Cohomology algebra of orbit spaces of free involutions on lens spaces.(English)Zbl 1292.57030

Let $$p\geq 2$$ be a positive integer and $$q_1,q_2,\dots, q_m$$ be integers prime to $$p$$. Let $$L^{2m-1}_p(q_1,q_2,\dots, q_m)$$ denote the orbit space of the action of the cyclic group $$G= \mathbb{Z}_2$$ on $$S^{2m-1}\subset \mathbb{C}^m$$ which sends $$(z_1,z_2,\dots, z_m)$$ to $$(e^{2\pi iq_1/p} z_1,\dots, e^{2\pi iq_m/p} z_m)$$.
The main theorem on this paper completely determines the possible cohomology algebras $$H^*(X/G;\mathbb{Z}_2)$$ arising from free $$G$$ actions on a finitistic space $$X$$ with the $$\mathbb{Z}_2$$ cohomology of $$L^{2m-1}_p(q_1,q_2,\dots, q_m)$$. There are five possible truncated-polynomial algebras with 1, 2, 3, or 4 generators. For example, if $$p$$ is odd then $$H^*(X/G;\mathbb{Z}_2)= \mathbb{Z}_2[x]/(x^{2m})$$, $$\deg(x)= 1$$, and if $$p$$ is even, $$p\not\equiv 0\pmod 4$$, then $$H^*(X/G; \mathbb{Z}_2)= \mathbb{Z}[x,y]/(x^A2, y^m)$$, $$\deg(x)= 1$$ and $$\deg(y)= 2$$. If $$p\equiv 0\pmod 4$$, the description of the cohomology algebra depends on differentials in the Leray spectral sequence of the Borel fibration $$X\to X_G\to B_G$$. The paper includes a Borsuk-Ulam type application of the main theorem. If $$m\geq 3$$ and $$X$$ is a finitistic space with the $$\mathbb{Z}_2$$ cohomology of $$L^{2m-1}_p(q_1,q_2,\dots, q_m)$$ and $$X$$ admits a free involution, then there does not exist a $$\mathbb{Z}_2$$ equivariant map $$S^n\to X$$ if $$n\geq 2m$$.

MSC:

 57S17 Finite transformation groups 55R20 Spectral sequences and homology of fiber spaces in algebraic topology 55M20 Fixed points and coincidences in algebraic topology
Full Text:

References:

 [1] R. Ashraf, Singular cohomology rings of some orbit spaces defined by free involution on $$\mathbb{CP}(2m + 1)$$, J. Algebra, 324 (2010), 1212-1218. · Zbl 1225.55001 [2] K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre, Fund. Math., 20 (1933), 177-190. · Zbl 0006.42403 [3] D. G. Bourgin, On some separation and mapping theorems, Comment. Math. Helv., 29 (1955), 199-214. · Zbl 0065.16604 [4] G. E. Bredon, Introduction to Compact Transformation Groups, Pure Appl. Math. (Amst.), 46 , Academic Press, New York, 1972. · Zbl 0246.57017 [5] G. E. Bredon, Sheaf Theory, 2nd ed., Grad. Texts in Math., 170 , Springer-Verlag, New York, 1997. [6] A. Borel, Seminar on Transformation Groups, Ann. of Math. Stud., 46 , Princeton University Press, Princeton, NJ, 1960. · Zbl 0091.37202 [7] P. E. Conner and E. E. Floyd, Fixed point free involutions and equivariant maps, Bull. Amer. Math. Soc., 66 (1960), 416-441. · Zbl 0106.16301 [8] A. Dold, Erzeugende der Thomschen algebra $$\mathfrak{N}_*$$, Math. Z., 65 (1956), 25-35. · Zbl 0071.17601 [9] A. Dold, Parametrized Borsuk-Ulam theorems, Comment. Math. Helv., 63 (1988), 275-285. · Zbl 0651.55002 [10] R. M. Dotzel, T. B. Singh and S. P. Tripathi, The cohomology rings of the orbit spaces of free transformation groups of the product of two spheres, Proc. Amer. Math. Soc., 129 (2001), 921-930. · Zbl 0962.57020 [11] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. · Zbl 1044.55001 [12] P. K. Kim, Periodic homeomorphisms of the 3-sphere and related spaces, Michigan Math. J., 21 (1974), 1-6. · Zbl 0287.55008 [13] B. S. Koikara and H. K. Mukerjee, A Borsuk-Ulam type theorem for a product of spheres, Topology Appl., 63 (1995), 39-52. · Zbl 0827.55009 [14] G. R. Livesay, Fixed point free involutions on the 3-sphere, Ann. of Math. (2), 72 (1960), 603-611. · Zbl 0096.17302 [15] L. Lyusternik and L. Šhnirel’man, Topological methods in variational problems and their application to the differential geometry of surfaces, Uspehi Matem. Nauk. (N.S.), 2 (1947), 166-217. [16] J. McCleary, A User’s Guide to Spectral Sequences, 2nd ed., Cambridge Stud. Adv. Math., 58 , Cambridge University Press, Cambridge, 2001. · Zbl 0959.55001 [17] J. Milnor, Groups which act on $$S^n$$ without fixed points, Amer. J. Math., 79 (1957), 623-630. · Zbl 0078.16304 [18] R. Myers, Free involutions on lens spaces, Topology, 20 (1981), 313-318. · Zbl 0508.57032 [19] M. Nakaoka, Parametrized Borsuk-Ulam theorems and characteristic polynomials, In: Topological Fixed Point Theory and Applications, Tianjin, 1988, Lecture Notes in Math., 1411 , Springer-Verlag, Berlin, 1989, pp.,155-170. · Zbl 0691.55003 [20] P. M. Rice, Free actions of $$Z_4$$ on $$S^3$$, Duke Math. J., 36 (1969), 749-751. · Zbl 0184.27402 [21] G. X. Ritter, Free $$Z_8$$ actions on $$S^3$$, Trans. Amer. Math. Soc., 181 (1973), 195-212. [22] G. X. Ritter, Free actions of cyclic groups of order $$2^n$$ on $$S^1 \times S^2$$, Proc. Amer. Math. Soc., 46 (1974), 137-140. · Zbl 0293.57019 [23] J. H. Rubinstein, Free actions of some finite groups on $$S^3$$. I, Math. Ann., 240 (1979), 165-175. · Zbl 0382.57019 [24] R. G. Swan, A new method in fixed point theory, Comment. Math. Helv., 34 (1960), 1-16. · Zbl 0144.22602 [25] H. K. Singh and T. B. Singh, On the cohomology of orbit space of free $$\mathbb{Z}_p$$-actions on lens spaces, Proc. Indian Acad. Sci. Math. Sci., 117 (2007), 287-292. · Zbl 1126.57015 [26] M. Singh, Orbit spaces of free involutions on the product of two projective spaces, Results Math., 57 (2010), 53-67. · Zbl 1198.57023 [27] M. Singh, Parametrized Borsuk-Ulam problem for projective space bundles, Fund. Math., 211 (2011), 135-147. · Zbl 1220.55002 [28] Y. Tao, On fixed point free involutions of $$S^1 \times S^2$$, Osaka Math. J., 14 (1962), 145-152. · Zbl 0105.17302 [29] C.-T. Yang, Continuous functions from spheres to euclidean spaces, Ann. of Math. (2), 62 (1955), 284-292. · Zbl 0067.15203 [30] C.-T. Yang, On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson. I, Ann. of Math. (2), 60 (1954), 262-282. · Zbl 0057.39104 [31] C.-T. Yang, On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson. II, Ann. of Math. (2), 62 (1955), 271-283. · Zbl 0067.15202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.