scholarly journals MULTIPLICATIVE SUB-HODGE STRUCTURES OF CONJUGATE VARIETIES

2014 ◽  
Vol 2 ◽  
Author(s):  
STEFAN SCHREIEDER
Keyword(s):  
Fundamental Group ◽  
Equivalence Class ◽  
Hodge Structure ◽  
Quadratic Extension ◽  
Cohomology Algebra ◽  
Hodge Conjecture ◽  
Hodge Structures ◽  
Birational Equivalence ◽  
Simply Connected

AbstractFor any subfield $K\subseteq \mathbb{C}$, not contained in an imaginary quadratic extension of $\mathbb{Q}$, we construct conjugate varieties whose algebras of $K$-rational ($p,p$)-classes are not isomorphic. This compares to the Hodge conjecture which predicts isomorphisms when $K$ is contained in an imaginary quadratic extension of $\mathbb{Q}$; additionally, it shows that the complex Hodge structure on the complex cohomology algebra is not invariant under the Aut($\mathbb{C}$)-action on varieties. In our proofs, we find simply connected conjugate varieties whose multilinear intersection forms on $H^{2}(-,\mathbb{R})$ are not (weakly) isomorphic. Using these, we detect nonhomeomorphic conjugate varieties for any fundamental group and in any birational equivalence class of dimension $\geq $10.

scholarly journals Hodge correlators

2019 ◽  
Vol 2019 (748) ◽  
pp. 1-138
Author(s):  
Alexander B. Goncharov
Keyword(s):  
Fundamental Group ◽  
Hodge Structure ◽  
Euler System ◽  
Feynman Integral ◽  
Mixed Hodge Structure ◽  
Modular Curves ◽  
Hodge Structures ◽  
Motivic Cohomology ◽  
Linear Differential ◽  
Mixed Hodge Structures

Abstract Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by twistor plane. We call them twistor connections. In particular, we define the canonical period map on variations of real mixed Hodge structures. We show that the obtained period functions satisfy a simple Maurer–Cartan type non-linear differential equation. Generalizing this, we suggest a DG-enhancement of the subcategory of Saito’s Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin–Selberg integrals. Examples of the motivic correlators include Beilinson’s elements in the motivic cohomology, e.g. the ones delivering the Beilinson–Kato Euler system on modular curves.

scholarly journals TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

2021 ◽  
pp. 1-8
Author(s):  
DANIEL KASPROWSKI ◽  
MARKUS LAND
Keyword(s):  
Fundamental Group ◽  
Homotopy Equivalent ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Canonical Map ◽  
Duality Space ◽  
Simply Connected Manifolds ◽  
Simply Connected ◽  
Poincaré Duality Space

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

Non-cocompact Group Actions and -Semistability at Infinity

2019 ◽  
Vol 72 (5) ◽  
pp. 1275-1303 ◽  
Author(s):  
Ross Geoghegan ◽  
Craig Guilbault ◽  
Michael Mihalik
Keyword(s):  
Fundamental Group ◽  
Group Actions ◽  
Fundamental Property ◽  
Companion Paper ◽  
Infinite Index ◽  
Locally Compact ◽  
Finitely Presented Groups ◽  
Simply Connected ◽  
Open Question ◽  
Finitely Presented

AbstractA finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.

scholarly journals The fundamental group of an algebra with a strongly simply connected Galois covering

2019 ◽  
Vol 18 (03) ◽  
pp. 1950052
Author(s):  
Diane Castonguay ◽  
Claudia Chaio ◽  
Sonia Trepode
Keyword(s):  
Fundamental Group ◽  
Galois Covering ◽  
Triangular Algebra ◽  
Simply Connected

In this work, we prove that if a triangular algebra [Formula: see text] admits a strongly simply connected universal Galois covering for a given presentation, then the fundamental group associated to this presentation is free.

scholarly journals On the boundary of moduli spaces of log Hodge structures, II: Nontrivial torsors

2014 ◽  
Vol 213 ◽  
pp. 1-20
Author(s):  
Tatsuki Hayama
Keyword(s):  
Moduli Spaces ◽  
Hodge Structure ◽  
Hodge Structures

AbstractIn this paper, we determine when a natural torsor arising in the work of Kato and Usui on partial compactification of period domains of pure Hodge structure is trivial, and we give an application to cycle spaces.

scholarly journals On Moduli Spaces of Positive Scalar Curvature Metrics on Highly Connected Manifolds

