Journal of the Mathematical Society of Japan

Cohomology algebra of orbit spaces of free involutions on lens spaces

Mahender SINGH

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Let $G$ be a group acting continuously on a space $X$ and let $X/G$ be its orbit space. Determining the topological or cohomological type of the orbit space $X/G$ is a classical problem in the theory of transformation groups. In this paper, we consider this problem for cohomology lens spaces. Let $X$ be a finitistic space having the mod 2 cohomology algebra of the lens space $L_p^{2m-1}$ $(q_1,\dots,q_m)$. Then we classify completely the possible mod 2 cohomology algebra of orbit spaces of arbitrary free involutions on $X$. We also give examples of spaces realizing the possible cohomology algebras. In the end, we give an application of our results to non-existence of $\mathbb{Z}_2$-equivariant maps $\mathbb{S}^n \to X$.

Article information

J. Math. Soc. Japan, Volume 65, Number 4 (2013), 1055-1078.

First available in Project Euclid: 24 October 2013

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Zentralblatt MATH identifier

Primary: 57S17: Finite transformation groups
Secondary: 55R20: Spectral sequences and homology of fiber spaces [See also 55Txx] 55M20: Fixed points and coincidences [See also 54H25]

cohomology algebra finitistic space index of involution Leray spectral sequence orbit space Smith-Gysin sequence


SINGH, Mahender. Cohomology algebra of orbit spaces of free involutions on lens spaces. J. Math. Soc. Japan 65 (2013), no. 4, 1055--1078. doi:10.2969/jmsj/06541055.

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  • R. Ashraf, Singular cohomology rings of some orbit spaces defined by free involution on $\mathbb{CP}(2m + 1)$, J. Algebra, 324 (2010), 1212–1218.
  • K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre, Fund. Math., 20 (1933), 177–190.
  • D. G. Bourgin, On some separation and mapping theorems, Comment. Math. Helv., 29 (1955), 199–214.
  • G. E. Bredon, Introduction to Compact Transformation Groups, Pure Appl. Math. (Amst.), 46, Academic Press, New York, 1972.
  • G. E. Bredon, Sheaf Theory, 2nd ed., Grad. Texts in Math., 170, Springer-Verlag, New York, 1997.
  • A. Borel, Seminar on Transformation Groups, Ann. of Math. Stud., 46, Princeton University Press, Princeton, NJ, 1960.
  • P. E. Conner and E. E. Floyd, Fixed point free involutions and equivariant maps, Bull. Amer. Math. Soc., 66 (1960), 416–441.
  • A. Dold, Erzeugende der Thomschen algebra $\mathfrak{N}_*$, Math. Z., 65 (1956), 25–35.
  • A. Dold, Parametrized Borsuk-Ulam theorems, Comment. Math. Helv., 63 (1988), 275–285.
  • 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.
  • A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
  • P. K. Kim, Periodic homeomorphisms of the 3-sphere and related spaces, Michigan Math. J., 21 (1974), 1–6.
  • B. S. Koikara and H. K. Mukerjee, A Borsuk-Ulam type theorem for a product of spheres, Topology Appl., 63 (1995), 39–52.
  • G. R. Livesay, Fixed point free involutions on the 3-sphere, Ann. of Math. (2), 72 (1960), 603–611.
  • 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.
  • J. McCleary, A User's Guide to Spectral Sequences, 2nd ed., Cambridge Stud. Adv. Math., 58, Cambridge University Press, Cambridge, 2001.
  • J. Milnor, Groups which act on $S^n$ without fixed points, Amer. J. Math., 79 (1957), 623–630.
  • R. Myers, Free involutions on lens spaces, Topology, 20 (1981), 313–318.
  • 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.
  • P. M. Rice, Free actions of $Z_4$ on $S^3$, Duke Math. J., 36 (1969), 749–751.
  • G. X. Ritter, Free $Z_8$ actions on $S^3$, Trans. Amer. Math. Soc., 181 (1973), 195–212.
  • 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.
  • J. H. Rubinstein, Free actions of some finite groups on $S^3$. I, Math. Ann., 240 (1979), 165–175.
  • R. G. Swan, A new method in fixed point theory, Comment. Math. Helv., 34 (1960), 1–16.
  • 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.
  • M. Singh, Orbit spaces of free involutions on the product of two projective spaces, Results Math., 57 (2010), 53–67.
  • M. Singh, Parametrized Borsuk-Ulam problem for projective space bundles, Fund. Math., 211 (2011), 135–147.
  • Y. Tao, On fixed point free involutions of $S^1 \times S^2$, Osaka Math. J., 14 (1962), 145–152.
  • C.-T. Yang, Continuous functions from spheres to euclidean spaces, Ann. of Math. (2), 62 (1955), 284–292.
  • C.-T. Yang, On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson. I, Ann. of Math. (2), 60 (1954), 262–282.
  • C.-T. Yang, On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson. II, Ann. of Math. (2), 62 (1955), 271–283.