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Free action of finite groups on spaces of cohomology type \((0,b)\). (English) Zbl 1423.57058

Consider a path connected topological space \(X\) such that the cohomology groups \(H^k(X; \mathbb Z)\cong \mathbb Z\) for \(k=0, n,2n,3n\) and vanish for all other values of \(k\). Choosing generators \(x,y,z\) in dimensions \(n,2n,3n\) by \(x,y,z\), respectively, one has \(x^2=ay, xy=bz\) for some integers \(a,b\). One says that \(X\) is of type \((a,b)\) (characterized by \(n\)). For example, \(\mathbb S^n\vee \mathbb S^{2n}\vee \mathbb S^{3n}\) is of type \((0,0)\) whereas \(\mathbb S^n\times \mathbb S^{2n}\) has type \((0,1)\).
The authors obtain restrictions on the possible existence of a free action of a finite group \(G\) on a space \(X\) of type \((a,b)\) characterised by \(n>1\). The authors further assume that \(X\) is a finitistic space. Recall that a space is finitistic if every open cover admits a refinement which has finite covering dimension.
The authors show, in Theorem 3.1, that when \(p\) is an odd prime and \(a\equiv 0\) mod \(p\), then \(\mathbb Z_p\times \mathbb Z_p\) cannot act on freely on \(X\). This improves a special case, namely when \(m=2n\), of a result of A. Heller [Ill. J. Math. 3, 98–100 (1959; Zbl 0084.38803)] who showed that \(\mathbb Z_p\times \mathbb Z_p\times \mathbb Z_p\) cannot act freely on \(\mathbb S^n\times \mathbb S^{m}\).
The authors also show, in Theorem 3.2, that if both \(a,b\) are even then \(\mathbb Z_2\times \mathbb Z_2\) cannot act freely on \(X\).
The proofs involve a careful analysis of the mod \(p\)-cohomology spectral sequence associated to the Borel fibration \(X\hookrightarrow X\times_G EG\to BG\) which converges to the mod \(p\) cohomology of \(X/G\); here \(EG\to BG\) is the projection of the classifying space of \(G\). As applications the authors obtain restrictions on finite groups that can possibly act freely on a space \(X\). For example, it is shown (in Theorem 3.8) that when \(n\) is even and \(a\equiv 0\equiv b~\text{mod~} p\) for a prime \(p\), the only finite group that can possibly act freely on \(X\) is \(\mathbb Z_2\).

MSC:

57S17 Finite transformation groups
55T10 Serre spectral sequences
55M20 Fixed points and coincidences in algebraic topology

Citations:

Zbl 0084.38803
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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. doi:10.1007/s00014-004-0810-4 · Zbl 1057.57019
[2] Borel, A., Seminar on transformation groups, Annals of mathametics studies, vol. 46 (Princeton University Press, Princeton, NJ, 1960). · Zbl 0091.37202
[3] Heller, A., A note on spaces with operators, Ill. J. Math., 3, 98-100, (1959) · Zbl 0084.38803
[4] Volovikov, A. Yu., On the index of G-spaces, Sb. Math., 191, 1259-1277, (2000) · Zbl 0987.57016 · doi:10.1070/SM2000v191n09ABEH000504
[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. doi:10.2140/agt.2013.13.2827 · Zbl 1275.55002
[6] Bredon, G. E., Introduction to compact transformation groups, (1972), Academic Press: Academic Press, New York · Zbl 0246.57017
[7] Toda, H., Note on cohomology ring of certain spaces, Proc. Amer. Math. Soc., 14, 89-95, (1963) · Zbl 0114.39701 · doi:10.1090/S0002-9939-1963-0150763-5
[8] James, I. M., Note on cup products, Proc. Amer. Math. Soc., 8, 374-383, (1957) · Zbl 0077.36501 · doi:10.1090/S0002-9939-1957-0091467-4
[9] Madsen, I., Thomas, C. B. and Wall, C. T. C., The topological spherical space form problem II existence of free actions, Topology15 (1976), 375-382. doi:10.1016/0040-9383(76)90031-8 · Zbl 0348.57019
[10] Davis, J. F. and Kirk, P., Lecture notes in algebraic topology, Graduate studies in mathematics, vol. 35 (American Mathematical Society, USA, 2001). doi:10.1090/gsm/035 · Zbl 1018.55001
[11] Mccleary, J., A user’s guide to spectral sequences, (2001), Cambridge University Press: Cambridge University Press, New York · Zbl 0959.55001
[12] Milnor, J., Groups which act on \(\mathbb{S}\)n without fixed point, Amer. J. Math., 79, 623-630, (1957) · Zbl 0078.16304 · doi:10.2307/2372566
[13] Rotman, J. J., An introduction to the theory of groups, (1995), Springer: Springer, New York · Zbl 0810.20001 · doi:10.1007/978-1-4612-4176-8
[14] Smith, P. A., Permutable periodic transformations, Proc. Natl. Acad. Sci. USA, 30, 105-108, (1944) · Zbl 0063.07092 · doi:10.1073/pnas.30.5.105
[15] Conner, P. E., On the action of a finite group on \(\mathbb{S}\)n × Sn, Ann. Math. Soc., 66, 586-588, (1957) · Zbl 0079.38904 · doi:10.2307/1969910
[16] Pergher, P. L. Q., Singh, H. K. and Singh, T. B., On ℤ_{2} and \(\mathbb{S}\)1 free actions on spaces of cohomology type (a,b), Houst. J. Math.36 (2010), 137-146. · Zbl 1226.57049
[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. doi:10.1090/S0002-9939-00-05668-9 · Zbl 0962.57020
[18] Dotzel, R. M. and Singh, T. B., ℤ_{p} actions on spaces of cohomology type (a,0), Pro. Amer. Math. Sec.113 (1991), 875-878. · Zbl 0739.57024
[19] Dotzel, R. M. and Singh, T. B., The cohomology rings of the orbit spaces of free ℤ_{p}-actions, Proc. Amer. Math. Soc.123 (1995), 3581-3585. · Zbl 0849.57031
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