The Seifert Fiber Space Conjecture and Torus Theorem for Nonorientable 3-Manifolds

1994 ◽  
Vol 37 (4) ◽  
pp. 482-489 ◽  
Author(s):  
Wolfgang Heil ◽  
Wilbur Whitten
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Fiber Space ◽  
Seifert Fiber Space

AbstractThe Seifert-fiber-space conjecture for nonorientable 3-manifolds states that if M denotes a compact, irreducible, nonorientable 3-manifold that is not a fake P2 x S1, if π1M is infinite and does not contain Z2 * Z2 as a subgroup, and if π1M does however contain a nontrivial, cyclic, normal subgroup, then M is a Seifert bundle. In this paper, we construct all compact, irreducible, nonorientable 3-manifolds (that do not contain a fake P2 × I) each of whose fundamental group contains Z2 * Z2 and an infinité cyclic, normal subgroup; none of these manifolds admits a Seifert fibration, but they satisfy Thurston's Geometrization Conjecture. We then reformulate the statement of the (nonorientable) SFS-conjecture and obtain a torus theorem for nonorientable manifolds.

RECOGNIZING NONORIENTABLE SEIFERT BUNDLES

1992 ◽  
Vol 01 (04) ◽  
pp. 471-475 ◽  
Author(s):  
WILBUR WHITTEN
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Fiber Space ◽  
Seifert Fiber Space

Roughly speaking, a compact, orientable, irreducible 3-manifold M with infinite fundamental group is a Seifert fiber space, if either 1) π1M contains a nontrivial, cyclic, normal subgroup (the so-called Seifert-fiber-space conjecture), 2) M is finitely covered by a Seifert fiber space, or 3) π1M is isomorphic to the group of a Seifert fiber space. Excluding a fake P2 × S1 where necessary, we show in this paper that similar results hold when M is nonorientable.

Elementary amenable groups and 4-manifolds with Euler characteristic 0

1991 ◽  
Vol 50 (1) ◽  
pp. 160-170 ◽  
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Euler Characteristic ◽  
Fundamental Groups ◽  
Normal Subgroups ◽  
Euler Characteristics ◽  
Amenable Groups ◽  
Amenable Subgroup ◽  
Finite Subgroups ◽  
Torsion Free

AbstractWe extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is asphencal and the group is virtually poly- Z of Hirsch length 4.

scholarly journals A Survey on Seifert Fiber Space Theorem

ISRN Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jean-Philippe Préaux
Keyword(s):  
Fiber Space ◽  
Seifert Fiber Space ◽  
History Of

We review the history of the proof of the Seifert fiber space theorem, as well as its motivations in 3-manifold topology and its generalizations.

scholarly journals A Note on Regular Coverings of Closed Orientable Surfaces

1961 ◽  
Vol 5 (2) ◽  
pp. 49-66 ◽  
Author(s):  
Jens Mennicke
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Finite Index ◽  
Orientable Surface ◽  
Fundamental Groups ◽  
Closed Orientable Surface ◽  
Regular Covering ◽  
Genus 2 ◽  
Image Position ◽  
Orientable Surfaces

The object of this note is to study the regular coverings of the closed orientable surface of genus 2.Let the closed orientable surfaceFhof genushbe a covering ofF2and letand f be the fundamental groups respectively. Thenis a subgroup of f of indexn = h − 1. A covering is called regular ifis normal in f.Conversely, letbe a normal subgroup of f of finite index. Then there is a uniquely determined regular coveringFhsuch thatis the fundamental group ofFh. The coveringFhis an orientable surface. Since the indexnofin f is supposed to be finite,Fhis closed, and its genus is given byn=h − 1.The fundamental group f can be defined by.

scholarly journals The fundamental groupoid as a topological groupoid

1975 ◽  
Vol 19 (3) ◽  
pp. 237-244 ◽  
Author(s):  
R. Brown ◽  
G. Danesh-Naruie
Keyword(s):  
Normal Subgroup ◽  
Topological Space ◽  
Fundamental Group ◽  
Initial Point ◽  
Covering Space ◽  
Topological Groupoid ◽  
Fundamental Groupoid ◽  
Path Connected ◽  
Totally Disconnected

