scholarly journals Sasaki-Einstein 7-Manifolds, Orlik Polynomials and Homology

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 947
Author(s):  
Ralph R. Gomez
Keyword(s):  
Betti Number ◽  
Homology Group ◽  
Integral Homology ◽  
Einstein Manifolds ◽  
The Third

In this article, we give ten examples of 2-connected seven dimensional Sasaki-Einstein manifolds for which the third homology group is completely determined. Using the Boyer-Galicki construction of links over particular Kähler-Einstein orbifolds, we apply a valid case of Orlik’s conjecture to the links so that one is able to explicitly determine the entire third integral homology group. We give ten such new examples, all of which have the third Betti number satisfy 10 ≤ b 3 ( L f ) ≤ 20 .

scholarly journals Computing the fundamental group of a higher-rank graph

2021 ◽  
pp. 1-12
Author(s):  
Sooran Kang ◽  
David Pask ◽  
Samuel B.G. Webster
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Klein Bottle ◽  
Fundamental Groups ◽  
The Third ◽  
Higher Rank ◽  
The One ◽  
Standard Presentation

Abstract We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about higher-rank graphs when we show that the abelianization of the fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.

scholarly journals Existence of Taut Foliations on Seifert Fibered Homology 3-spheres

2014 ◽  
Vol 66 (1) ◽  
pp. 141-169
Author(s):  
Shanti Caillat-Gibert ◽  
Daniel Matignon
Keyword(s):  
Homology Group ◽  
Integral Homology

AbstractThis paper concerns the problem of existence of taut foliations among 3-manifolds. From the work of David Gabai we know that a closed 3-manifold with non-trivial second homology group admits a taut foliation. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite different if they are integral or rational but non-integral homology 3-spheres. Concerning integral homology 3-spheres, we can see that all but the 3-sphere and the Poincaré 3-sphere admit a taut foliation. Concerning non-integral homology 3-spheres, we prove there are infinitely many that admit a taut foliation, and infinitely many without a taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology 3-spheres.

scholarly journals Lescop's invariant and gauge theory

2015 ◽  
Vol 24 (09) ◽  
pp. 1550050 ◽  
Author(s):  
Prayat Poudel
Keyword(s):  
Gauge Theory ◽  
Euler Characteristic ◽  
Betti Number ◽  
Floer Homology ◽  
Integral Homology ◽  
Similar Relationship

Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds with positive first Betti number by establishing a similar relationship between the Lescop invariant of the manifold and its instanton Floer homology. The proof uses surgery techniques.

scholarly journals Double Solids, Categories and Non-Rationality

2013 ◽  
Vol 57 (1) ◽  
pp. 145-173 ◽  
Author(s):  
Atanas Iliev ◽  
Ludmil Katzarkov ◽  
Victor Przyjalkowski
Keyword(s):  
Category Theory ◽  
Mirror Symmetry ◽  
Homology Group ◽  
Hochschild Homology ◽  
Singular Points ◽  
Double Covering ◽  
Homological Mirror Symmetry ◽  
New Approach ◽  
The Third

AbstractThis paper suggests a new approach to questions of rationality of 3-folds based on category theory. Following work by Ballard et al., we enhance constructions of Kuznetsov by introducing Noether–Lefschetz spectra: an interplay between Orlov spectra and Hochschild homology. The main goal of this paper is to suggest a series of interesting examples where the above techniques might apply. We start by constructing a sextic double solid X with 35 nodes and torsion in H3(X, ℤ). This is a novelty: after the classical example of Artin and Mumford, this is the second example of a Fano 3-fold with a torsion in the third integer homology group. In particular, X is non-rational. We consider other examples as well: V10 with 10 singular points, and the double covering of a quadric ramified in an octic with 20 nodal singular points. After analysing the geometry of their Landau–Ginzburg models, we suggest a general non-rationality picture based on homological mirror symmetry and category theory.

scholarly journals On the capability of groups

1998 ◽  
Vol 41 (3) ◽  
pp. 487-495 ◽  
Author(s):  
Graham Ellis
Keyword(s):  
Automorphism Group ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Integral Homology ◽  
The Third ◽  
Inner Automorphism

We show how the third integral homology of a group plays a role in determining whether a given group is isomorphic to an inner automorphism group. Various necessary conditions, and sufficient conditions, for the existence of such an isomorphism are obtained.

