Construction of group actions on fourmanifolds
Author:
Allan L. Edmonds
Journal:
Trans. Amer. Math. Soc. 299 (1987), 155170
MSC:
Primary 57N13; Secondary 57N15, 57S17, 57S25
MathSciNet review:
869405
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: It is shown that any cyclic group of odd prime order acts on any closed, simply connected topological manifold, inducing the identity on integral homology. The action is locally linear except perhaps at one isolated fixed point. In the case of primes greater than three a more careful argument is used to show that the action can be constructed to be locally linear.
 [G]
Glen
E. Bredon, Introduction to compact transformation groups,
Academic Press, New York, 1972. Pure and Applied Mathematics, Vol. 46. MR 0413144
(54 #1265)
 [S]
Sylvain
E. Cappell and Julius
L. Shaneson, The codimension two placement problem and homology
equivalent manifolds, Ann. of Math. (2) 99 (1974),
277–348. MR 0339216
(49 #3978)
 [A]
L. Edmonds, Preliminary notes on finite group actions on fourmanifolds, unpublished notes, 1984.
 [M]
Michael
Hartley Freedman, The topology of fourdimensional manifolds,
J. Differential Geom. 17 (1982), no. 3,
357–453. MR
679066 (84b:57006)
 1.
Michael
H. Freedman, The disk theorem for fourdimensional manifolds,
2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 647–663. MR 804721
(86m:57016)
 [M]
Marvin
J. Greenberg and John
R. Harper, Algebraic topology, Mathematics Lecture Note
Series, vol. 58, Benjamin/Cummings Publishing Co. Inc. Advanced Book
Program, Reading, Mass., 1981. A first course. MR 643101
(83b:55001)
 [F]
F.
Hirzebruch, W.
D. Neumann, and S.
S. Koh, Differentiable manifolds and quadratic forms, Marcel
Dekker Inc., New York, 1971. Appendix II by W. Scharlau; Lecture Notes in
Pure and Applied Mathematics, Vol. 4. MR 0341499
(49 #6250)
 [F]
Hirzebruch and D. Zagier, The AtiyahSinger index theorem and elementary number theory, Publish or Perish, Boston, 1973.
 [W]
W.
C. Hsiang and R.
H. Szczarba, On embedding surfaces in fourmanifolds,
Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin,
Madison, Wis., 1970), Amer. Math. Soc., Providence, R.I., 1971,
pp. 97–103. MR 0339239
(49 #4000)
 [S]
Sławomir
Kwasik, On the symmetries of the fake
𝐶𝑃², Math. Ann. 274 (1986),
no. 3, 385–389. MR 842620
(87h:57034), http://dx.doi.org/10.1007/BF01457223
 [S]
Sławomir
Kwasik and Pierre
Vogel, Asymmetric fourdimensional manifolds, Duke Math. J.
53 (1986), no. 3, 759–764. MR 860670
(88c:57038), http://dx.doi.org/10.1215/S001270948605341X
 2.
, Nonlocally smoothable topological symmetries of four manifolds, Preprint, 1985.
 [J]
John
Milnor, A duality theorem for Reidemeister torsion, Ann. of
Math. (2) 76 (1962), 137–147. MR 0141115
(25 #4526)
 3.
J.
Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 0196736
(33 #4922), http://dx.doi.org/10.1090/S000299041966114842
 [L]
Laurence
R. Taylor, Relative Rochlin invariants, Topology Appl.
18 (1984), no. 23, 259–280. MR 769295
(86g:57027), http://dx.doi.org/10.1016/01668641(84)900142
 [C]
C.
T. C. Wall, Surgery on compact manifolds, Academic Press,
London, 1970. London Mathematical Society Monographs, No. 1. MR 0431216
(55 #4217)
 [S]
Steven
H. Weintraub, Semifree 𝑍_{𝑝}actions on
highlyconnected manifolds, Math. Z. 145 (1975),
no. 2, 163–185. MR 0400268
(53 #4103)
 4.
Steven
H. Weintraub, Topological realization of equivariant intersection
forms, Pacific J. Math. 73 (1977), no. 1,
257–280. MR 0494175
(58 #13100)
 [G]
 E. Bredon, Introduction to compact transformation groups, Academic Press, New York 1972. MR 0413144 (54:1265)
 [S]
 E. Cappell and J. L. Shaneson, The codimension two placement problem and homology equivalent manifolds, Ann. of Math. (2) 99 (1974), 277348. MR 0339216 (49:3978)
 [A]
 L. Edmonds, Preliminary notes on finite group actions on fourmanifolds, unpublished notes, 1984.
 [M]
 H. Freedman, The topology of fourdimensional manifolds, J. Differential Geom. 17 (1982), 357453. MR 679066 (84b:57006)
 1.
 , The disk theorem for fourdimensional manifolds, Proc. Internat. Congr. Math., Warsaw, Poland, 1983, pp. 647663. MR 804721 (86m:57016)
 [M]
 J. Greenberg and J. R. Harper, Algebraic topology: a first course. Benjamin/Cummings, Reading, Mass., 1981. MR 643101 (83b:55001)
 [F]
 Hirzebruch, W. D. Neumann, and S. S. Koh, Differentiable manifolds and quadratic forms, Marcel Dekker, New York, 1971. MR 0341499 (49:6250)
 [F]
 Hirzebruch and D. Zagier, The AtiyahSinger index theorem and elementary number theory, Publish or Perish, Boston, 1973.
 [W]
 C. Hsiang and R. H. Szczarba, On embedding surfaces in fourmanifolds, Proc. Sympos. Pure Math., vol. 22, Amer. Math. Soc., Providence, R.I., 1971, pp. 97103. MR 0339239 (49:4000)
 [S]
 Kwasik, On symmetries of the fake , Preprint, 1985. MR 842620 (87h:57034)
 [S]
 Kwasik and P. Vogel, Asymmetric fourdimensional manifolds, Preprint, 1984. MR 860670 (88c:57038)
 2.
 , Nonlocally smoothable topological symmetries of four manifolds, Preprint, 1985.
 [J]
 W. Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962), 137147. MR 0141115 (25:4526)
 3.
 , Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358426. MR 0196736 (33:4922)
 [L]
 R. Taylor, Relative Rochlin invariant, Topology Appl. 18 (1984), 259280. MR 769295 (86g:57027)
 [C]
 T. C. Wall, Surgery on compact manifolds, Academic Press, London, 1970. MR 0431216 (55:4217)
 [S]
 H. Weintraub, Semifree actions on highlyconnected manifolds, Math. Z. 145 (1975), 163185. MR 0400268 (53:4103)
 4.
 , Topological realization of equivariant intersection forms, Pacific J. Math. 73 (1977), 257280. MR 0494175 (58:13100)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
57N13,
57N15,
57S17,
57S25
Retrieve articles in all journals
with MSC:
57N13,
57N15,
57S17,
57S25
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708694057
PII:
S 00029947(1987)08694057
Article copyright:
© Copyright 1987 American Mathematical Society