Let X be a topological space. Then we may define the fundamental groupoid πX and also the quotient groupoid (πX)/N for N any wide, totally disconnected, normal subgroupoid N of πX (1). The purpose of this note is to show that if X is locally path-connected and semi-locally 1-connected, then the topology of X determines a “lifted topology” on (πX)/N so that it becomes a topological groupoid over X. With this topology the subspace which is the fibre of the initial point map ∂′: (πX)/N→X over x in X, is the usual covering space of X determined by the normal subgroup N{x} of the fundamental group π(X, x).

scholarly journals A Group of Automorphisms of the Homotopy Groups

1951 ◽  
Vol 2 ◽  
pp. 73-82
Author(s):  
Hiroshi Uehara
Keyword(s):  
Normal Subgroup ◽  
Topological Space ◽  
Fundamental Group ◽  
Homotopy Group ◽  
Factor Group ◽  
Complete Analysis ◽  
Homotopy Groups ◽  
Group Of Automorphisms ◽  
Arcwise Connected

It is well known that the fundamental group π1(X) of an arcwise connected topological space X operates on the n-th homotopy group πn(X) of X as a group of automorphisms. In this paper I intend to construct geometrically a group 𝒰(X) of automorphisms of πn(X), for every integer n ≥ 1, which includes a normal subgroup isomorphic to π1(X) so that the factor group of 𝒰(X) by π1(X) is completely determined by some invariant Σ(X) of the space X. The complete analysis of the operation of the group on πn(X) is given in §3, §4, and §5,

scholarly journals Pin(2)-equivariant Seiberg–Witten Floer homology of Seifert fibrations

2019 ◽  
Vol 156 (2) ◽  
pp. 199-250 ◽  
Author(s):  
Matthew Stoffregen
Keyword(s):  
Isomorphism Class ◽  
Floer Homology ◽  
Heegaard Floer Homology ◽  
Homology Sphere ◽  
Integral Homology ◽  
Fiber Space ◽  
Rational Homology ◽  
Seifert Fiber Space ◽  
Homology Cobordism ◽  
Seifert Fiber Spaces

We compute the $\text{Pin}(2)$-equivariant Seiberg–Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu’s conjecture that $\unicode[STIX]{x1D6FD}=-\bar{\unicode[STIX]{x1D707}}$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $\unicode[STIX]{x1D6F4}(a_{1},\ldots ,a_{n})$ are not homology cobordant to any $-\unicode[STIX]{x1D6F4}(b_{1},\ldots ,b_{n})$. We then use the same invariants to give an example of an integral homology sphere not homology cobordant to any Seifert fiber space. We also show that the $\text{Pin}(2)$-equivariant Seiberg–Witten Floer spectrum provides homology cobordism obstructions distinct from $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$. In particular, we identify an $\mathbb{F}[U]$-module called connected Seiberg–Witten Floer homology, whose isomorphism class is a homology cobordism invariant.

On the fundamental group of an orbit space

1965 ◽  
Vol 61 (3) ◽  
pp. 639-646 ◽  
Author(s):  
M. A. Armstrong
Keyword(s):  
Fixed Point ◽  
Normal Subgroup ◽  
Fundamental Group ◽  
Simplicial Complex ◽  
Orbit Space ◽  
Fixed Point Set ◽  
Point Set ◽  
Simply Connected

Introduction. Let K be a connected simplicial complex, finite or infinite, its polyhedron ((2), page 45) being the space X. Then X is connected. Suppose further that X is simply connected. For any group G of simplicial transformations of X, H will denote the normal subgroup generated by elements which have a non-empty fixed-point set.

Canonical Submersion and Lie Homomorphisms Normal Subgroup

2020 ◽  
Vol 23 (1) ◽  
pp. 97-101
Author(s):  
Mikhail Petrichenko ◽  
Dmitry W. Serow
Keyword(s):  
Normal Subgroup ◽  
Canonical System ◽  
System Of Equations ◽  
Core Group ◽  
The Core ◽  
Switch Group

Normal subgroup module f (module over the ring F = [ f ] 1; 2-diffeomorphisms) coincides with the kernel Ker Lf derivations along the field. The core consists of the trivial homomorphism (integrals of the system v = x = f (t; x )) and bundles with zero switch group Lf , obtained from the condition ᐁ( ω × f ) = 0. There is the analog of the Liouville for trivial immersion. In this case, the core group Lf derivations along the field replenished elements V ( z ), such that ᐁz = ω × f. Hence, the core group Lf updated elements helicoid (spiral) bundles, in particular, such that f = ᐁU. System as an example Crocco shown that the canonical system does not permit the trivial embedding: the canonical system of equations are the closure of the class of systems that permit a submersion.

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