Some examples of pseudofree S1-actions on homotopy (4m+1)-spheres

Shin-ichiro KAKUTANI

doi:10.2969/jmsj/03410061

Some examples of pseudofree S¹-actions on homotopy (4m+1)-spheres

Shin-ichiro KAKUTANI

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1982 Volume 34 Issue 1 Pages 61-81

DOI https://doi.org/10.2969/jmsj/03410061

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Published: 1982 Received: May 22, 1980 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) M.F. Atiyah, K-theory, Benjamin, 1967.
2) M.F. Atiyah, Elliptic operators and compact groups, Lecture Notes in Math., 401, Springer-Verlag, 1974.
3) M.F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes, II, Applications, Ann. of Math., 88 (1968), 451-491.
4) M.F. Atiyah and I.M. Singer, The index of elliptic operators, III, Ann. of Math., 87 (1968), 546-604.
5) G.E. Bredon, Introduction to compact transformation groups, Academic Press, 1972.
6) W. Browder, Surgery on simply-connected manifolds, Ergebn. Math., 65, Springer- Verlag, 1972.
7) W.C. Hsiang, A note on free differentiable actions of S¹ and S³ on homotopy spheres, Ann, of Math., 83 (1966), 266-272.
8) W. Iberkleid, Pseudo-linear spheres, Michigan Math. J., 25 (1978), 359-370.
9) S. Kakutani, Equivariant KO-rings and J-groups of spheres which have linear pseudofree S¹-actions, Osaka J. Math., 18 (1981), 533-554.
10) T. Kawasaki, Cohomology of twisted projective spaces and lens complexes, Math. Ann., 206 (1973), 243-248.
11) C.N. Lee and A.G. Wasserman, On the groups JO(G), Mem. Amer. Math. Soc., 159 (1975).
12) M.N. Mahammed, A propos de la K-théorie des espaces lenticulaires, C. R. Acad. Sci. Paris, 271 (1970), 639-642.
13) A. Meyerhoff and T. Petrie, Quasi equivalence of G modules, Topology, 15 (1976). 69-75.
14) J. Milnor, Differential topology, Lecture notes, Princeton Univ., 1958.
15) D. Montgomery and C.T. Yang, Differentiable pseudo-free circle actions, Proc. Nat. Acad. Sci. U.S.A., 68 (1971), 894-896.
16) D. Montgomery and C.T. Yang, Differentiable pseudo-free circle actions on homotopy seven spheres, Proc. of the Second Conf, on Compact Transformation Groups, Amherst, Lecture Notes in Math., 298, Springer-Verlag (1971), 41-101.
17) H. Ozeki and F. Uchida, Principal circle actions on a product of spheres, Osaka. J. Math., 9 (1972), 379-390.
18) T. Petrie, Equivariant quasi-equivalence, transversality and normal cobordism, Proc. Internat. Congr. Math. (Vancouver 1974), 537-541.
19) T. Petrie, Pseudoequivalences of G-manifolds, Proc. Sympos. Pure Math., 32 (1978), 169-210.
20) B.J. Sanderson, Immersions and embeddings of projective spaces, Proc. London Math. Soc., 14 (1964), 137-153.
21) G. Segal, Equivariant K-theory, Publ. Math. Inst. des Hautes Études Scient., 34 (1968), 129-151.
22) A.G. Wasserman, Equivariant differential topology, Topology, 8 (1969), 127-150.

Right : [1] M. F. Atiyah, K-theory, Benjamin, 1967.
[2] M. F. Atiyah, Elliptic operators and compact groups, Lecture Notes in Math., 401, Springer-Verlag, 1974.
[3] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes, II, Applications, Ann. of Math., 88 (1968), 451-491.
[4] M. F. Atiyah and I. M. Singer, The index of elliptic operators, III, Ann. of Math., 87 (1968), 546-604.
[5] G. E. Bredon, Introduction to compact transformation groups, Academic Press, 1972.
[6] W. Browder, Surgery on simply-connected manifolds, Ergebn. Math., 65, Springer-Verlag, 1972.
[7] W. C. Hsiang, A note on free differentiable actions of S¹ and S³ on homotopy spheres, Ann, of Math., 83 (1966), 266-272.
[8] W. Iberkleid, Pseudo-linear spheres, Michigan Math. J., 25 (1978), 359-370.
[9] S. Kakutani, Equivariant KO-rings and J-groups of spheres which have linear pseudofree S¹-actions, Osaka J. Math., 18 (1981), 533-554.
[10] T. Kawasaki, Cohomology of twisted projective spaces and lens complexes, Math. Ann., 206 (1973), 243-248.
[11] C. N. Lee and A. G. Wasserman, On the groups JO(G), Mem. Amer. Math. Soc., 159 (1975).
[12] M. N. Mahammed, A propos de la K-théorie des espaces lenticulaires, C. R. Acad. Sci. Paris, 271 (1970), 639-642.
[13] A. Meyerhoff and T. Petrie, Quasi equivalence of G modules, Topology, 15 (1976). 69-75.
[14] J. Milnor, Differential topology, Lecture notes, Princeton Univ., 1958.
[15] D. Montgomery and C. T. Yang, Differentiable pseudo-free circle actions, Proc. Nat. Acad. Sci. U. S. A., 68 (1971), 894-896.
[16] D. Montgomery and C. T. Yang, Differentiable pseudo-free circle actions on homotopy seven spheres, Proc. of the Second Conf. on Compact Transformation Groups, Amherst, Lecture Notes in Math., 298, Springer-Verlag (1971), 41-101.
[17] H. Ozeki and F. Uchida, Principal circle actions on a product of spheres, Osaka. J. Math., 9 (1972), 379-390.
[18] T. Petrie, Equivariant quasi-equivalence, transversality and normal cobordism, Proc. Internat. Congr. Math. (Vancouver 1974), 537-541.
[19] T. Petrie, Pseudoequivalences of G-manifolds, Proc. Sympos. Pure Math., 32 (1978), 169-210.
[20] B. J. Sanderson, Immersions and embeddings of projective spaces, Proc. London Math. Soc., 14 (1964), 137-153.
[21] G. Segal, Equivariant K-theory, Publ. Math. Inst. des Hautes Études Scient., 34 (1968), 129-151.
[22] A. G. Wasserman, Equivariant differential topology, Topology, 8 (1969), 127-150.

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