Actions of Finite Groups on R n+k with Fixed Set R k

Hambleton, Ian ; Madsen, Ib

Canadian journal of mathematics, Tome 38 (1986) no. 4, pp. 781-860

Résumé

In this paper we study the existence problem for topological actions of finite groups on euclidean spaces R n+k which are free outside a fixed point set R k (embedded as a vector subspace). We refer to such an action as a semi-free action on (R n+k , R n ) and note that all our actions will be assumed orientation-preserving.Suppose the finite group π acts semi-freely on (R n+k , R n ), then it acts freely on (R n+k – R n ) = S n–l × R k+1. Since this space is homotopy equivalent to S n–l, π will have periodic integral cohomology and n will be a multiple of the period. In fact the orbit space is a finitely-dominated Poincaré complex of formal dimension n – 1 with π 1 W = π and as considered by Swan [41].

Détail

DOI : 10.4153/CJM-1986-041-x

Bibliographie

[1] 1. Anderson, D. and Pedersen, E. K., Semi-free actions of finite groups on spheres, Preprint, Odense (1982). Google Scholar

[2] 2. Bentzen, S., On the finiteness obstruction, Preprint, Aarhus University (1982). Google Scholar

[3] 3. Bentzen, S., Some numerical results on space forms, Preprint, Aarhus (1982). Google Scholar

[4] 4. Bredon, G. E., Introduction to compact transformation groups (Academic Press, New York, 1972). Google Scholar

[5] 5. Carter, D., Localizations in lower algebraic K-theory, Comm. in Algebra 8 (1980),603–622. Google Scholar

[6] 6. Carter, D., Lower K-theory of finite groups, Preprint. Google Scholar | DOI

[7] 7. Cassel, J. W. S. and Fröhlich, A., Algebraic number theory (Acad. Press, New York, 1967). Google Scholar

[8] 8. Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, N.Y., 1956). Google Scholar

[9] 9. Davis, J., Evaluation of the Swan finiteness obstruction, Thesis, Stanford (1981). Google Scholar

[10] 10. Dress, A., A characterization of solvable groups, Math. Z. 110 (1969),213–217. Google Scholar

[11] 11. Dress, A., Induction and structure theorems for orthogonal representations affinité groups, Ann. of Math. 102 (1975),291–325. Google Scholar

[12] 12. Farrell, F. T. and Hsiang, W. C, Manifolds with π = G × T, Amer. J. Math. 95 (1973),813–848. Google Scholar

[13] 13. Fröhlich, A., Arithmetic and Galois modules over arithmetic orders, J. reine angew. Math. 286/287 (1916), 380–440. Google Scholar

[14] 14. Hambleton, I. and Madsen, I., Semi-free actions on n with one fixed point, Current Trends in Algebraic Topology 1, CMS Conference Proceedings 2 (1982),89–134. Google Scholar

[15] 15. Hambleton, I. and Madsen, I., Local surgery obstructions and space forms, Preprint, I.H.E.S. (1983). Google Scholar

[16] 16. Hsiang, W. C. and Shaneson, J., Fake tori, Proceedings, Topology of Manifolds (Markham Publ. Company, Chicago, 1969). Google Scholar PubMed

[17] 17. Hodgson, J., General position in the Poincaré duality category, Invent. Math. 24 (1974),311–334. Google Scholar

[18] 18. Jones, L., Patch spaces: a geometric representation for Poincaré spaces, Ann. of Math. 97 (1973),306–343. Google Scholar

[19] 19. Laitinen, E. and Madsen, I., The L-theory of groups with periodic cohomology I, Preprint, Aarhus University (1981). Google Scholar

[20] 20. Lee, R., Semi-characteristic classes, Topology 12 (1973),183–199. Google Scholar

[21] 21. Madsen, I. Reidemeister torsion, surgery invariants and spherical space forms, Proc. LMS 46 (1983),193–240. Google Scholar

[22] 22. Madsen, I. Smooth spherical space forms, Proceedings, Evanston (1977),303–352 (Springer Lecture Notes, 657). Google Scholar

[23] 23. Milnor, J., Whitehead torsion, Bull. AMS 72 (1966),358–426. Google Scholar

