scholarly journals On computing homology gradients over finite fields

2016 ◽  
Vol 162 (3) ◽  
pp. 507-532 ◽  
Author(s):  
ŁUKASZ GRABOWSKI ◽  
THOMAS SCHICK
Keyword(s):  
Finite Fields ◽  
Group Ring ◽  
Spectral Measure ◽  
Irrational Number ◽  
Betti Numbers ◽  
Universal Cover ◽  
Specific Method ◽  
Arbitrary Characteristic ◽  
Cw Complex ◽  
Atiyah Conjecture

AbstractRecently the so-called Atiyah conjecture about l2-Betti numbers has been disproved. The counterexamples were found using a specific method of computing the spectral measure of a matrix over a complex group ring. We show that in many situations the same method allows to compute homology gradients, i.e. generalisations of l2-Betti numbers to fields of arbitrary characteristic. As an application we point out that (i) the homology gradient over any field of characteristic different than 2 can be an irrational number, and (ii) there exists a finite CW-complex with the property that the homology gradients of its universal cover taken over different fields have infinitely many different values.

VANISHING RESULTS FOR L2-BETTI NUMBERS AND L2-EULER CHARACTERISTICS AND THEIR APPLICATIONS

2008 ◽  
Vol 19 (01) ◽  
pp. 21-26 ◽  
Author(s):  
JANG HYUN JO
Keyword(s):  
Normal Subgroup ◽  
Euler Characteristic ◽  
Betti Numbers ◽  
Universal Cover ◽  
Euler Characteristics ◽  
Cw Complex

A CW-complex X is called a [G,m]-complex if X is an m-dimensional complex with π1(X) ≅ G and the universal cover [Formula: see text] is (m - 1)-connected. We show that if G has an infinite amenable normal subgroup, then the asphericity of a [G,m]-complex X is equivalent to the vanishing of L2-Euler characteristic of [Formula: see text]. This result corresponds to a generalization and a variation of earlier several works. Also, we show that the L2-Betti numbers of a group which belongs to the class of groups K𝔉 eventually vanish. As a byproduct, we give an example of a group which belongs to the class of groups H𝔉 but does not belong to the class of groups K𝔉.

scholarly journals The number of idempotents in abelian group rings

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 719-723
Author(s):  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Group Rings ◽  
Arbitrary Characteristic

Suppose that R is a commutative unitary ring of arbitrary characteristic and G is a multiplicative abelian group. Our main theorem completely determines the cardinality of the set id(RG), consisting of all idempotent elements in the group ring RG. It is explicitly calculated only in terms associated with R, G and their divisions. This result strengthens previous estimates obtained in the literature recently.

scholarly journals A Gorenstein numerical semi-group ring having a transcendental series of Betti numbers

2015 ◽  
Vol 219 (3) ◽  
pp. 591-621
Author(s):  
Clas Löfwall ◽  
Samuel Lundqvist ◽  
Jan-Erik Roos
Keyword(s):  
Group Ring ◽  
Betti Numbers ◽  
Semi Group

A Note on Relations Between the Zeta-Functions of Galois Coverings of Curves Over Finite Fields

1990 ◽  
Vol 33 (3) ◽  
pp. 282-285 ◽  
Author(s):  
Amilcar Pacheco
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Group Ring ◽  
Algebraic Curve ◽  
Zeta Functions ◽  
Finite Subgroup ◽  
Rational Group ◽  
Galois Coverings ◽  
Curves Over Finite Fields ◽  
Special Case

AbstractLet C be a complete irreducible nonsingular algebraic curve defined over a finite field k. Let G be a finite subgroup of the group of automorphisms Aut(C) of C. We prove that certain idempotent relations in the rational group ring Q[G] imply other relations between the zeta-functions of the quotient curves C/H, where H is a subgroup of G. In particular we generalize some results of Kani in the special case of curves over finite fields.

