Group actions and group extensions

Yalçin, Ergün

doi:10.1090/S0002-9947-00-02485-5

Group actions and group extensions

Author: Ergün Yalçin
Journal: Trans. Amer. Math. Soc. 352 (2000), 2689-2700
MSC (1991): Primary 57S25; Secondary 20J06, 20C15
DOI: https://doi.org/10.1090/S0002-9947-00-02485-5
Published electronically: February 24, 2000
MathSciNet review: 1661282
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study finite group extensions represented by special cohomology classes. As an application, we obtain some restrictions on finite groups which can act freely on a product of spheres or on a product of real projective spaces. In particular, we prove that if acts freely on , then $r \leq k$ .

1. A. Adem and D.J. Benson, Abelian groups acting on products of spheres, Math. Z. 228 (1998), 705-712. CMP 99:01
2. Alejandro Adem and William Browder, The free rank of symmetry of (𝑆ⁿ)^{𝑘}, Invent. Math. 92 (1988), no. 2, 431–440. MR 936091, https://doi.org/10.1007/BF01404462
3. Christopher Allday, Elementary abelian 𝑝-group actions on lens spaces, Topology Hawaii (Honolulu, HI, 1990) World Sci. Publ., River Edge, NJ, 1992, pp. 1–11. MR 1181477
4. David J. Benson and Jon F. Carlson, Complexity and multiple complexes, Math. Z. 195 (1987), no. 2, 221–238. MR 892053, https://doi.org/10.1007/BF01166459
5. William Browder, Cohomology and group actions, Invent. Math. 71 (1983), no. 3, 599–607. MR 695909, https://doi.org/10.1007/BF02095996
6. Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original. MR 1324339
7. Gunnar Carlsson, On the nonexistence of free actions of elementary abelian groups on products of spheres, Amer. J. Math. 102 (1980), no. 6, 1147–1157. MR 595008, https://doi.org/10.2307/2374182
8. Gunnar Carlsson, On the rank of abelian groups acting freely on (𝑆ⁿ)^{𝑘}, Invent. Math. 69 (1982), no. 3, 393–400. MR 679764, https://doi.org/10.1007/BF01389361
9. Larry W. Cusick, Elementary abelian 2-groups that act freely on products of real projective spaces, Proc. Amer. Math. Soc. 87 (1983), no. 4, 728–730. MR 687651, https://doi.org/10.1090/S0002-9939-1983-0687651-4
10. Leonard Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224–239. MR 137742, https://doi.org/10.1090/S0002-9947-1961-0137742-1
11. Daniel H. Gottlieb, Kyung B. Lee, and Murad Özaydin, Compact group actions and maps into 𝐾(𝜋,1)-spaces, Trans. Amer. Math. Soc. 287 (1985), no. 1, 419–429. MR 766228, https://doi.org/10.1090/S0002-9947-1985-0766228-2
12. Marvin J. Greenberg, Lectures on forms in many variables, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0241358
13. Howard Hiller, Zbigniew Marciniak, Chih-Han Sah, and Andrzej Szczepański, Holonomy of flat manifolds with 𝑏₁=0. II, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 213–220. MR 891616, https://doi.org/10.1093/qmath/38.2.213
14. Joseph J. Rotman, An introduction to the theory of groups, 4th ed., Graduate Texts in Mathematics, vol. 148, Springer-Verlag, New York, 1995. MR 1307623

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 57S25, 20J06, 20C15

Retrieve articles in all journals with MSC (1991): 57S25, 20J06, 20C15

Additional Information

Ergün Yalçin
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: eyalcin@math.indiana.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02485-5
Keywords: Group extensions, special classes, products of spheres, cohomology of groups
Received by editor(s): January 30, 1998
Published electronically: February 24, 2000
Additional Notes: Partially supported by NATO grants of the Scientific and Technical Research Council of Turkey
Article copyright: © Copyright 2000 American Mathematical Society