Homological stability for moduli spaces of high dimensional manifolds. II.

*(English)*Zbl 1412.57026This 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”.

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”.

Reviewer: Ben Knudsen (Somerville)

##### 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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