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FREE ACTION OF FINITE GROUPS ON SPACES OF COHOMOLOGY TYPE (0, b)

Part of: Classical Topics in Algebraic Topology Spectral sequences

Published online by Cambridge University Press:  28 January 2018

K. SOMORJIT SINGH ,
HEMANT KUMAR SINGH  and
TEJ BAHADUR SINGH
K. SOMORJIT SINGH
Affiliation:
Department of Mathematics, University of Delhi, Delhi110007, India e-mail: ksomorjitmaths@gmail.com, hemantksingh@maths.du.ac.in, tej_b_singh@yahoo.co.in
HEMANT KUMAR SINGH
Affiliation:
Department of Mathematics, University of Delhi, Delhi110007, India e-mail: ksomorjitmaths@gmail.com, hemantksingh@maths.du.ac.in, tej_b_singh@yahoo.co.in
TEJ BAHADUR SINGH
Affiliation:
Department of Mathematics, University of Delhi, Delhi110007, India e-mail: ksomorjitmaths@gmail.com, hemantksingh@maths.du.ac.in, tej_b_singh@yahoo.co.in
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Abstract

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Let G be a finite group acting freely on a finitistic space X having cohomology type (0, b) (for example, $\mathbb S$n × $\mathbb S$2n is a space of type (0, 1) and the one-point union $\mathbb S$n ∨ $\mathbb S$2n ∨ $\mathbb S$3n is a space of type (0, 0)). It is known that a finite group G that contains ℤp ⊕ ℤp ⊕ ℤp, p a prime, cannot act freely on $\mathbb S$n × $\mathbb S$2n. In this paper, we show that if a finite group G acts freely on a space of type (0, 1), where n is odd, then G cannot contain ℤp ⊕ ℤp, p an odd prime. For spaces of cohomology type (0, 0), we show that every p-subgroup of G is either cyclic or a generalized quaternion group. Moreover, for n even, it is shown that ℤ2 is the only group that can act freely on X.

MSC classification

Primary: 57S99: None of the above, but in this section
Secondary: 55T10: Serre spectral sequences 55M20: Fixed points and coincidences
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 3 , September 2018 , pp. 673 - 680
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

1. Adem, A., Davis, J. F. and Unlu, O., Fixity and free group actions on products of spheres, Comment. Math. Helv. 79 (2004), 758–778.Google Scholar
2. Borel, A., Seminar on transformation groups, Annals of mathametics studies, vol. 46 (Princeton University Press, Princeton, NJ, 1960).Google Scholar
3. Heller, A., A note on spaces with operators, Ill. J. Math. 3 (1959), 98–100.Google Scholar
4. Volovikov, A. Yu., On the index of G-spaces, Sb. Math. 191 (2000), 1259–1277.CrossRefGoogle Scholar
5. Mattos, D. D., Pergher, P. L. Q. and Santos, E. L. D., Borsuk-Ulam theorems and their parametrized versions for spaces of type (a,b), Algebraic Geom. Topol. 13 (2013), 2827–2843.Google Scholar
6. Bredon, G. E., Introduction to compact transformation groups (Academic Press, New York, 1972).Google Scholar
7. Toda, H., Note on cohomology ring of certain spaces, Proc. Amer. Math. Soc. 14 (1963), 89–95.CrossRefGoogle Scholar
8. James, I. M., Note on cup products, Proc. Amer. Math. Soc. 8 (1957), 374–383.Google Scholar
9. 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
10. Davis, J. F. and Kirk, P., Lecture notes in algebraic topology, Graduate studies in mathematics, vol. 35 (American Mathematical Society, USA, 2001).Google Scholar
11. McCleary, J., A user's guide to spectral sequences, 2nd edition (Cambridge University Press, New York, 2001).Google Scholar
12. Milnor, J., Groups which act on $\mathbb{S}$n without fixed point, Amer. J. Math. 79 (1957), 623–630.CrossRefGoogle Scholar
13. Rotman, J. J., An introduction to the theory of groups, 4th edition (Springer, New York, 1995).Google Scholar
14. Smith, P. A., Permutable periodic transformations, Proc. Natl. Acad. Sci. USA 30 (1944), 105–108.Google Scholar
15. Conner, P. E., On the action of a finite group on $\mathbb{S}$n × Sn, Ann. Math. Soc. 66 (1957), 586–588.Google Scholar
16. Pergher, P. L. Q., Singh, H. K. and Singh, T. B., On ℤ2 and $\mathbb{S}$1 free actions on spaces of cohomology type (a,b), Houst. J. Math. 36 (2010), 137–146.Google Scholar
17. Dotzel, R. M., Singh, T. B. and Tripathi, S. P., The cohomology rings of the orbit spaces of free transformation groups of the product of two spheres, Proc. Amer. Math. Soc. 129 (2000), 921–930.Google Scholar
18. Dotzel, R. M. and Singh, T. B., ℤp actions on spaces of cohomology type (a,0), Pro. Amer. Math. Sec. 113 (1991), 875–878.Google Scholar
19. Dotzel, R. M. and Singh, T. B., The cohomology rings of the orbit spaces of free ℤp-actions, Proc. Amer. Math. Soc. 123 (1995), 3581–3585.Google Scholar