Topological rigidity and actions on contractible manifolds with discrete singular set
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- by Frank Connolly, James F. Davis and Qayum Khan HTML | PDF
- Trans. Amer. Math. Soc. Ser. B 2 (2015), 113-133
Abstract:
The problem of equivariant rigidity is the $\Gamma$-homeomorphism classification of $\Gamma$-actions on manifolds with compact quotient and with contractible fixed sets for all finite subgroups of $\Gamma$. In other words, this is the classification of cocompact $E_{\mathrm {fin}} \Gamma$-manifolds.
We use surgery theory, algebraic $K$-theory, and the Farrell–Jones Conjecture to give this classification for a family of groups which satisfy the property that the normalizers of nontrivial finite subgroups are themselves finite. More generally, we study cocompact proper actions of these groups on contractible manifolds and prove that the $E_{\mathrm {fin}}$ condition is always satisfied.
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Additional Information
- Frank Connolly
- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
- MR Author ID: 51035
- Email: connolly.1@nd.edu
- James F. Davis
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 194576
- Email: jfdavis@indiana.edu
- Qayum Khan
- Affiliation: Department of Mathematics, Saint Louis University, St. Louis, Missouri 63103
- MR Author ID: 817046
- ORCID: 0000-0002-4987-3211
- Email: khanq@slu.edu
- Received by editor(s): April 18, 2014
- Received by editor(s) in revised form: August 2, 2015, and October 26, 2015
- Published electronically: November 25, 2015
- Additional Notes: The authors were partly supported by the NSF (DMS-0601234, DMS-1210991, DMS-0904276)
- © Copyright 2015 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 2 (2015), 113-133
- MSC (2010): Primary 57S30, 57R91; Secondary 19J05, 19J25
- DOI: https://doi.org/10.1090/btran/9
- MathSciNet review: 3427570