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FREE FINITE GROUP ACTIONS ON RATIONAL HOMOLOGY 3-SPHERES

Part of: Connections with homological algebra and category theory

Published online by Cambridge University Press:  13 September 2019

ALEJANDRO ADEM  and
IAN HAMBLETON
ALEJANDRO ADEM
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada; adem@math.ubc.ca
IAN HAMBLETON
Affiliation:
Department of Mathematics & Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada; hambleton@mcmaster.ca

Abstract

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We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.

MSC classification

Primary: 57S17: Finite transformation groups
Secondary: 20J06: Cohomology of groups
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e28
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

References

Adem, A., ‘Cohomological exponents of Z G-lattices’, J. Pure Appl. Algebra 58 (1989), 1–5.Google Scholar
Adem, A. and Milgram, R. J., Cohomology of Finite Groups, 2nd edn., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 309 (Springer-Verlag, Berlin, 2004).Google Scholar
Bentzen, S. and Madsen, I., ‘On the Swan subgroup of certain periodic groups’, Math. Ann. 264 (1983), 447–474.Google Scholar
Browder, W., ‘Cohomology and group actions’, Invent. Math. 71 (1983), 599–607.Google Scholar
Browder, W. and Hsiang, W. C., ‘Some problems on homotopy theory manifolds and transformation groups’, inAlgebraic and Geometric Topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, CA, 1976), Part 2, Proceedings of Symposia in Pure Mathematics, XXXII (American Mathematical Society, Providence, RI, 1978), 251–267.Google Scholar
Brown, K. S., Cohomology of Groups, Graduate Texts in Mathematics, 87 (Springer-Verlag, New York–Berlin, 1982).Google Scholar
Cairns, S. S., ‘A simple triangulation method for smooth manifolds’, Bull. Amer. Math. Soc. (N.S.) 67 (1961), 389–390.Google Scholar
Carlson, J. F., Townsley, L., Valeri-Elizondo, L. and Zhang, M., Cohomology Rings of Finite Groups, Algebra and Applications, 3 (Kluwer Academic Publishers, Dordrecht, 2003), With an appendix: Calculations of cohomology rings of groups of order dividing 64 by Carlson, Valeri-Elizondo and Zhang.Google Scholar
Cavendish, W., ‘On finite derived quotients of 3-manifold groups’, Algebr. Geom. Topol. 15 (2015), 3355–3369.Google Scholar
Cooper, D. and Long, D. D., ‘Free actions of finite groups on rational homology 3-spheres’, Topology Appl. 101 (2000), 143–148.Google Scholar
Davis, J. F., ‘Evaluation of odd-dimensional surgery obstructions with finite fundamental group’, Topology 27 (1988), 179–204.Google Scholar
Evens, L. and Priddy, S., ‘The cohomology of the semidihedral group’, inConference on Algebraic Topology in Honor of Peter Hilton (Saint John’s, Nfld., 1983), Contemporary Mathematics, 37 (American Mathematical Society, Providence, RI, 1985), 61–72.Google Scholar
Gorenstein, D., Finite Groups, 2nd edn., (Chelsea Publishing Co., New York, 1980).Google Scholar
Hambleton, I., ‘Some examples of free actions on products of spheres’, Topology 45 (2006), 735–749.Google Scholar
Hambleton, I. and Madsen, I., ‘Local surgery obstructions and space forms’, Math. Z. 193 (1986), 191–214.Google Scholar
Handel, D., ‘On products in the cohomology of the dihedral groups’, Tohoku Math. J. (2) 45 (1993), 13–42.Google Scholar
Hilton, P. J. and Stammbach, U., A Course in Homological Algebra, 2nd edn., Graduate Texts in Mathematics, 4 (Springer-Verlag, New York, 1997).Google Scholar
Hopf, H., ‘Zum Clifford–Kleinschen Raumproblem’, Math. Ann. 95 (1926), 313–339.Google Scholar
Lee, R., ‘Semicharacteristic classes’, Topology 12 (1973), 183–199.Google Scholar
Lott, J., ‘The work of Grigory Perelman’, inInternational Congress of Mathematicians. Vol. I (Eur. Math. Soc., Zürich, 2007), 66–76.Google Scholar
Mac Lane, S., ‘Homology’, inClassics in Mathematics (Springer-Verlag, Berlin, 1995), Reprint of the 1975 edition.Google Scholar
Madsen, I., ‘Reidemeister torsion, surgery invariants and spherical space forms’, Proc. Lond. Math. Soc. (3) 46 (1983), 193–240.Google Scholar
Madsen, I., Thomas, C. B. and Wall, C. T. C., ‘The topological spherical space form problem. II. Existence of free actions’, Topology 15 (1976), 375–382.Google Scholar
Madsen, I., Thomas, C. B. and Wall, C. T. C., ‘Topological spherical space form problem. III. Dimensional bounds and smoothing’, Pacific J. Math. 106 (1983), 135–143.Google Scholar
Milgram, R. J., ‘The cohomology of the Mathieu group M 23 ’, J. Group Theory 3 (2000), 7–26.Google Scholar
Milnor, J., ‘Groups which act on S n without fixed points’, Amer. J. Math. 79 (1957), 623–630.Google Scholar
Mislin, G., ‘Finitely dominated nilpotent spaces’, Ann. of Math. (2) 103 (1976), 547–556.Google Scholar
Pardon, W., ‘Mod 2 semicharacteristics and the converse to a theorem of Milnor’, Math. Z. 171 (1980), 247–268.Google Scholar
Reznikov, A., ‘Three-manifolds class field theory (homology of coverings for a nonvirtually b 1 -positive manifold)’, Selecta Math. (N.S.) 3 (1997), 361–399.Google Scholar
Swan, R. G., ‘The p-period of a finite group’, Illinois J. Math. 4 (1960), 341–346.Google Scholar
Swan, R. G., ‘Periodic resolutions for finite groups’, Ann. of Math. (2) 72 (1960), 267–291.Google Scholar
Wall, C. T. C., ‘Periodic projective resolutions’, Proc. Lond. Math. Soc. (3) 39 (1979), 509–553.Google Scholar
Wall, C. T. C., ‘On the structure of finite groups with periodic cohomology’, inLie Groups: Structure, Actions, and Representations, Progress in Mathematics, 306 (Birkhäuser/Springer, New York, 2013), 381–413.Google Scholar