On 3-Manifolds with Sufficiently Large Decompositions

1971 ◽  
Vol 23 (4) ◽  
pp. 746-748
Author(s):  
Wolfgang Heil
Keyword(s):  
Fibre Bundle ◽  
Fundamental Groups ◽  
Connected Sum ◽  
Peripheral Structure ◽  
Image Position

In [6] it is shown that two (compact) orientable 3-manifolds which are irreducible, boundary irreducible and sufficiently large are homeomorphic if and only if there exists an isomorphism between the fundamental groups which respects the peripheral structure. In this note we extend this theorem to reducible 3-manifolds.Any compact 3-manifold M has a decomposition into prime manifolds [1; 4].1Here the connected sum of two bounded manifolds N1, N2 is denned by removing 3-balls B1 B2 in int N1, int N2, respectively, and glueing the resulting boundary spheres together. The M1's which occur in the decomposition (1) are either irreducible or handles (i.e., a fibre bundle over S1 with fibre S2). If (1) contains a fake 3-sphere, we assume it to be Mn.

Knot surgery and primeness

1986 ◽  
Vol 99 (1) ◽  
pp. 89-102 ◽  
Author(s):  
Francisco González Acuña ◽  
Hamish Short
Keyword(s):  
Dehn Surgery ◽  
Connected Sum ◽  
Low Dimensional Topology ◽  
Knot Surgery ◽  
Low Dimensional ◽  
Image Position

The aim of this paper is to prove some new results towards answering the question: When does Dehn surgery on a knot give a non-prime manifold? This question has been raised on several occasions (see for instance [5] or [4]; concerning the latter see below). Recall that a 3-manifold is prime if, in any connected sum decompositionone of M1, M2 is S3. (For standard definitions of low-dimensional topology see [2] or [16].)

scholarly journals n-QUASI-ISOTOPY I: QUESTIONS OF NILPOTENCE

2005 ◽  
Vol 14 (05) ◽  
pp. 571-602 ◽  
Author(s):  
SERGEY A. MELIKHOV ◽  
DUŠAN REPOVŠ
Keyword(s):  
Fundamental Group ◽  
Piecewise Linear ◽  
Equivalence Relations ◽  
Connected Sum ◽  
Structure Preserving ◽  
Singular Link ◽  
Peripheral Structure ◽  
Fibered Link ◽  
Link Homotopy ◽  
Baer’S Theorem

It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for (piecewise-linear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close C0-approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-isotopy, k∈ℕ; all of them are weaker than isotopy (in the sense of Milnor). We prove that every link can be cancelled up to peripheral structure preserving isomorphism of any quotient of the fundamental group, functorially invariant under k-quasi-isotopy; functoriality means that the isomorphism between the quotients for links related by any allowable crossing change fits in the commutative diagram with the fundamental group of the complement to the intermediate singular link. The proof invokes Baer's theorem on the join of subnormal locally nilpotent subgroups. On the other hand, the integral generalized ( lk ≠ 0) Sato–Levine invariant [Formula: see text] is invariant under 1-quasi-isotopy, but is not determined by any quotient of the fundamental group (endowed with the peripheral structure), functorially invariant under 1-quasi-isotopy — in contrast to Waldhausen's theorem.As a byproduct, we use [Formula: see text] to determine the image of the Kirk–Koschorke invariant [Formula: see text] of fibered link maps.

Closed braids which are not prime knots

1979 ◽  
Vol 86 (3) ◽  
pp. 421-426 ◽  
Author(s):  
H. R. Morton
Keyword(s):  
Braid Group ◽  
Connected Sum ◽  
Image Position

An element B ∈ Bn, the braid group on n strings, which can be written asin terms of the standard generators of Bn is called a split braid. It is easy to see that the resulting closed braid is the connected sum of the closed braids Ĉ and on k + 1 and n − k strings respectively (figure 1).

scholarly journals Non-orthogonalisable vector fields on spheres

1984 ◽  
Vol 27 (3) ◽  
pp. 275-281
Author(s):  
Martin Raussen
Keyword(s):  
Vector Fields ◽  
Fibre Bundle ◽  
Basis Vector ◽  
Image Position

A (k – l)-field on Sn-1 may be given as a section ϕ of the fibre bundlewith fibre Vn-1, k-1 or, equivalently, as a semi-orthogonal map, i.e., a mapwhich is isometric in the second variable and such that for the basis vector e1∈Rk and every x∈Rn

A rational invariant for certain infinite discrete groups

1993 ◽  
Vol 113 (3) ◽  
pp. 473-478
Author(s):  
F. E. A. Johnson
Keyword(s):  
Euler Characteristic ◽  
Discrete Groups ◽  
Fundamental Groups ◽  
Intersection Form ◽  
Absolute Value ◽  
Rational Invariant ◽  
The Absolute ◽  
Image Position ◽  
Aspherical Manifolds