Abelian Groups with a Vanishing Homology Group

1969 ◽  
Vol 21 ◽  
pp. 406-409 ◽  
Author(s):  
James A. Schafer
Keyword(s):  
Homology Group ◽  
Abelian Groups ◽  
Additive Group ◽  
Rational Numbers ◽  
Complete Characterization ◽  
Integral Homology ◽  
Positive Dimension ◽  
Homology Groups

In this paper, we wish to characterize those abelian groups whose integral homology groups vanish in some positive dimension. We obtain a complete characterization provided the dimension in which the homology vanishes is odd; in fact, we prove that the only abelian groups which possess a vanishing homology group in an odd dimension are, up to isomorphism, subgroups of Qn, where Q denotes the additive group of rational numbers. The case of vanishing in an even dimension is much more complicated. We exhibit a class of groups whose homology vanishes in even dimensions and is otherwise very nice, namely the subgroups of Q/Z, and then show that unless we impose further restrictions, there exist abelian groups which possess the homology of subgroups of Q/Z without being isomorphic to a subgroup of Q/Z.

scholarly journals Arrangements of lines and monodromy of associated Milnor fibers

2016 ◽  
Vol 25 (12) ◽  
pp. 1642014 ◽  
Author(s):  
Mario Salvetti ◽  
Matteo Serventi
Keyword(s):  
Betti Number ◽  
Open Problem ◽  
Homology Group ◽  
Commutator Subgroup ◽  
Milnor Fiber ◽  
Double Points ◽  
Arrangements Of Lines

Consider an arrangement [Formula: see text] of homogeneous hyperplanes in [Formula: see text] with complement [Formula: see text]. The (co)homology of [Formula: see text] with twisted coefficients is strictly related to the cohomology of the Milnor fiber associated to the natural fibration onto [Formula: see text] endowed with the geometric monodromy. It is still an open problem to understand in general the cohomology of the Milnor fiber, even for dimension 1. In Sec. 1, we show that all questions about the first homology group are detected by a precise group, which is a quotient ot the commutator subgroup of [Formula: see text] by the commutator of its length zero subgroup, which didn’t appear in the literature before. In Sec. 2, we state a conjecture of [Formula: see text]-monodromicity for the first homology, which is of a different nature with respect to the known results. Let [Formula: see text] be the graph of double points of [Formula: see text] we conjecture that if [Formula: see text] is connected, then the geometric monodromy acts trivially on the first homology of the Milnor fiber (so the first Betti number is combinatorially determined in this case). This conjecture depends only on the combinatorics of [Formula: see text]. We show the truth of the conjecture under some stronger hypotheses.

Families of smooth hypersurfaces on certain compact homogeneous complex manifolds

1983 ◽  
Vol 93 (2) ◽  
pp. 315-321
Author(s):  
Ciprian Borcea
Keyword(s):  
Complex Manifold ◽  
Betti Number ◽  
Homology Group ◽  
Cell Decomposition ◽  
Homogeneous Manifold ◽  
Holomorphic Maps ◽  
Algebraic Sets ◽  
Homogeneous Complex ◽  
Homogeneous Complex Manifold ◽  
Lower Dimensional

Let X be a compact connected homogeneous complex manifold, which is Kāhlerian and has the second Betti number equal to one: b2(X) = 1; dimcX ≥ 3.It is known that these conditions imply the following: X is a projective-rational homogeneous manifold (see (3)); X has an ‘algebraic cell-decomposition’: the 2s-dimensional closed cells are s-dimensional irreducible algebraic sets in X and they form a basis for the 2s-homology group of X, s = 1, 2, …, dimcX (see (1)); there are no holomorphic maps of X on lower dimensional (normal) analytic spaces except constants (see (9)).

scholarly journals On the Computation of Certain Homotopical Functors

1998 ◽  
Vol 1 ◽  
pp. 25-41 ◽  
Author(s):  
Graham Ellis
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Integral Homology ◽  
Finitely Presented Groups ◽  
The Third ◽  
Finite Module ◽  
Nonabelian Tensor ◽  
A Finite Group ◽  
Finitely Presented

AbstractThis paper provides details of a Magma computer program for calculating various homotopy-theoretic functors, defined on finitely presented groups. A copy of the program is included as an Add-On. The program can be used to compute: the nonabelian tensor product of two finite groups, the first homology of a finite group with coefficients in the arbirary finite module, the second integral homology of a finite group relative to its normal subgroup, the third homology of the finite p-group with coefficients in Zp, Baer invariants of a finite group, and the capability and terminality of a finite group. Various other related constructions can also be computed.

scholarly journals ON SASAKIAN–EINSTEIN GEOMETRY

2000 ◽  
Vol 11 (07) ◽  
pp. 873-909 ◽  
Author(s):  
CHARLES P. BOYER ◽  
KRZYSZTOF GALICKI
Keyword(s):  
Betti Number ◽  
Einstein Manifolds ◽  
Smooth Manifolds ◽  
Topological Monoid

We introduce a multiplication ⋆ (we call it a join) on the space of all compact Sasakian-Einstein orbifolds [Formula: see text] and show that [Formula: see text] has the structure of a commutative associative topological monoid. The set [Formula: see text] of all compact regular Sasakian–Einstein manifolds is then a submonoid. The set of smooth manifolds in [Formula: see text] is not closed under this multiplication; however, the join [Formula: see text] of two Sasakian–Einstein manifolds is smooth under some additional conditions which we specify. We use this construction to obtain many old and new examples of Sasakain–Einstein manifolds. In particular, in every odd dimension greater that five we obtain spaces with arbitrary second Betti number.

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