Crossed complexes and chain complexes with operators

1990 ◽  
Vol 107 (1) ◽  
pp. 33-57 ◽  
Author(s):  
Ronald Brown ◽  
Philip J. Higgins
Keyword(s):  
Fundamental Group ◽  
Algebraic Topology ◽  
Chain Complex ◽  
Universal Cover ◽  
Cw Complex ◽  
Chain Complexes

Chain complexes with a group of operators are a well known tool in algebraic topology, where they arise naturally as the chain complex of cellular chains of the universal cover of a reduced CW-complex X. The group of operators here is the fundamental group of X.

scholarly journals Symmetry and parity in Frobenius action on cohomology

2011 ◽  
Vol 148 (1) ◽  
pp. 295-303 ◽  
Author(s):  
Junecue Suh
Keyword(s):  
Finite Fields ◽  
Characteristic Zero ◽  
Betti Numbers ◽  
Newton Polygons ◽  
Smooth Variety ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Crystalline Cohomology ◽  
Lefschetz Theorem ◽  
Hard Lefschetz Theorem

AbstractWe prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincaré duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz.

scholarly journals Volume and macroscopic scalar curvature

2021 ◽  
Author(s):  
Sabine Braun ◽  
Roman Sauer
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Upper Bound ◽  
Betti Numbers ◽  
Universal Cover ◽  
Positive Scalar Curvature ◽  
Simplicial Volume ◽  
Positive Scalar ◽  
Curvature Bound

AbstractWe prove the macroscopic cousins of three conjectures: (1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, (2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, (3) a conjectural bound of $$\ell ^2$$ ℓ 2 -Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover.

scholarly journals Novikov homology, jump loci and Massey products

2014 ◽  
Vol 12 (9) ◽  
Author(s):  
Toshitake Kohno ◽  
Andrei Pajitnov
Keyword(s):  
Finite Number ◽  
Spectral Sequence ◽  
Cohomology Class ◽  
Betti Numbers ◽  
Local System ◽  
Generic Point ◽  
Massey Products ◽  
Cw Complex ◽  
Local Coefficients ◽  
Novikov Ring

AbstractLet X be a finite CW complex, and ρ: π 1(X) → GL(l, ℂ) a representation. Any cohomology class α ∈ H 1(X, ℂ) gives rise to a deformation γ t of ρ defined by γ t (g) = ρ(g) exp(t〈α, g〉). We show that the cohomology of X with local coefficients γ gen corresponding to the generic point of the curve γ is computable from a spectral sequence starting from H*(X, ρ). We compute the differentials of the spectral sequence in terms of the Massey products and show that the spectral sequence degenerates in case when X is a Kähler manifold and ρ is semi-simple.If α ∈ H 1(X, ℝ) one associates to the triple (X, ρ, α) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the local system γ gen. We investigate the dependence of these numbers on α and prove that they are constant in the complement to a finite number of proper vector subspaces in H 1(X, ℝ).

scholarly journals Commutative group algebras and Prufer groups

2010 ◽  
Vol 41 (1) ◽  
pp. 85-95
Author(s):  
P. V. Danchev
Keyword(s):  
Group Ring ◽  
Quotient Group ◽  
Commutative Group ◽  
Group Algebras ◽  
Direct Factor ◽  
Perfect Field ◽  
Characteristic P ◽  
Arbitrary Characteristic ◽  
Direct Sum ◽  
Normalized Units

Suppose $G$ is a multiplicatively written abelian $p$-group, where $p$ is a prime, and $F$ is a field of arbitrary characteristic. The main results in this paper are that none of the Sylow $p$-group of all normalized units $S(FG)$ in the group ring $FG$ and its quotient group $S(FG)/G$ cannot be Prufer groups. This contrasts a classical conjecture for which $S(FG)/G$ is a direct factor of a direct sum of generalized Prufer groups whenever $F$ is a perfect field of characteristic $p$.

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