involutions of some manifolds
Author:
Myung Mi Myung
Journal:
Proc. Amer. Math. Soc. 33 (1972), 576581
MSC:
Primary 57C99; Secondary 55C35, 57E30
MathSciNet review:
0295363
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let and be PL involutions of connected, oriented, closed, irreducible 3manifolds and , respectively. Let , be a fixed point of such that near the fixed point sets of are of the same dimension. Then we obtain a PL involution on induced by by taking the connected sum of and along neighborhoods of . In this paper, we study the possibility for a PL involution h on having a 2dimensional fixed point set to be of the form , where are lens spaces. It is shown that: (1) if is orientable, then and h is the obvious involution, (2) if the fixed point set F contains a projective plane, then projective 3space, and in this case, F is the disjoint union of two projective planes and h is unique up to PL equivalences, (3) if F contains a Klein bottle K, then F is the disjoint union of a Klein bottle and two points.
 [1]
J. Alexander, On the subdivision of 3space by a polyhedron, Proc. Nat. Acad. Sci. U.S.A. 10 (1924), 68.
 [2]
Armand
Borel, Seminar on transformation groups, With contributions by
G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. Annals of Mathematics
Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. MR 0116341
(22 #7129)
 [3]
Kyung
Whan Kwun, Scarcity of orientationreversing 𝑃𝐿
involutions of lens spaces, Michigan Math. J. 17
(1970), 355–358. MR 0279814
(43 #5535)
 [4]
Kyung
Whan Kwun, Nonexistence of orientation reversing
involutions on some manifolds., Proc. Amer.
Math. Soc. 23
(1969), 725–726. MR 0247629
(40 #893), http://dx.doi.org/10.1090/S00029939196902476294
 [5]
Kyung
Whan Kwun, Piecewise linear involutions of
𝑆¹×𝑆²., Michigan Math. J.
16 (1969), 93–96. MR 0242161
(39 #3495)
 [6]
G.
R. Livesay, Involutions with two fixed points on the
threesphere, Ann. of Math. (2) 78 (1963),
582–593. MR 0155323
(27 #5257)
 [7]
G.
R. Livesay, Fixed point free involutions on the 3sphere, Ann.
of Math. (2) 72 (1960), 603–611. MR 0116343
(22 #7131)
 [8]
J.
Milnor, A unique decomposition theorem for 3manifolds, Amer.
J. Math. 84 (1962), 1–7. MR 0142125
(25 #5518)
 [9]
Edwin
H. Spanier, Algebraic topology, McGrawHill Book Co., New
YorkToronto, Ont.London, 1966. MR 0210112
(35 #1007)
 [1]
 J. Alexander, On the subdivision of 3space by a polyhedron, Proc. Nat. Acad. Sci. U.S.A. 10 (1924), 68.
 [2]
 A. Borel, Seminar on transformation groups, Ann. of Math. Studies, no. 46, Princeton Univ. Press, Princeton, N.J., 1960. MR 22 #7129. MR 0116341 (22:7129)
 [3]
 K. W. Kwun, Scarcity of orientation reversing PL involutions of lens spaces (to appear). MR 0279814 (43:5535)
 [4]
 K. W. Kwun, Nonexistence of orientation reversing involutions on some manifolds, Proc. Amer. Math. Soc. 23 (1969), 725726. MR 40 #893. MR 0247629 (40:893)
 [5]
 , Piecewise linear involutions of , Michigan Math. J. 16 (1969), 9396. MR 39 #3495. MR 0242161 (39:3495)
 [6]
 G. R. Livesay, Involutions with two fixed points on the three sphere, Ann. of Math. (2) 78 (1963), 582593. MR 27 #5257. MR 0155323 (27:5257)
 [7]
 , Fixedpointfree involutions on the 3sphere, Ann. of Math. (2) 72 (1960), 603611. MR 22 #7131. MR 0116343 (22:7131)
 [8]
 J. W. Milnor, A unique decomposition theorem for 3manifolds, Amer. J. Math. 84 (1962), 17. MR 25 #5518. MR 0142125 (25:5518)
 [9]
 E. H. Spanier, Algebraic topology, McGrawHill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
57C99,
55C35,
57E30
Retrieve articles in all journals
with MSC:
57C99,
55C35,
57E30
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197202953637
PII:
S 00029939(1972)02953637
Keywords:
PL involution,
lens space,
connected sum
Article copyright:
© Copyright 1972
American Mathematical Society
