Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T23:35:03.968Z Has data issue: false hasContentIssue false
  • English
  • Français

FREE ACTIONS OF SOME COMPACT GROUPS ON MILNOR MANIFOLDS

Part of: Classical Topics in Algebraic Topology Fiber spaces and bundles

Published online by Cambridge University Press:  31 October 2018

PINKA DEY  and
MAHENDER SINGH
PINKA DEY
Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, SAS Nagar, Manauli (PO), Punjab 140306, India e-mails: pinkadey11@gmail.com, mahender@iisermohali.ac.in
MAHENDER SINGH
Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, SAS Nagar, Manauli (PO), Punjab 140306, India e-mails: pinkadey11@gmail.com, mahender@iisermohali.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we investigate free actions of some compact groups on cohomology real and complex Milnor manifolds. More precisely, we compute the mod 2 cohomology algebra of the orbit space of an arbitrary free ℤ2 and $\mathbb{S}^1$-action on a compact Hausdorff space with mod 2 cohomology algebra of a real or a complex Milnor manifold. As applications, we deduce some Borsuk–Ulam type results for equivariant maps between spheres and these spaces. For the complex case, we obtain a lower bound on the Schwarz genus, which further establishes the existence of coincidence points for maps to the Euclidean plane.

MSC classification

Primary: 57S25: Groups acting on specific manifolds 57S10: Compact groups of homeomorphisms
Secondary: 55R20: Spectral sequences and homology of fiber spaces 55M20: Fixed points and coincidences
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 3 , September 2019 , pp. 727 - 742
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Allday, C. and Puppe, V., Cohomological methods in transformation groups, Cambridge Studies in Advanced Mathematics, vol. 32 (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
Bartsch, T., Topological methods for variational problems with symmetries, Lecture Notes in Mathematics, vol. 1560 (Springer-Verlag, Berlin, 1993).CrossRefGoogle Scholar
Borel, A., Seminar on transformation groups, With contributions by Bredon, G., Floyd, E. E., Montgomery, D., Palais, R.. Annals of Mathematics Studies, volume 46 (Princeton University Press, Princeton, N.J., 1960).Google Scholar
Bredon, G. E., Introduction to compact transformation groups, Pure and Applied Mathematics, vol. 46 (Academic Press, New York-London, 1972).Google Scholar
Bukhshtaber, V. M. and Raĭ, N., Toric manifolds and complex cobordisms, Uspekhi Mat. Nauk 53 (2(320)) (1998), 139–140.Google Scholar
Coelho, F. R. C., D. de Mattos and E. L. dos Santos, On the existence of G-equivariant maps, Bull. Braz. Math. Soc. (N.S.) 43 (3) (2012), 407–421.CrossRefGoogle Scholar
Conner, P. E. and Floyd, E. E., Fixed point free involutions and equivariant maps, Bull. Amer. Math. Soc. 66 (1960), 416–441.CrossRefGoogle Scholar
Conner, P. E. and Floyd, E. E., Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 33 (Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964).Google Scholar
de Mattos, D., Pergher, P. L. Q. and dos Santos, E. L., Borsuk–Ulam theorems and their parametrized versions for spaces of type (a, b), Algebr. Geom. Topol. 13 (5) (2013), 2827–2843.CrossRefGoogle Scholar
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 (3) (2001), 921–930.CrossRefGoogle Scholar
Gálvez, I. and Tonks, A., Differential operators and the Witten genus for projective spaces and Milnor manifolds, Math. Proc. Cambridge Philos. Soc. 135 (1) (2003), 123–131.Google Scholar
Hatcher, A., Algebraic topology (Cambridge University Press, Cambridge, 2002).Google Scholar
Hirsch, M. W. and Milnor, J., Some curious involutions of spheres, Bull. Amer. Math. Soc. 70 (1964), 372–377.CrossRefGoogle Scholar
Jahren, B. and Kwasik, S., Free involutions on S 1 × Sn, Math. Ann. 351 (2) (2011), 281–303.CrossRefGoogle Scholar
Kamata, M. and Ono, K., On the multiple points of the self-transverse immersions of the real projective space and the Milnor manifold, Kyushu J. Math. 60 (2) (2006), 331–344.CrossRefGoogle Scholar
McCleary, J., A user's guide to spectral sequences, Cambridge Studies in Advanced Mathematics, 2nd edn. vol. 58 (Cambridge University Press, Cambridge, 2001).Google Scholar
Milnor, J., On the Stiefel–Whitney numbers of complex manifolds and of spin manifolds, Topology 3 (1965), 223–230.CrossRefGoogle Scholar
Mukerjee, H. K., Classification of homotopy real Milnor manifolds, Topology Appl. 139 (1–3) (2004), 151–184.CrossRefGoogle Scholar
Myers, R., Free involutions on lens spaces, Topology 20 (3) (1981), 313–318.CrossRefGoogle Scholar
Oliver, R., A proof of the Conner conjecture, Ann. Math. (2) 103 (3) (1976), 637–644.CrossRefGoogle Scholar
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), Houston J. Math. 36 (1) (2010), 137–146.Google Scholar
Pfister, A. and Stolz, S., On the level of projective spaces, Comment. Math. Helv. 62 (2) (1987), 286–291.CrossRefGoogle Scholar
Rice, P. M., Free actions of Z 4 on S 3, Duke Math. J. 36 (1969), 749–751.CrossRefGoogle Scholar
Ritter, G. X., Free Z 8 actions on S 3, Trans. Amer. Math. Soc. 181 (1973), 195–212.Google Scholar
Ritter, G. X., Free actions of cyclic groups of order 2n on S 1 × S 2, Proc. Amer. Math. Soc. 46 (1974), 137–140.Google Scholar
Rubinstein, J. H., Free actions of some finite groups on S 3, I. Math. Ann. 240 (2) (1979), 165–175.CrossRefGoogle Scholar
Singh, M., Orbit spaces of free involutions on the product of two projective spaces, Results Math. 57 (1–2) (2010), 53–67.CrossRefGoogle Scholar
Singh, M., Cohomology algebra of orbit spaces of free involutions on lens spaces, J. Math. Soc. Japan 65 (4) (2013), 1055–1078.CrossRefGoogle Scholar
Singh, M., Free 2-rank of symmetry of products of Milnor manifolds, Homology Homotopy Appl. 16 (1) (2014), 65–81.CrossRefGoogle Scholar
Tao, Y., On fixed point free involutions of S 1 × S 2, Osaka Math. J. 14 (1962), 145–152.Google Scholar
Tollefson, J. L., Involutions on S 1 × S 2 and other 3-manifolds, Trans. Amer. Math. Soc. 183 (1973), 139–152.Google Scholar
Schwarz, A. S., The genus of a fibre space, Trudy Moskov Mat. Obšč. 11 (1962), 99–126. Translation in Amer. Math. Soc. Trans., 55, 1966, 49–140.Google Scholar
Yu, A.. Volovikov, On the index of G-spaces, Mat. Sb. 191 (9) (2000), 3–22.Google Scholar
Yang, C.-T., On theorems of Borsuk–Ulam, Kakutani-Yamabe-Yujobô and Dyson. II. Ann. Math. (2) 62 (1955), 271–283.CrossRefGoogle Scholar