2020 ◽  
Author(s):  
Michael Wiemeler
Keyword(s):  
Fundamental Group ◽  
Scalar Curvature ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Positive Scalar Curvature ◽  
Spin Manifold ◽  
Homotopy Groups ◽  
Positive Scalar ◽  
Simply Connected ◽  
Highly Connected Manifolds

Abstract Let $M$ be a simply connected spin manifold of dimension at least six, which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on $M$ has non-trivial higher homotopy groups. Moreover, denote by $\mathcal{M}_0^+(M)$ the moduli space of positive scalar curvature metrics on $M$ associated to the group of orientation-preserving diffeomorphisms of $M$. We show that if $M$ belongs to a certain class of manifolds that includes $(2n-2)$-connected $(4n-2)$-dimensional manifolds, then the fundamental group of $\mathcal{M}_0^+(M)$ is non-trivial.

scholarly journals MIXED HODGE STRUCTURES WITH MODULUS

2020 ◽  
pp. 1-35
Author(s):  
Florian Ivorra ◽  
Takao Yamazaki
Keyword(s):  
Hodge Structure ◽  
Mixed Hodge Structure ◽  
Hodge Structures ◽  
Classical Notion ◽  
Mixed Hodge Structures

We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension, we attach mixed Hodge structures with modulus. We combine this construction with an equivalence between the category of level one mixed Hodge structures with modulus and the category of Laumon 1-motives to generalize Kato–Russell’s Albanese varieties with modulus to 1-motives.

scholarly journals On the fundamental group of a Lie semigroup

1992 ◽  
Vol 34 (3) ◽  
pp. 379-394 ◽  
Author(s):  
Karl-Hermann Neeb
Keyword(s):  
Fundamental Group ◽  
Lie Group ◽  
Unit Group ◽  
Universal Covering ◽  
Covering Group ◽  
Lie Semigroup ◽  
Lie Semigroups ◽  
Inclusion Mapping ◽  
Image Position ◽  
Simply Connected

The simplest type of Lie semigroups are closed convex cones in finite dimensional vector spaces. In general one defines a Lie semigroup to be a closed subsemigroup of a Lie group which is generated by one-parameter semigroups. If W is a closed convex cone in a vector space V, then W is convex and therefore simply connected. A similar statement for Lie semigroups is false in general. There exist generating Lie semigroups in simply connected Lie groups which are not simply connected (Example 1.15). To find criteria for cases when this is true, one has to consider the homomorphisminduced by the inclusion mapping i:S→G, where S is a generating Lie semigroup in the Lie group G. Our main results concern the description of the image and the kernel of this mapping. We show that the image is the fundamental group of the largest covering group of G, into which S lifts, and that the kernel is the fundamental group of the inverse image of 5 in the universal covering group G. To get these results we construct a universal covering semigroup S of S. If j: H(S): = S ∩ S-1 →S is the inclusion mapping of the unit group of S into S, then it turns out that the kernel of the induced mappingmay be identfied with the fundamental group of the unit group H(S)of S and that its image corresponds to the intersection H(S)0 ⋂π1(S), where π1(s) is identified with a central subgroup of S.

FIXED POINT FREE CIRCLE ACTIONS AND FINITENESS THEOREMS

2000 ◽  
Vol 02 (01) ◽  
pp. 75-86 ◽  
Author(s):  
FUQUAN FANG ◽  
XIAOCHUN RONG
Keyword(s):  
Fixed Point ◽  
Fundamental Group ◽  
Fundamental Groups ◽  
Vanishing Theorem ◽  
Fixed Point Set ◽  
Circle Actions ◽  
Finite Fundamental Group ◽  
Finiteness Theorems ◽  
Point Set ◽  
Simply Connected

We prove a vanishing theorem of certain cohomology classes for an 2n-manifold of finite fundamental group which admits a fixed point free circle action. In particular, it implies that any Tk-action on a compact symplectic manifold of finite fundamental group has a non-empty fixed point set. The vanishing theorem is used to prove two finiteness results in which no lower bound on volume is assumed. (i) The set of symplectic n-manifolds of finite fundamental groups with curvature, λ ≤ sec ≤ Λ, and diameter, diam ; ≤ d, contains only finitely many diffeomorphism types depending only on n, λ, Λ and d. (ii) The set of simply connected n-manifolds (n ≤ 6) with λ ≤ sec ≤ Λ and diam ≤ d contains only finitely many diffeomorphism types depending only on n, λ, Λ and d.

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