,” asserts that a certain moduli space of highly connected null-bordisms \(\partial W^{2n}=P\), perhaps with tangential structure, becomes a cobordism invariant at the level of homology after stabilization by the connect-sum addition of countably many copies of \(S^n\times S^n\). The companion theorem identifies this stable object, again up to homology, as the infinite loop space associated to a certain Thom spectrum depending on the choice of tangential structure, strengthening a theorem of [Zbl 1377.55012].
The computational power of these results arises in pairing them with the prequel [Zbl 1395.57044], which shows that the components of the moduli spaces in question, which are (Borel constructions over) classifying spaces of diffeomorphism groups, exhibit homological stability for the operation of connect-sum with \(S^n\times S^n\). Thus, the results above apply to these classifying spaces in a range of degrees depending on “genus”.

MSC:
57R90 Other types of cobordism
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57R56 Topological quantum field theories (aspects of differential topology)
55P47 Infinite loop spaces
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[1] Adams, J. F., A variant of {E}. {H}. {B}rown’s representability theorem, Topology. Topology. An International Journal of Mathematics, 10, 185-198, (1971) · Zbl 0197.19604
[2] Berglund, Alexander; Madsen, Ib, Homological stability of diffeomorphism groups, (2012) · Zbl 1295.57038
[3] Berglund, Alexander; Madsen, Ib, Rational homotopy theory of automorphisms of highly connected manifolds, (2014) · Zbl 1295.57038
[4] Binz, E.; Fischer, H. R., Differential Geometric Methods in Mathematical Physics. The manifold of embeddings of a closed manifold, Lecture Notes in Phys., 139, 310-325, (1981)
[5] Botvinnik, Boris; Ebert, Johannes; Randal-Williams, Oscar, Infinite loop spaces and positive scalar curvature, (2017) · Zbl 1377.53067
[6] Galatius, S.; Tillmann, Ulrike; Madsen, Ib; Weiss, Michael, The homotopy type of the cobordism category, Acta Math.. Acta Mathematica, 202, 195-239, (2009) · Zbl 1221.57039
[7] Galatius, S.; Randal-Williams, Oscar, Monoids of moduli spaces of manifolds, Geom. Topol.. Geometry & Topology, 14, 1243-1302, (2010) · Zbl 1205.55007
[8] Galatius, S.; Randal-Williams, Oscar, Homological stability for moduli spaces of high dimensional manifolds. {I}, J. Amer. Math. Soc., 50 pp. pp., (2017) · Zbl 1395.57044
[9] Galatius, S.; Randal-Williams, Oscar, Stable moduli spaces of high-dimensional manifolds, Acta Math.. Acta Mathematica, 212, 257-377, (2014) · Zbl 1377.55012
[10] Goerss, Paul; Schemmerhorn, Kristen, Interactions Between Homotopy Theory and Algebra. Model categories and simplicial methods, Contemp. Math., 436, 3-49, (2007) · Zbl 1134.18007
[11] Hatcher, Allen; Vogtmann, Karen, Homology stability for outer automorphism groups of free groups, Algebr. Geom. Topol.. Algebraic & Geometric Topology, 4, 1253-1272, (2004) · Zbl 1093.20020
[12] Kervaire, Michel A., Le th\'eor\`eme de {B}arden-{M}azur-{S}tallings, Comment. Math. Helv.. Commentarii Mathematici Helvetici, 40, 31-42, (1965) · Zbl 0135.41503
[13] Madsen, Ib; Weiss, Michael, The stable moduli space of {R}iemann surfaces: {M}umford’s conjecture, Ann. of Math.. Annals of Mathematics. Second Series, 165, 843-941, (2007) · Zbl 1156.14021
[14] McDuff, D.; Segal, G., Homology fibrations and the ““‘group-completion””’ theorem, Invent. Math.. Inventiones Mathematicae, 31, 279-284, (1975/76) · Zbl 0306.55020
[15] Miller, Jeremy; Palmer, Martin, A twisted homology fibration criterion and the twisted group-completion theorem, Q. J. Math.. The Quarterly Journal of Mathematics, 66, 265-284, (2015) · Zbl 1326.55005
[16] Quinn, Frank, The stable topology of {\(4\)}-manifolds, Topology Appl.. Topology and its Applications, 15, 71-77, (1983) · Zbl 0536.57006
[17] Randal-Williams, Oscar, Resolutions of moduli spaces and homological stability, J. Eur. Math. Soc. \((\)JEMS\()\). Journal of the European Mathematical Society (JEMS), 18, 1-81, (2016) · Zbl 1366.55011
[18] Tillmann, Ulrike, On the homotopy of the stable mapping class group, Invent. Math.. Inventiones Mathematicae, 130, 257-275, (1997) · Zbl 0891.55019
[19] Wahl, Nathalie, Homological stability for the mapping class groups of non-orientable surfaces, Invent. Math.. Inventiones Mathematicae, 171, 389-424, (2008) · Zbl 1140.55007
[20] Wall, C. T. C., Finiteness conditions for {\({\rm CW}\)}-complexes, Ann. of Math.. Annals of Mathematics. Second Series, 81, 56-69, (1965) · Zbl 0152.21902
[21] Weiss, Michael, Dalian notes on {P}ontryagin classes, (2015)
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