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Central stability homology

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Abstract

We give a new categorical way to construct the central stability homology of Putman and Sam and explain how it can be used in the context of representation stability and homological stability. In contrast to them, we cover categories with infinite automorphism groups. We also connect central stability homology to Randal-Williams and Wahl’s work on homological stability. We also develop a criterion that implies that functors that are polynomial in the sense of Randal-Williams and Wahl are centrally stable in the sense of Putman.

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Notes

  1. Only information on homology is used in [16], but in all cases vanishing homotopy groups are used to imply this condition.

  2. In the published version [16] of this paper, they slightly changed the definition.

  3. In the published version [16] of this paper, they slightly changed the definition. They require the slightly stronger condition that if V has polynomial degree \(\le r\) in ranks \(>d\), \({{\,\mathrm{coker}\,}}V\) has polynomial degree \(\le r-1\) in ranks \(>d-1\) (instead of \(>d\)).

References

  1. Church, T., Ellenberg, J.S.: Homology of FI-modules. Geom. Topol. 21(4), 2373–2418 (2017)

    Article  MathSciNet  Google Scholar 

  2. Church, T., Ellenberg, J.S., Farb, B.: FI-modules and stability for representations of symmetric groups. Duke Math. J. 164(9), 1833–1910 (2015)

    Article  MathSciNet  Google Scholar 

  3. Church, T., Ellenberg, J.S., Farb, B., Nagpal, R.: FI-modules over Noetherian rings. Geom. Topol. 18(5), 2951–2984 (2014)

    Article  MathSciNet  Google Scholar 

  4. Church, T., Farb, B.: Representation theory and homological stability. Adv. Math. 245, 250–314 (2013)

    Article  MathSciNet  Google Scholar 

  5. Djament, A.: Des propriétés de finitude des foncteurs polynomiaux. Fundam. Math. 233(3), 197–256 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Gan, W.L., Li, L.: On central stability. Bull. Lond. Math. Soc. 49(3), 449–462 (2017)

    Article  MathSciNet  Google Scholar 

  7. Kan, D.M.: Adjoint functors. Trans. Am. Math. Soc. 87, 294–329 (1958)

    Article  MathSciNet  Google Scholar 

  8. Mac Lane, S.: Categories for the Working Mathematician, vol. 5 of Graduate Texts in Mathematics, second edn. Springer, New York (1998)

    MATH  Google Scholar 

  9. Miller, J., Patzt, P., Wilson, J.C.H.: Polynomial VIC- and SI-modules. Adv. Math. (2019) (to appear)

  10. Nakaoka, M.: Decomposition theorem for homology groups of symmetric groups. Ann. Math. 2(71), 16–42 (1960)

    Article  MathSciNet  Google Scholar 

  11. Putman, A., Sam, S.V.: Representation stability and finite linear groups. Duke Math. J. 166(13), 2521–2598 (2017)

    Article  MathSciNet  Google Scholar 

  12. Putman, A.: Stability in the homology of congruence subgroups. Invent. Math. 202(3), 987–1027 (2015)

    Article  MathSciNet  Google Scholar 

  13. Patzt, P., Xiaolei, W.: Stability results for Houghton groups. Algebr. Geom. Topol. 16(4), 2365–2377 (2016)

    Article  MathSciNet  Google Scholar 

  14. Quillen, D.: Personal Notes, August 9, Clay Library (1970). http://www.claymath.org/library/Quillen/Working_papers/quillen%201971/1971-11.pdf

  15. Randal-Williams, O., Wahl, N.: Homological stability for automorphism groups. Preprint (2015). arXiv:1409.3541v3

  16. Randal-Williams, O., Wahl, N.: Homological stability for automorphism groups. Adv. Math. 318, 534–626 (2017)

    Article  MathSciNet  Google Scholar 

  17. van der Kallen, W.: Homology stability for linear groups. Invent. Math. 60(3), 269–295 (1980)

    Article  MathSciNet  Google Scholar 

  18. Weibel, C.A.: An Introduction to Homological Algebra, vol. 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

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Acknowledgements

The author was supported by the Berlin Mathematical School and the Dahlem Research School. The author also wants to thank Aurélien Djament, Daniela Egas Santander, Reiner Hermann, Henning Krause, Daniel Lütgehetmann, Jeremy Miller, Holger Reich, Steven Sam, Elmar Vogt, Nathalie Wahl, Jenny Wilson for helpful conversations. Special thanks to Reiner Hermann and his invitation to NTNU where the idea for this project was born. The author would also like to thank the anonymous referee for many helpful suggestions and corrections.

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  1. Department of Mathematics, Purdue University, West Lafayette, IN, USA

    Peter Patzt

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Patzt, P. Central stability homology. Math. Z. 295, 877–916 (2020). https://doi.org/10.1007/s00209-019-02365-y

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  • DOI: https://doi.org/10.1007/s00209-019-02365-y

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