[24] 24. Milnor, J., Groups which act on S without fixed points, Amer. J. Math. 79 (1957),623–630. Google Scholar

[25] 25. Milgram, R. J., Odd index subgroups of units in cyclotomic fields and applications, Proceedings, Evanston (1980),269–298 (Springer Lecture Notes, 854). Google Scholar

[26] 26. Milgram, R. J., Patching techniques in surgery and the solution of the compact space form problem, Preprint, Stanford University (1981). Google Scholar

[27] 27. Madsen, I., Thomas, C. B. and Wall, C. T. C., The topological spherical space form problem II, Topology 75 (1978),375–382. Google Scholar

[28] 28. Madsen, I., Thomas, C. B. and Wall, C. T. C., The topological spherical space form problem III, Smoothings and dimensional bounds, Pac. J. Math, (to appear). Google Scholar | DOI

[29] 29. Maumary, S., Proper surgery groups and Wall-Novikov groups, Proceedings (Springer Lecture Notes 343, 1973),526–539. Google Scholar

[30] 30. Oliver, R., SK for finite group rings: II, Math. Scand. 47 (1980),195–231. Google Scholar

[31] 31. Pardon, W., Mod 2 semi-characteristic and the converse to a theorem of Milnor, Math. Z. 171 (1980), 247–268. Google Scholar

[32] 32. Pardon, W., The exact sequences of a localization for Witt groups II. Numerical invariants of odd-dimensional surgery obstructions, Pac. J. Math, (to appear). Google Scholar | DOI

[33] 33. Pedersen, E. K. and Ranicki, A. A., Projective surgery theory, Topology 19 (1980),239–254. Google Scholar

[34] 34. Ranicki, A. A., Algebraic L-theory I. Foundations, Proc. LMS 27 (1973), 101–125. Google Scholar

[35] 35. Ranicki, A. A., Algebraic L-theory II. Laurent extensions, Proc. LMS 27 (1973), 126–158. Google Scholar

[36] 36. Ranicki, A. A., The algebraic theory of surgery I, Proc. LMS 40 (1980), 87–192. Google Scholar

[37] 37. Ranicki, A. A., Exact sequences in the algebraic theory of surgery, Math. Notes 26 (Princeton University Press, Princeton, 1981). Google Scholar

[38] 38. Serre, J. P., Local fields, Graduate Texts in Mathematics 67 (Springer-Verlag, New York-Heidelberg-Berlin, 1979). Google Scholar

[39] 39. Siebenmann, L., The obstruction to finding a boundary for an open manifold of dimension greater than five, Thesis, Princeton University (1965). Google Scholar

[40] 40. Siebenmann, L., A total Whitehead torsion obstruction to fibering over a circle, Comm. Math. Helv. 45 (1970), 1–48. Google Scholar

[41] 41. Swan, R., Periodic resolutions for finite groups, Ann. of Math. 72 (1960), 167–191. Google Scholar

[42] 42. Taylor, L., Surgery on paracompact manifolds, Thesis, Berkeley (1972). Google Scholar

[43] 43. Thomas, C. B. and Wall, C. T. C., On the structure of finite groups with periodic cohomology, Preprint, Liverpool University (1979). Google Scholar

[44] 44. Wall, C. T. C., The classification of hermitian forms II and III, Invent. Math. (1972), 199–141 and (1973), 59–71. Google Scholar

[45] 45. Wall, C. T. C., The classification of hermitian forms IV, Invent. Math. 23 (1974), 241–260. Google Scholar

[46] 46. Wall, C. T. C., The classification of hermitian forms VI, Ann. of Math. 103 (1976), 1–80. Google Scholar

[47] 47. Wall, C. T. C., Periodic projective resolutions, Proc. LMS 39 (1979), 509–553. Google Scholar

[48] 48. Wall, C. T. C., Norms of units in group ring, Proc. LMS 29 (1974), 593–632. Google Scholar

[49] 49. Wall, C. T. C., Foundations of algebraic L-theory, Proceedings (Springer Lecture Notes, 343, 1972). Google Scholar

[50] 50. Wolf, J., Spaces of constant curvature, 3rd. Ed. (Publish or Perish, Boston, 1977). Google Scholar