Functions on Manifolds: Algebraic and
                    Topological Aspects

Sharko, V. V.

doi:10.1090/mmono/131

Cited by 33 publications

(49 citation statements)

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“…The algebra C ∞ (M ) comes equipped with a countable family of semi-norms, defined as usual in terms of local partial derivatives, making it into a nuclear Fréchet space whose product is continuous with respect to the resulting topology [23]. The norm Proof.…”

Section: Torus-equivariant Classical Geometrymentioning

confidence: 99%

Moduli spaces of instantons on toric noncommutative manifolds

Brain

Landi

Suijlekom

2013

Advances in Theoretical and Mathematical Physics

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We study analytic aspects of U(n) gauge theory over a toric noncommutative manifold M θ . We analyse moduli spaces of solutions to the selfdual Yang-Mills equations on U(2) vector bundles over four-manifolds M θ , showing that each such moduli space is either empty or a smooth Hausdorff manifold whose dimension we explicitly compute. In the special case of the four-sphere S 4 θ we find that the moduli space of U (2) instantons with fixed second Chern number k is a smooth manifold of dimension 8k − 3. e-print archive:

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Section: Torus-equivariant Classical Geometrymentioning

confidence: 99%

Moduli spaces of instantons on toric noncommutative manifolds

Brain

Landi

Suijlekom

2013

Advances in Theoretical and Mathematical Physics

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“…As we were able to conclude from [2], [24], [26] and discussions with colleagues, many natural questions about round functions remain unanswered even in low dimensions. Thus it seemed reasonable to begin by discussing round functions on low-dimensional manifolds.…”

Section: Introductionmentioning

confidence: 76%

“…For example, it is still unclear how to describe the class of compact closed manifolds which possess round functions. Some results about general (not necessarily Morse) round functions may be found in [1], [2], [24], [26], but to the best of our knowledge there exists no systematic exposition of this topic in the literature. The present paper may be considered as an attempt to fill this gap and provide a framework for further investigations.…”

Section: Introductionmentioning

confidence: 99%

Remarks on minimal round functions

Khimshiashvili

Siersma

2003

Geometry and Topology of Caustics – Caustics '02

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Abstract. We describe the structure of minimal round functions on compact closed surfaces and three-dimensional manifolds. The minimal possible number of critical loops is determined and typical non-equisingular round function germs are interpreted in the spirit of isolated line singularities. We also discuss a version of Lusternik-Schnirelmann theory suitable for round functions. Introduction.A differentiable function on a differentiable manifold is called a round function if its critical set is a union of embedded one-dimensional submanifolds. In general it is not assumed that the critical submanifolds are non-degenerate critical submanifolds in the sense of R. Bott [3] so this is a straightforward extension of the notion of round Morse function introduced by W. Thurston [30].Our main concern in this paper are round functions on a given compact closed (i.e., without boundary) smooth manifold with the minimal possible number of critical submanifolds (critical loops). By analogy with the terminology of [29] they will be called minimal round functions as in [21]. We are especially interested in describing possible changes of the topology of their Lebesgue sets (preimages of semi-infinite intervals) and typical local models of their singular behaviour near critical loops. Our setting and approach are much in the spirit of F. Takens' paper [29] which contains a comprehensive

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“…There are examples (of stably-free modules) where this inequality is strict [3]. Recall that a Λ-module M is called stably free if the direct sum of M and some free Λ-module F k is free.…”

Section: Stable Invariants Of Finitely Generated Modules and L 2 -Modmentioning

confidence: 99%

“…Using numerical invariants of free cochain and Hilbert complexes of the manifold M n (see [3,4]), we give an answer to this problem for i = 0, 1, n − 2, n − 1 and 3 ≤ i ≤ n − 4 in the case where dim M n ≥ 6. Thus, the only unsettled case is i = 2 (and n − 3 by duality).…”

Section: Introductionmentioning

confidence: 99%