scholarly journals Genus Integration, Abelianization, and Extended Monodromy

2019 ◽  
Author(s):  
Ivan Contreras ◽  
Rui Loja Fernandes
Keyword(s):  
Path Space ◽  
Lie Algebroid ◽  
Connected Manifold ◽  
Arbitrary Genus ◽  
Space Construction ◽  
Monodromy Groups ◽  
Simply Connected

Abstract Given a Lie algebroid we discuss the existence of a smooth abelian integration of its abelianization. We show that the obstructions are related to the extended monodromy groups introduced recently in [9]. We also show that this groupoid can be obtained by a path-space construction, similar to the Weinstein groupoid of [6], but where the underlying homotopies are now supported in surfaces with arbitrary genus. As an application, we show that the prequantization condition for a (possibly non-simply connected) manifold is equivalent to the smoothness of an abelian integration. Our results can be interpreted as a generalization of the classical Hurewicz theorem.

scholarly journals Singularity of the varieties of representations of lattices in solvable Lie groups

2016 ◽  
Vol 08 (02) ◽  
pp. 273-285 ◽  
Author(s):  
Hisashi Kasuya
Keyword(s):  
Lie Group ◽  
Vector Bundles ◽  
Similar Technique ◽  
Trivial Representation ◽  
Nilpotent Lie Algebra ◽  
Solvable Lie Groups ◽  
Analytic Germ ◽  
Space Construction ◽  
Simply Connected ◽  
Solvable Lie Group

For a lattice [Formula: see text] of a simply connected solvable Lie group [Formula: see text], we describe the analytic germ in the variety of representations of [Formula: see text] at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by [Formula: see text]. By this description, under certain assumption, we study the singularity of the analytic germ in the variety of representations of [Formula: see text] at the trivial representation by using the Kuranishi space construction. By a similar technique, we also study deformations of holomorphic structures of trivial vector bundles over complex parallelizable solvmanifolds.

scholarly journals Yang–Mills theory and jumping curves

2015 ◽  
Vol 26 (05) ◽  
pp. 1550029
Author(s):  
Yasha Savelyev
Keyword(s):  
Vector Bundle ◽  
Holomorphic Vector Bundle ◽  
Chain Complex ◽  
Complex Manifolds ◽  
Smooth Manifolds ◽  
Hermitian Vector Bundle ◽  
Connected Manifold ◽  
Classical Sense ◽  
Yang Mills ◽  
Simply Connected

We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang–Mills theory over S2 to show that any non-trivial, smooth Hermitian vector bundle E over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-trivial holomorphic vector bundle must have jumping curves in the classical sense (when c1(E) is zero). We also use this to give a new proof of a theorem of Gromov on the norm of curvature of unitary connections, and make the theorem slightly sharper. Lastly we define a sequence of new non-trivial integer invariants of smooth manifolds, connected to this theory of smooth jumping curves, and make some computations of these invariants. Our methods include an application of the recently developed Morse–Bott chain complex for the Yang–Mills functional over S2.

scholarly journals Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

2020 ◽  
Author(s):  
Eric Schippers ◽  
Mohammad Shirazi ◽  
Wolfgang Staubach
Keyword(s):  
Riemann Surface ◽  
Bergman Space ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Dirichlet Space ◽  
Holomorphic Functions ◽  
Finite Collection ◽  
Approximation Theorems ◽  
Arbitrary Genus ◽  
Simply Connected

Abstract We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface $$\Sigma $$ Σ , and the union $$\mathcal {O}$$ O of a finite collection of simply connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $$\mathcal {O}$$ O to the Bergman space of holomorphic forms on $$\Sigma $$ Σ is an isomorphism onto the exact one-forms, when restricted to the orthogonal complement of the set of forms on all of R. We then apply this to prove versions of the Plemelj–Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $$\Sigma $$ Σ by elements of Bergman space and Dirichlet space on fixed regions in R containing $$\Sigma $$ Σ .

scholarly journals In simply connected cotangent bundles, exact Lagrangian cobordisms are h-cobordisms

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Hiro Lee Tanaka
Keyword(s):  
Connected Manifold ◽  
Cotangent Bundles ◽  
Lagrangian Cobordisms ◽  
Simply Connected ◽  
Lagrangian Cobordism

Abstract Let Q be a simply connected manifold. We show that every exact Lagrangian cobordism between compact, exact Lagrangians in T*Q is an h-cobordism. This is a corollary of the Abouzaid–Kragh Theorem.

A Note on Involutions with a Finite Number of Fixed Points

1968 ◽  
Vol 20 ◽  
pp. 1522-1530
Author(s):  
John D. Miller
Keyword(s):  
Finite Number ◽  
Fixed Points ◽  
Connected Manifold ◽  
Simply Connected

LetMbe a smooth, closed, simply connected manifold of dimension greater than 5. LetTbe an involution onMwith a positive, finite number of fixed points. Our aim in this paper is to prove the following theorem (which is somewhat like that of Wasserman (7)).

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Grzegorz Graff ◽  
Agnieszka Kaczkowska
Keyword(s):  
Natural Number ◽  
Homotopy Class ◽  
Periodic Points ◽  
Topological Invariant ◽  
Periodic Sequence ◽  
Connected Manifold ◽  
Lefschetz Numbers ◽  
Simply Connected Manifolds ◽  
Simply Connected

AbstractLet f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f.In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH 2(M; ℚ) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.

scholarly journals The space of immersed surfaces in a manifold

2013 ◽  
Vol 154 (3) ◽  
pp. 419-438 ◽  
Author(s):  
OSCAR RANDAL–WILLIAMS
Keyword(s):  
Stable Range ◽  
Connected Manifold ◽  
Immersed Surfaces ◽  
Rational Cohomology ◽  
Simply Connected

AbstractWe study the cohomology of the space of immersed genus g surfaces in a simply-connected manifold. We compute the rational cohomology of this space in a stable range which goes to infinity with g. In fact, in this stable range we are also able to obtain information about torsion in the cohomology of this space, as long as we localise away from (g-1).

scholarly journals Integrable lifts for transitive Lie algebroids

2018 ◽  
Vol 29 (09) ◽  
pp. 1850062 ◽  
Author(s):  
Iakovos Androulidakis ◽  
Paolo Antonini
Keyword(s):  
Extra Dimensions ◽  
Lie Algebroids ◽  
The Other ◽  
Lie Algebroid ◽  
Base Manifold ◽  
Lie Groupoid ◽  
Finite Dimensional ◽  
Other Hand ◽  
De Rham ◽  
Simply Connected

Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid is the quotient of a finite-dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an “Almeida–Molino” integrable lift when the base manifold is simply connected. On the other hand, we notice that the classical de Rham isomorphism provides a universal integrable algebroid. Using it we construct a “de Rham” integrable lift for any given transitive Abelian Lie algebroid.

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