Involutions With Two Fixed Points on the Three-Sphere

Livesay, G. R.

doi:10.2307/1970543

Cited by 45 publications

(30 citation statements)

References 3 publications

(4 reference statements)

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“…As in [1] the fact that M is homeomorphic to P2 X / follows immediately. We divide the proof of this theorem into a number of steps, working throughout in the PL category.…”

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confidence: 76%

“…Let r: M -» P be a retraction such that r restricted to each component of dM is a homeomorphism. r can be chosen transverse to a simple closed noncontractible curve a in P. Then a-"'(a) is a compact 2-manifold embedded in M. Exactly as in [1] it follows that r~\a) contains a component K which is orientable, one-sided and has 3A" equal to two noncontractible simple closed curves, one in each component of dM.Let p: S2 X I -» M be the double covering and let g: S2 X I -» 52 X / be the covering transformation. Let L denote p~liK).…”

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confidence: 99%

“…Let M be a compact connected nonorientable 3-manifold with dM consisting of two copies of P2 and 7A,(M) = Z2. Using the analysis of the Heegaard splittings of 53 in [4], we prove the following Theorem [1]. // the orientable two-fold cover of M is homeomorphic to S2 X I then there is an annulus A embedded in M with A n dM = dA consisting of two simple closed noncontractible curves, one in each component of dM.…”

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confidence: 99%

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Heegaard Splittings and a Theorem of Livesay

Rubinstein¹

1976

Proceedings of the American Mathematical Society

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Abstract. Let M be a nonorientable 3-manifold which is double covered by S2 X I. We give a short proof of the theorem of Livesay [1] that M is homeomorphic to P2 x / (where P2 denotes the projective plane).0. Let M be a compact connected nonorientable 3-manifold with dM consisting of two copies of P2 and 7A,(M) = Z2. Using the analysis of the Heegaard splittings of 53 in [4], we prove the following Theorem [1]. // the orientable two-fold cover of M is homeomorphic to S2 X I then there is an annulus A embedded in M with A n dM = dA consisting of two simple closed noncontractible curves, one in each component of dM.As in [1] the fact that M is homeomorphic to P2 X / follows immediately. We divide the proof of this theorem into a number of steps, working throughout in the PL category.1. Let r: M -» P be a retraction such that r restricted to each component of dM is a homeomorphism. r can be chosen transverse to a simple closed noncontractible curve a in P. Then a-"'(a) is a compact 2-manifold embedded in M. Exactly as in [1] it follows that r~\a) contains a component K which is orientable, one-sided and has 3A" equal to two noncontractible simple closed curves, one in each component of dM.Let p: S2 X I -» M be the double covering and let g: S2 X I -» 52 X / be the covering transformation. Let L denote p~liK). If W is the closure of a component of S2 X I -L, then W u gW = S2 X I and W n gW = L.Clearly we can assume without loss of generality that K is incompressible, i.e., there is no disk D embedded in M with D n K = dD and 3D a noncontractible curve in K. Also it will be supposed that genus K > 0, i.e., K is not an annulus.2. We show that K incompressible implies W is a handlebody. By [2] there are disjoint simple closed noncontractible curves C,, . . . , Cm in dW such that the normal closure of the elements of tr^dW) given by joining each C, to the base point along some path for 1 < i < aai is Ker d> (where d>; trx(W) is induced by the inclusion map).

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“…As in [1] the fact that M is homeomorphic to P2 X / follows immediately. We divide the proof of this theorem into a number of steps, working throughout in the PL category.…”

mentioning

confidence: 76%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Heegaard Splittings and a Theorem of Livesay

Rubinstein¹

1976

Proceedings of the American Mathematical Society

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“…In order to classify the involutions on a sohd torus we begin with the following lemma, which is an immediate consequence of Theorem 1 in [5]. Lemma (3.1).…”

Section: Figurementioning

confidence: 99%

Involutions with isolated fixed points on orientable 3-dimensional flat space forms

Luft¹,

Sjerve²

1984

Trans. Amer. Math. Soc.

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Abstract. In this paper we completely classify (up to conjugacy) all involutions i: M -» M, where M is an orientable connected flat 3-dimensional space form, such that i has fixed points but only finitely many. If Mx,..., M6 are the 6 space forms then only Mx, M2, M6 admit such involutions. Moreover, they are unique up to conjugacy. The main idea behind the proof is to find incompressible tori T ç M so that either i(T) = T or t(T) n T = 0 and then cut M into simpler pieces. These results lead to a complete classification of 3-manifolds containing Z © Z © Z in their fundamental groups.

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“…All orientation-reversing PL involutions of lens spaces [6], [8] and all PL involutions of P3 [2], [6] are known. As consequences of Theorem A and [3], we have Theorem B.…”

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confidence: 99%

Orientation-reversing periodic PL maps of lens spaces

Kim¹

1977

Proc. Amer. Math. Soc.

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Abstract. We complete the classification of all orientation-reversing PL maps of period 4k (k > 1) on lens spaces. All Z4-actions on the projective 3-space are also classified.1. Introduction. The 3-sphere and the projective 3-space are the only 3-dimensional lens spaces Lip, q) which admit orientation-reversing PL maps of period « > 4 [4]. In [4] all orientation-reversing PL maps of period 4k on the 3-sphere S3 are classified for all k. In this paper we show:

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Involutions With Two Fixed Points on the Three-Sphere

Cited by 45 publications

References 3 publications

Heegaard Splittings and a Theorem of Livesay

Heegaard Splittings and a Theorem of Livesay

Involutions with isolated fixed points on orientable 3-dimensional flat space forms

Orientation-reversing periodic PL maps of lens spaces

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