Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-29T18:57:11.871Z Has data issue: false hasContentIssue false
  • English
  • Français

Rigid cyclic group actions on cohomology complex projective spaces

Published online by Cambridge University Press:  24 October 2008

Karl Heinz Dovermann
Karl Heinz Dovermann
Affiliation:
Department of Mathematics, University of Hawaii at Manoa
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we discuss smooth and locally smooth cyclic actions on cohomology complex projective spaces. We shall show that in low dimensions the presence of a codimension-two fixed-point component forces many algebraic invariants to behave as they do in the linear examples.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 3 , May 1987 , pp. 487 - 507
Copyright
Copyright © Cambridge Philosophical Society 1987

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

[1]Borel, A.. Seminar on Transformation Groups. Ann. of Math. Studies 46 (Princeton University Press, 1960).Google Scholar
[2]Bredon, G.. The cohomology ring structure of a fixed point set. Ann. of Math. 80 (1964), 524–537.CrossRefGoogle Scholar
[3]Bredon, G.. Introduction to Compact Transformation Groups (Academic Press, 1972).Google Scholar
[4]Bredon, G.. Examples of differentiable group actions. Topology 3 (1965), 115–122.CrossRefGoogle Scholar
[5]Cappell, S. and Shaneson, J.. Piecewise linear embeddings and their singularities. Ann. of Math. 103 (1976), 163–228.CrossRefGoogle Scholar
[6]Cappell, S. and Shaneson, J.. Branched cyclic coverings. Ann. of Math. Studies, 84 (1975), 165–173.Google Scholar
[7]Dejter, I.. Smooth S1 manifolds in the homotopy type of ℂP 3. Michigan Math. J. (1976), 89–95.Google Scholar
[8]Dovermann, K. H.. Symmetries of complex projective spaces. RIMS Kokyuroku 567, Kyoto (1985), 19–28.Google Scholar
[9]Dovermann, K. H. and Masuda, M.. Exotic cyclic actions on homotopy complex projective spaces. Preprint 1986.CrossRefGoogle Scholar
[10]Dovermann, K. H., Masuda, M. and Suh, D. Y.. Rigid versus non-rigid ℤm actions on complex projective spaces. Preprint 1986.Google Scholar
[11]Hirzebruch, F. and Zagier, D.. The Atiyah–Singer Theorem and Elementary Number Theory (Publish or Perish, 1974).Google Scholar
[12]Hsiang, W. Y.. Cohomology Theory of Transformation Groups. Ergeb. der Math. Bd 85 (Springer, 1975).CrossRefGoogle Scholar
[13]Hsiang, W. C. and Szczarba, R. H.. On embedding surfaces in four-manifolds. Proc. of Symp. in Pure Math., vol. XXII (Amer. Math. Soc, 1971), 97–103.CrossRefGoogle Scholar
[14]James, D.. Smooth S 1 actions on homotopy ℂP 4's. Michigan Math. J. 32 (1985), 259–266.CrossRefGoogle Scholar
[15]Katz, G.. Witt analogs of the Burnside ring and integrality theorems II, preprint.Google Scholar
[16]Kirby, R. C. and Siebenmann, L. C.. Normal bundles for codimension 2 locally flat imbeddings. Proc. Topology Conference, Park City, Utah, Feb. 1974, Lecture Notes in Math., vol. 438 (Springer, 1975).Google Scholar
[17]Kirby, R. C. and Siebenmann, L. C.. Foundational Essays on Topological Manifolds, Smoothings and Triangulations. Ann. of Math. Studies 88 (Princeton University Press, 1977).CrossRefGoogle Scholar
[18]Kneser, H.. Die Deformationssätze der einfach zusammenhängenden Flächen. Math. Z. 25 (1926), 362–372.CrossRefGoogle Scholar
[19]Madsen, I. and Milgram, J.. Classifying Spaces for Surgery and Cobordism of Manifolds. Ann. of Math. Studies 92 (Princeton University Press, 1979).Google Scholar
[20]Masuda, M.. Smooth involutions on homotopy ℂP 3. Amer. J. Math. 106 (1984), 1487–1501.CrossRefGoogle Scholar
[21]Masuda, M.. Smooth group actions on cohomology complex projective spaces with a fixed point component of codimension 2, preprint (1986).Google Scholar
[22]Masuda, M. and Tsai, Y.. Tangential representations of cyclic group actions on homotopy complex projective spaces. Osaka J. Math. 22 (1985), 907–919.Google Scholar
[23]Petrie, T.. Smooth S 1 actions on homotopy complex projective spaces and related topics. Bull. Amer. Math. Soc. 78 (1972), 105–153.CrossRefGoogle Scholar
[24]Petrie, T.. Exotic S 1 actions on ℂP 3 and related topics. Invent. Math. 17 (1972), 317–327.CrossRefGoogle Scholar
[25]Petrie, T.. Torus actions on homotopy complex projective spaces. Invent. Math. 20 (1973), 139–146.CrossRefGoogle Scholar
[26]Sullivan, D.. Triangulating homotopy equivalences, thesis, Princeton University, 1965.Google Scholar
[27]Sullivan, D.. Triangulating and smoothing homotopy equivalences and homeomorphisms, geometric topology seminar notes, Princeton University, 1967.Google Scholar
[28]Wall, C. T.C.. Surgery on Compact Manifolds (Academic Press, 1970).Google Scholar
[29]Weinberger, S.. Constructions of group actions: a survey of some recent developments. Proc. of the Conference on Transformation Groups in Boulder, Colorado, 1983, Contemporary Math. vol. 36 (Amer. Math. Soc, 1985), 269–298.CrossRefGoogle Scholar