Homology of Branched Coverings of 3-Manifolds

1992 ◽  
Vol 44 (1) ◽  
pp. 119-134
Author(s):  
John Hempel
Keyword(s):  
Classical Result ◽  
Quantitative Information ◽  
Cyclic Cover ◽  
Branched Covers ◽  
Branched Coverings ◽  
Homology Groups

AbstractWe give a relation between the homology groups H1() and H1 (M) for a branched cyclic cover → M of arbitrary closed, oriented 3-manifolds which generalizes a classical result of Plans on covers of S3 branched over a knot and provides other quantitative information as well. We include a general "free calculus" procedure for computing homology groups of branched covers and reinterpret the results in this computational setting.

scholarly journals On the cyclic coverings of the knot 52

1999 ◽  
Vol 42 (3) ◽  
pp. 575-587 ◽  
Author(s):  
P. Bandieri ◽  
A. C. Kim ◽  
M. Mulazzani
Keyword(s):  
Fundamental Groups ◽  
Branched Coverings ◽  
Homology Groups ◽  
Cyclic Coverings

We construct a family of hyperbolic 3-manifolds whose fundamental groups admit a cyclic presentation. We prove that all these manifolds are cyclic branched coverings of S3 over the knot 52 and we compute their homology groups. Moreover, we show that thecyclic presentations correspond to spines of the manifolds.

New invariants of periodic knots

1997 ◽  
Vol 122 (2) ◽  
pp. 281-290 ◽  
Author(s):  
SWATEE NAIK
Keyword(s):  
Branched Covers ◽  
Periodic Action ◽  
Homology Groups ◽  
Cyclic Covers ◽  
New Criteria

For knots in S3 we obtain new criteria for periodicity. We show that the Casson–Gordon invariants of a periodic knot are preserved under the periodic action lifted to the cyclic covers. As an application, we consider a family of knots with the Seifert form of a period 3 knot, and using Casson–Gordon invariants show that knots in this family do not have period 3. We also obtain periodicity criteria in terms of the homology groups of cyclic branched covers of S3.

ALEXANDER POLYNOMIALS AND ORDERS OF HOMOLOGY GROUPS OF BRANCHED COVERS OF KNOTS

2009 ◽  
Vol 18 (07) ◽  
pp. 973-984 ◽  
Author(s):  
SE-GOO KIM
Keyword(s):  
Prime Power ◽  
Alexander Polynomial ◽  
Branched Covers ◽  
Splitting Property ◽  
Homology Groups ◽  
Alexander Polynomials ◽  
Knot Concordance ◽  
Branched Cover ◽  
Linearly Independent

Fox showed that the order of homology of a cyclic branched cover of a knot is determined by its Alexander polynomial. We find examples of knots with relatively prime Alexander polynomials such that the first homology groups of their q-fold cyclic branched covers are of the same order for every prime power q. Furthermore, we show that these knots are linearly independent in the knot concordance group using the polynomial splitting property of the Casson–Gordon–Gilmer invariants.

scholarly journals On the homology of branched coverings of 3-manifolds

2014 ◽  
Vol 213 ◽  
pp. 21-39 ◽  
Author(s):  
Jun Ueki
Keyword(s):  
Topological Invariant ◽  
Ideal Class ◽  
Class Groups ◽  
Branched Covers ◽  
Branched Coverings ◽  
3 Dimensional ◽  
Genus Theory ◽  
Ideal Class Groups ◽  
Galois Covers ◽  
Insight Into

AbstractFollowing the analogies between 3-manifolds and number rings in arithmetic topology, we study the homology of branched covers of 3-manifolds. In particular, we show some analogues of Iwasawa’s theorems on ideal class groups and unit groups, Hilbert’s Satz 90, and some genus-theory–type results in the context of 3-dimensional topology. We also prove that the 2-cycles valued Tate cohomology of branched Galois covers is a topological invariant, and we give a new insight into the analogy between 2-cycle groups and unit groups.

scholarly journals Annihilation of torsion in homology of finite m-AQ quandles

2016 ◽  
Vol 25 (12) ◽  
pp. 1642012
Author(s):  
Józef H. Przytycki ◽  
Seung Yeop Yang
Keyword(s):  
Finite Groups ◽  
Classical Result ◽  
Torsion Subgroup ◽  
Homology Groups

It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. Similarly, we have proved that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. However, it does not hold for connected quandles in general. In this paper, we define an [Formula: see text]-almost quasigroup ([Formula: see text]-AQ) quandle which is a generalization of a quasigroup quandle and study annihilation of torsion in its rack and quandle homology groups.

