## Homological stability for moduli spaces of high dimensional manifolds. I

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Søren Galatius and Oscar Randal-Williams
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## Abstract:

We prove a homological stability theorem for moduli spaces of simply connected manifolds of dimension $2n > 4$, with respect to forming connected sum with $S^n \times S^n$. This is analogous to Harer’s stability theorem for the homology of mapping class groups. Combined with previous work of the authors, it gives a calculation of the homology of the moduli spaces of manifolds diffeomorphic to connected sums of $S^n \times S^n$ in a range of degrees.## References

- M. F. Atiyah and F. Hirzebruch,
*Bott periodicity and the parallelizability of the spheres*, Proc. Cambridge Philos. Soc.**57**(1961), 223–226. MR**126282**, DOI 10.1017/s0305004100035088 - Anthony Bak,
*On modules with quadratic forms*, Algebraic $K$-Theory and its Geometric Applications (Conf., Hull, 1969) Springer, Berlin, 1969, pp. 55–66. MR**0252431** - Anthony Bak,
*$K$-theory of forms*, Annals of Mathematics Studies, No. 98, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. MR**632404** - E. Binz and H. R. Fischer,
*The manifold of embeddings of a closed manifold*, Differential geometric methods in mathematical physics (Proc. Internat. Conf., Tech. Univ. Clausthal, Clausthal-Zellerfeld, 1978) Lecture Notes in Phys., vol. 139, Springer, Berlin-New York, 1981, pp. 310–329. With an appendix by P. Michor. MR**613004** - Alexander Berglund and Ib Madsen,
*Homological stability of diffeomorphism groups*, Pure Appl. Math. Q.**9**(2013), no. 1, 1–48. MR**3126499**, DOI 10.4310/PAMQ.2013.v9.n1.a1 - Søren K. Boldsen,
*Improved homological stability for the mapping class group with integral or twisted coefficients*, Math. Z.**270**(2012), no. 1-2, 297–329. MR**2875835**, DOI 10.1007/s00209-010-0798-y - Jean Cerf,
*Topologie de certains espaces de plongements*, Bull. Soc. Math. France**89**(1961), 227–380 (French). MR**140120**, DOI 10.24033/bsmf.1567 - Ruth Charney,
*A generalization of a theorem of Vogtmann*, Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), 1987, pp. 107–125. MR**885099**, DOI 10.1016/0022-4049(87)90019-3 - Søren Galatius and Oscar Randal-Williams,
*Homological stability for moduli spaces of high dimensional manifolds*, arXiv:1203.6830v2, 2012. - Søren Galatius and Oscar Randal-Williams,
*Detecting and realising characteristic classes of manifold bundles*, Algebraic topology: applications and new directions, Contemp. Math., vol. 620, Amer. Math. Soc., Providence, RI, 2014, pp. 99–110. MR**3290088**, DOI 10.1090/conm/620/12365 - Søren Galatius and Oscar Randal-Williams,
*Stable moduli spaces of high-dimensional manifolds*, Acta Math.**212**(2014), no. 2, 257–377. MR**3207759**, DOI 10.1007/s11511-014-0112-7 - Søren Galatius and Oscar Randal-Williams,
*Abelian quotients of mapping class groups of highly connected manifolds*, Math. Ann.**365**(2016), no. 1-2, 857–879. MR**3498929**, DOI 10.1007/s00208-015-1300-2 - Søren Galatius and Oscar Randal-Williams,
*Homological stability for moduli spaces of high dimensional manifolds. II*, Annals of Mathematics, to appear. arXiv:1601.00232, 2016. - John L. Harer,
*Stability of the homology of the mapping class groups of orientable surfaces*, Ann. of Math. (2)**121**(1985), no. 2, 215–249. MR**786348**, DOI 10.2307/1971172 - Morris W. Hirsch,
*Immersions of manifolds*, Trans. Amer. Math. Soc.**93**(1959), 242–276. MR**119214**, DOI 10.1090/S0002-9947-1959-0119214-4 - Allen Hatcher and Karen Vogtmann,
*Tethers and homology stability for surfaces*, arXiv:1508.04334, 2015. - Allen Hatcher and Nathalie Wahl,
*Stabilization for mapping class groups of 3-manifolds*, Duke Math. J.**155**(2010), no. 2, 205–269. MR**2736166**, DOI 10.1215/00127094-2010-055 - Nikolai V. Ivanov,
*On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients*, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991) Contemp. Math., vol. 150, Amer. Math. Soc., Providence, RI, 1993, pp. 149–194. MR**1234264**, DOI 10.1090/conm/150/01290 - John Jones and Elmer Rees,
*Kervaire’s invariant for framed manifolds*, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 141–147. MR**520501** - Matthias Kreck,
*Surgery and duality*, Ann. of Math. (2)**149**(1999), no. 3, 707–754. MR**1709301**, DOI 10.2307/121071 - Hendrik Maazen,
*Homology stability for the general linear group*, Utrecht thesis, 1979. - J. P. May and J. Sigurdsson,
*Parametrized homotopy theory*, Mathematical Surveys and Monographs, vol. 132, American Mathematical Society, Providence, RI, 2006. MR**2271789**, DOI 10.1090/surv/132 - David Mumford,
*Towards an enumerative geometry of the moduli space of curves*, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328. MR**717614** - Ib Madsen and Michael Weiss,
*The stable moduli space of Riemann surfaces: Mumford’s conjecture*, Ann. of Math. (2)**165**(2007), no. 3, 843–941. MR**2335797**, DOI 10.4007/annals.2007.165.843 - Oscar Randal-Williams,
*‘Group-completion’, local coefficient systems and perfection*, Q. J. Math.**64**(2013), no. 3, 795–803. MR**3094500**, DOI 10.1093/qmath/hat024 - Oscar Randal-Williams,
*Resolutions of moduli spaces and homological stability*, J. Eur. Math. Soc. (JEMS)**18**(2016), no. 1, 1–81. MR**3438379**, DOI 10.4171/JEMS/583 - Edwin H. Spanier,
*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112** - C. T. C. Wall,
*On the orthogonal groups of unimodular quadratic forms*, Math. Ann.**147**(1962), 328–338. MR**138565**, DOI 10.1007/BF01440955 - C. T. C. Wall,
*Surgery on compact manifolds*, London Mathematical Society Monographs, No. 1, Academic Press, London-New York, 1970. MR**0431216** - Michael Weiss,
*What does the classifying space of a category classify?*, Homology Homotopy Appl.**7**(2005), no. 1, 185–195. MR**2175298**, DOI 10.4310/HHA.2005.v7.n1.a10

## Additional Information

**Søren Galatius**- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- ORCID: 0000-0002-1015-7322
- Email: galatius@stanford.edu
**Oscar Randal-Williams**- Affiliation: Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
- MR Author ID: 852236
- Email: o.randal-williams@dpmms.cam.ac.uk
- Received by editor(s): March 23, 2016
- Received by editor(s) in revised form: February 7, 2017
- Published electronically: June 23, 2017
- Additional Notes: The first author was partially supported by NSF grants DMS-1105058 and DMS-1405001, and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 682922).

The second author was supported by EPSRC grant EP/M027783/1 and the Herchel Smith Fund.

Both authors were supported by ERC Advanced Grant No. 228082, and the Danish National Research Foundation through the Centre for Symmetry and Deformation. - © Copyright 2017 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**31**(2018), 215-264 - MSC (2010): Primary 57R90; Secondary 57R15, 57R56, 55P47
- DOI: https://doi.org/10.1090/jams/884
- MathSciNet review: 3718454

Dedicated: Dedicated to Ulrike Tillmann