involutions of some -manifolds

Author:
Myung Mi Myung

Journal:
Proc. Amer. Math. Soc. **33** (1972), 576-581

MSC:
Primary 57C99; Secondary 55C35, 57E30

MathSciNet review:
0295363

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Abstract: Let and be PL involutions of connected, oriented, closed, irreducible 3-manifolds 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 2-dimensional 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 3-space, 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*3-*space by a polyhedron*, Proc. Nat. Acad. Sci. U.S.A.**10**(1924), 6-8.**[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****[3]**Kyung Whan Kwun,*Scarcity of orientation-reversing 𝑃𝐿 involutions of lens spaces*, Michigan Math. J.**17**(1970), 355–358. MR**0279814****[4]**Kyung Whan Kwun,*Nonexistence of orientation reversing involutions on some manifolds.*, Proc. Amer. Math. Soc.**23**(1969), 725–726. MR**0247629**, 10.1090/S0002-9939-1969-0247629-4**[5]**Kyung Whan Kwun,*Piecewise linear involutions of 𝑆¹×𝑆².*, Michigan Math. J.**16**(1969), 93–96. MR**0242161****[6]**G. R. Livesay,*Involutions with two fixed points on the three-sphere*, Ann. of Math. (2)**78**(1963), 582–593. MR**0155323****[7]**G. R. Livesay,*Fixed point free involutions on the 3-sphere*, Ann. of Math. (2)**72**(1960), 603–611. MR**0116343****[8]**J. Milnor,*A unique decomposition theorem for 3-manifolds*, Amer. J. Math.**84**(1962), 1–7. MR**0142125****[9]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0295363-7

Keywords:
PL involution,
lens space,
connected sum

Article copyright:
© Copyright 1972
American Mathematical Society