scholarly journals Non-Left-Orderable 3-Manifold Groups

2005 ◽  
Vol 48 (1) ◽  
pp. 32-40 ◽  
Author(s):  
Mieczysław K. Dąbkowski ◽  
Józef H. Przytycki ◽  
Amir A. Togha
Keyword(s):  
Special Interest ◽  
Branched Covers ◽  
Branched Coverings ◽  
Figure Eight Knot ◽  
Torsion Free

AbstractWe show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of S3 branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of S3 branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the 52 knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

scholarly journals On the homology of branched coverings of 3-manifolds

2014 ◽  
Vol 213 ◽  
pp. 21-39 ◽  
Author(s):  
Jun Ueki
Keyword(s):  
Topological Invariant ◽  
Ideal Class ◽  
Class Groups ◽  
Branched Covers ◽  
Branched Coverings ◽  
3 Dimensional ◽  
Genus Theory ◽  
Ideal Class Groups ◽  
Galois Covers ◽  
Insight Into

AbstractFollowing the analogies between 3-manifolds and number rings in arithmetic topology, we study the homology of branched covers of 3-manifolds. In particular, we show some analogues of Iwasawa’s theorems on ideal class groups and unit groups, Hilbert’s Satz 90, and some genus-theory–type results in the context of 3-dimensional topology. We also prove that the 2-cycles valued Tate cohomology of branched Galois covers is a topological invariant, and we give a new insight into the analogy between 2-cycle groups and unit groups.

Notes on More Fibonacci Groups

2008 ◽  
Vol 15 (04) ◽  
pp. 699-706 ◽  
Author(s):  
Kwang-Woo Jeong ◽  
Moon-Ok Wang
Keyword(s):  
Infinite Family ◽  
Geometrical Properties ◽  
Branched Coverings ◽  
Homology Groups ◽  
Split Extensions

The Fibonacci manifolds Mn introduced by Helling, Kim and Mennicke are generalized in an obvious way suggested by a tessellation of triangles on the boundary of 3-balls. We construct an infinite family of hyperbolic 3-manifolds by a tessellation of (2ℓ +1)-gons on the boundary of a combinatorial polyhedron P(2ℓ +1, n). We prove that these manifolds are the n-fold cyclic branched coverings of S3 over the 2-bridge knot b(4ℓ2+1, 2ℓ), and investigate some topological and geometrical properties of the manifolds together with their first homology groups and split extensions.

Some Results of the Spectroscopic Study of Four Close Binary Systems of Early Spectral Types

1965 ◽  
Vol 5 ◽  
pp. 120-130
Author(s):  
T. S. Galkina
Keyword(s):  
Spectroscopic Study ◽  
Spectral Type ◽  
Spectroscopic Investigation ◽  
Binary Systems ◽  
Quantitative Information ◽  
Close Binary ◽  
Physical Parameters ◽  
Absorption And Emission ◽  
Close Binary Systems ◽  
Quantitative Estimates

It is necessary to have quantitative estimates of the intensity of lines (both absorption and emission) to obtain the physical parameters of the atmosphere of components.Some years ago at the Crimean observatory we began the spectroscopic investigation of close binary systems of the early spectral type with components WR, Of, O, B to try and obtain more quantitative information from the study of the spectra of the components.

The direct determination of magnetic domain wall profiles using a split detector STEM

1978 ◽  
Vol 36 (1) ◽  
pp. 466-467
Author(s):  
J.N. Chapman ◽  
P.E. Batson ◽  
E.M. Waddell ◽  
R.P. Ferrier
Keyword(s):  
Electron Microscope ◽  
Domain Wall ◽  
Transmission Electron Microscope ◽  
Domain Walls ◽  
Photographic Plate ◽  
Magnetic Domain ◽  
Direct Determination ◽  
Quantitative Information ◽  
Scanning Transmission Electron Microscope ◽  
Transmission Electron

By far the most commonly used mode of Lorentz microscopy in the examination of ferromagnetic thin films is the Fresnel or defocus mode. Use of this mode in the conventional transmission electron microscope (CTEM) is straightforward and immediately reveals the existence of all domain walls present. However, if such quantitative information as the domain wall profile is required, the technique suffers from several disadvantages. These include the inability to directly observe fine image detail on the viewing screen because of the stringent illumination coherence requirements, the difficulty of accurately translating part of a photographic plate into quantitative electron intensity data, and, perhaps most severe, the difficulty of interpreting this data. One solution to the first-named problem is to use a CTEM equipped with a field emission gun (FEG) (Inoue, Harada and Yamamoto 1977) whilst a second is to use the equivalent mode of image formation in a scanning transmission electron microscope (STEM) (Chapman, Batson, Waddell, Ferrier and Craven 1977), a technique which largely overcomes the second-named problem as well.

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