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

Luft, E.; Sjerve, Denis

doi:10.1090/s0002-9947-1984-0748842-2

Cited by 6 publications

(7 citation statements)

References 9 publications

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“…All possible involutions with only isolated fixed points on closed, orientable, flat three-dimensional space forms and their orbit spaces have been classified in the work of Kwun and Tollefson [24] and Luft and Sjerve [26]. Theorems 1.1 and 1.3 settle Conjectures 1.10 and 1.11 in [30].…”

Section: Introduction and Resultsmentioning

confidence: 99%

On Three-Dimensional Alexandrov Spaces

Galaz-García

Guijarro

2014

Int Math Res Notices

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We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspension of RP 2 ; we use this to classify, up to homeomorphism, closed, positively curved Alexandrov spaces of dimension three. Second, we classify closed three-dimensional Alexandrov spaces of nonnegative curvature. Third, we study the well-known Poincaré Conjecture in dimension three, in the context of Alexandrov spaces, in the two forms it is usually formulated for manifolds. We first show that the only closed three-dimensional Alexandrov space that is also a homotopy sphere is the 3-sphere; then we give examples of closed, geometric, simply connected three-dimensional Alexandrov spaces for five of the eight Thurston geometries, proving along the way the impossibility of getting such examples for the Nil, SL2(R) and Sol geometries. We conclude the paper by proving the analogue of the geometrization conjecture for closed threedimensional Alexandrov spaces.

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Section: Introduction and Resultsmentioning

confidence: 99%

On Three-Dimensional Alexandrov Spaces

Galaz-García

Guijarro

2014

Int Math Res Notices

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“…These induced involutions are the canonical ones. For a more explicit description of Mf 9 M% and M* see [3]. In conclusion we have the following hierarchy of coverings:…”

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confidence: 81%

“…(2) This theorem explicitly describes the way in which the groups act by affine motions on R 3 . In order to put a metric of constant curvature…”

Section: Flat 3-dimensional Space Formsmentioning

confidence: 99%

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3-manifolds with subgroupsZ⊕Z⊕Zin their fundamental groups

Luft¹,

Sjerve²

1984

Pacific J. Math.

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In this paper we characterize those 3-manifolds M 3 satisfying ZΘZΘZC ^i(Λf). All such manifolds M arise in one of the following ways:Here Λf 0 is any 3-manifold in (I), (II) and any 3-manifold having P 2 components in its boundary in (III). R is a flat space form and R* is obtained from R and some involution t: R -> R with fixed points, but only finitely many, as follows: if C,,..., C n are disjoint 3-cells around the fixed points then R* is the 3-manifold obtained from (R -int(C, U UQ))/ί by identifying some pairs of projective planes in the boundary.

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“…The geminus M G is the disk sum of two copies of (P 2 ×I ): The quadripus M Q is the orbit manifold of M = S 1 ×S 1 ×I under the orientationreversing involution τ (z 1 , z 2 , t) = (z 1 ,z 2 , 1 − t) with the interiors of invariant 3-ball neighborhoods of the four fixed points removed (see [13,15]). Its boundary consists of 4 projective planes and one incompressible torus.…”

Section: Cat Vsolv (M) and Cat G (M)mentioning

confidence: 99%

Categorical group invariants of 3-manifolds

2014

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For a given class G of groups, a 3-manifold M is of G-category ≤ k if it can be covered by k open subsets such that for each path-component W of the subsets the image of its fundamental group π 1 (W ) → π(M) belongs to G. The smallest number k such that M admits such a covering is the G-category, cat G (M). If M is closed, it has G-category between 1 and 4. We characterize all closed 3-manifolds of G-category 1, 2, and 3 for various classes G.

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

Cited by 6 publications

References 9 publications

On Three-Dimensional Alexandrov Spaces

On Three-Dimensional Alexandrov Spaces

3-manifolds with subgroupsZ⊕Z⊕Zin their fundamental groups

Categorical group invariants of 3-manifolds

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