-axial and actions on homotopy spheres

Author:
R. D. Ball

Journal:
Trans. Amer. Math. Soc. **292** (1985), 51-79

MSC:
Primary 57S15

DOI:
https://doi.org/10.1090/S0002-9947-1985-0805953-1

MathSciNet review:
805953

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Abstract: Let or and or . For each integer a family of -axial homotopy spheres is constructed. Each has fixed point set of dimension and orbit space of dimension (resp. ) if (resp. ). is . is trivial if and is a homotopy theoretically defined subgroup of sections of an bundle depending only on and if . Assume that and satisfy the mild restriction , (1). It is shown that the above family is universal for -axial homotopy spheres and provides a classification analogous to the classification of fibre bundles: for each -axial homotopy sphere there is a and a unique equivariant stratified map . is equivariantly diffeomorphic to the pullback of via the map of orbit spaces. If then is unique (and ). If then is unique modulo the image of

**[1]**R. D. Ball, -*axial*and*actions on homotopy spheres*, Princeton Univ. Thesis, 1981.**[2]**-,*On the equivariant diffeomorphism classification of regular**actions on homotopy spheres*, (to appear).**[3]**G. E. Bredon,*Introduction to compact transformation groups*, Academic Press, New York, 1972. MR**0413144 (54:1265)****[4]**-,*Biaxial actions*, mimeographed notes.**[5]**W. Browder and F. Quinn,*A surgery theory for*-*manifolds and stratified sets*, Manifolds, University of Tokyo Press, Tokyo, 1973, pp. 27-36. MR**0375348 (51:11543)****[6]**M. W. Davis,*Smooth actions of the classical groups*, Princeton Univ. Ph. D. Thesis, 1974.**[7]**-,*Multiaxial actions on manifolds*, Lecture Notes in Math., vol. 643, Springer-Verlag, Berlin and New York, 1978. MR**488195 (80e:57046)****[8]**-,*Universal*-*manifolds*, Amer. J. Math.**103**(1981), 103-141.**[9]**-,*Smooth*-*manifolds as collections of fibre bundles*, Pacific J. Math.**77**(1978), 315-363. MR**510928 (80b:57034)****[10]**-,*Some group actions on homotopy spheres of dimension seven and fifteen*, Amer. J. Math.**104**(1982), 59-90. MR**648481 (83g:57027)****[11]**M. W. Davis and W. C. Hsiang,*Concordance classes of regular**and**actions on homotopy spheres*, Ann. of Math. (2)**105**(1977), 325-341. MR**0438368 (55:11282)****[12]**M. W. Davis, W. C . Hsiang and W. Y. Hsiang,*Differential actions of compact simple Lie groups on homotopy spheres and euclidean spaces*, Proc. Stanford Topology Conf., Amer. Math. Soc., Providence, R. I.**[13]**M. Davis, W. C. Hsiang and J. W. Morgan,*Concordance classes of regular*-*actions on homotopy spheres*, Acta Math.**144**(1980), 154-221. MR**573451 (83i:57030)****[14]**D. Gromoll and W. Meyer,*An exotic sphere with nonnegative sectional curvature*, Ann. of Math. (2)**100**(1974), 447-490. MR**0375151 (51:11347)****[15]**W. C. Hsiang and W. Y. Hsiang,*Differentiable actions of compact connected classical groups*. II, Ann. of Math. (2)**92**(1970), 189-223. MR**0265511 (42:420)****[16]**R. C. Kirby and L. C. Siebenmann,*Foundational essays on topological manifolds, smoothings and triangulations*, Ann. of Math. Studies, No. 88, Princeton Univ. Press, Princeton, N. J, 1977. MR**0645390 (58:31082)****[17]**J. W. Milnor,*On manifolds homeomorphic to the*-*sphere*, Ann. of Math. (2)**64**(1956), 399-405. MR**0082103 (18:498d)****[18]**G. W. Schwarz,*Smooth functions invariant under the action of a compact Lie group*, Topology**14**(1975), 63-68. MR**0370643 (51:6870)****[19]**-,*Covering smooth homotopies of orbit spaces*, Inst. Hautes Études Sci. Publ. Math.**51**(1980), 38-132.**[20]**N. E. Steenrod,*The topology of fibre bundles*, Ann. of Math. Studies, No. 14, Princeton Univ. Press, Princeton, N. J., 1950.**[21]**H. Weyl,*The classical groups*, Ann. of Math. Studies, No. 1, Princeton Univ. Press, Princeton, N. J., 1939. MR**1488158 (98k:01049)**

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DOI:
https://doi.org/10.1090/S0002-9947-1985-0805953-1

Article copyright:
© Copyright 1985
American Mathematical Society