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.

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.

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.

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