We introduce a rational-valued invariant which is capable of distinguishing between the commensurability classes of certain discrete groups, namely, the fundamental groups of smooth closed orientable aspherical manifolds of dimensional 4k(k ≥ 1) whose Euler characteristic χ(Λ) is non-zero. The invariant in question is the quotientwhere Sign (Λ) is the absolute value of the signature of the intersection formand [Λ] is a generator of H4k(Λ; ℝ).

scholarly journals Fundamental groups and Euler characteristics of sphere-like digital images

2016 ◽  
Vol 17 (2) ◽  
pp. 139 ◽  
Author(s):  
Laurence Boxer ◽  
P. Christopher Staecker
Keyword(s):  
Fundamental Group ◽  
Euler Characteristic ◽  
Digital Image ◽  
Digital Images ◽  
Current Paper ◽  
Dimensional Sphere ◽  
Fundamental Groups ◽  
Euler Characteristics ◽  
Connected Sum ◽  
Open Questions

The current paper focuses on fundamental groups and Euler characteristics of various digital models of the 2-dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler characteristics (which are not always equal). We consider the connected sum of digital surfaces and investigate how this operation relates to the fundamental group and Euler characteristic. We also consider two related but dierent notions of a digital image having "no holes," and relate this to the triviality of the fundamental group. Many of our results have origins in the paper [15] by S.-E. Han, which contains many errors. We correct these errors when possible, and leave some open questions. We also present some original results.

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 Topological 4-manifolds with right-angled Artin fundamental groups

2019 ◽  
Vol 11 (04) ◽  
pp. 777-821
Author(s):  
Ian Hambleton ◽  
Alyson Hildum
Keyword(s):  
Fundamental Group ◽  
Cohomological Dimension ◽  
Fundamental Groups ◽  
Connected Sum ◽  
Dimension Formula ◽  
Assembly Map ◽  
Torsion Free

We classify closed, spin[Formula: see text], topological [Formula: see text]-manifolds with fundamental group [Formula: see text] of cohomological dimension [Formula: see text] (up to [Formula: see text]-cobordism), after stabilization by connected sum with at most [Formula: see text] copies of [Formula: see text]. In general, we must also assume that [Formula: see text] satisfies certain [Formula: see text]-theory and assembly map conditions. Examples for which these conditions hold include the torsion-free fundamental groups of [Formula: see text]-manifolds and all right-angled Artin groupswhose defining graphs have no [Formula: see text]-cliques.

Congruence Subgroups of the Picard Group

1980 ◽  
Vol 32 (6) ◽  
pp. 1474-1481 ◽  
Author(s):  
Benjamin Fine
Keyword(s):  
Riemann Surfaces ◽  
Picard Group ◽  
Structure Theory ◽  
Fundamental Groups ◽  
Congruence Subgroups ◽  
Free Products ◽  
Abstract Group ◽  
Linear Fractional Transformations ◽  
Amalgamated Subgroup ◽  
Image Position

The Picard group Γ = PSL2 (Z [i]) is the group of linear fractional transformationswith ad – bc = ± 1 and a, b, c, d Gaussian integers.Γ is of interest as an abstract group and in automorphic function theory. In an earlier paper [1], a decomposition of Γ as a free product with amalgamated subgroup was given and this was utilized to investigate Fuchsian subgroups. Karrass and Solitar used a similar decomposition to characterize abelian and nilpotent subgroups. Maskit [6], Mennicke [7] and Fine [2], used Γ to generate faithful representations of Fundamental Groups of Riemann Surfaces while more recently Wielenberg [10] represented certain knot and link groups as subgroups of Γ. In this paper, we will examine the structure of the congruence subgroups of Γ. Our technique will be to use the decomposition cited above [1], together with the Karrass-Solitar subgroup structure theory for free products with amalgamations [3]. Finally, we give a conjecture and some results concerning Fuchsian subgroups which are contained in congruence subgroups.

scholarly journals On the generalised Todd genus of flag bundles

1974 ◽  
Vol 19 (1) ◽  
pp. 35-38
Author(s):  
S. A. Ilori
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Fibre Bundle ◽  
Flag Manifold ◽  
Complex Algebraic Variety ◽  
Flag Bundle ◽  
Image Position ◽  
Dimensional Projective Space ◽  
Complex Projective ◽  
Nested System

Let V be a complex algebraic variety. Given integers a1, …, am such thatone defines a (a1, …, am)-flag as a nested systemof subspaces of Sn, the n-dimensional complex projective space. The set of all such flags is called an incomplete flag-manifold in Sn, and is denoted by W(al, …, am). Also let E be a complex n-dimensional vector bundle over V. Then we denote by E(a1, …, am−1, n; V) an associated fibre bundle of E with fibre W(a1 − 1, …, am−1 − 1, n − 1). E(a1, …, am−1 − 1, n; V) is called an incomplete flag bundle of E over V (cf. (2), (3)). In Section 10.3 and Section 14.4 of (1), the generalised Todd genus Ty(W(0, n)) and Ty(W(0, 1, …, n)) of the n-dimensional projective space W(0, n) and the flag manifold W(0, 1, …, n) (or F(n+l)) were calculated. Here we compute Ty(W(a1, …, am)) and also Ty(E(a1, …, am−1, n; V)).

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