Topic

# Degree of a continuous mapping

About: Degree of a continuous mapping is a research topic. Over the lifetime, 139 publications have been published within this topic receiving 3381 citations. The topic is also known as: degree (continuous map) & degree of a mapping.

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01 Jan 1965

TL;DR: The fundamental theorem of algebra and its application to smooth manifolds and smooth maps was proved by Sard and Brown as discussed by the authors, and the Brouwer degree modulo 2 of a mapping was shown to be equivalent to the Hopf theorem.

Abstract: Preface1Smooth manifolds and smooth maps1Tangent spaces and derivatives2Regular values7The fundamental theorem of algebra82The theorem of Sard and Brown10Manifolds with boundary12The Brouwer fixed point theorem133Proof of Sard's theorem164The degree modulo 2 of a mapping20Smooth homotopy and smooth isotopy205Oriented manifolds26The Brouwer degree276Vector fields and the Euler number327Framed cobordism the Pontryagin construction42The Hopf theorem508Exercises52AppClassifying 1-manifolds55Bibliography59Index63

1,358 citations

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TL;DR: Brezis and Nirenberg as mentioned in this paper considered a class of maps u from a bounded domain Ω ⊂ R into R. In classical degree theory, for u ∈ C(Ω,R), the degree of u at a point

Abstract: II.0. Introduction This is a continuation of H. Brezis and L. Nirenberg [1] (= [BNI]), and we will often refer to concepts and results in that paper. There, we extended degree theory to VMO maps between compact n-dimensional oriented manifolds without boundaries. In this paper we consider a class of maps u from a bounded domain Ω ⊂ R into R. In classical degree theory, for u ∈ C(Ω,R), the degree of u at a point

292 citations

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TL;DR: In this paper, it was shown that a compact complex manifold is projective if and only if it admits a Kahler form whose cohomology class is integral, which suggests that Kahler geometry is an extension of projective geometry, obtained by relaxing the integrality condition on the Kahler class.

Abstract: The celebrated Kodaira theorem [6] says that a compact complex manifold is projective if and only if it admits a Kahler form whose cohomology class is integral. This suggests that Kahler geometry is an extension of projective geometry, obtained by relaxing the integrality condition on a Kahler class. This point of view, together with the many restrictive conditions on the topology of Kahler manifolds provided by Hodge theory (the strongest one being the formality theorem [4]), would indicate that compact Kahler manifolds and complex projective ones cannot be distinguished by topological invariants. This is supported by the results known for Kahler surfaces, for which a much stronger statement is known, as a consequence of Kodaira’s classification : recall first that two compact complex manifolds X and X ′ are said to be deformation equivalent if there exist a family π : X → B, where B is a connected analytic space and π is smooth and proper, and two points b, b′ ∈ B such that Xb ∼= X, Xb′ ∼= X ′.

104 citations

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TL;DR: In this article, the action of the diffeomorphism group of a non-compact paracompact oriented manifold on the space of C00 volume forms on the manifold and the existence of volume-form-preserving embedding dings of such manifolds into euclidean spaces are discussed.

Abstract: The theorem of J. Moser that any two volume elements of equal total volume on a compact manifold are diffeomorphism-equivalent is extended to noncompact manifolds: A necessary and sufficient condition (equal total and same end behavior) is given for diffeomorphism equiva- lence of two volume forms on a noncompact manifold. Results on the existence of embeddings and immersions with the property of inducing a given volume form are also given. Generalizations to nonorientable mani- folds and manifolds with boundary are discussed. The topics of this paper are the action of the diffeomorphism group of a noncompact paracompact oriented manifold on the space of C00 volume forms on the manifold and the existence of volume-form-preserving embed- dings of such manifolds into euclidean spaces. The results are essentially generalizations to the case of noncompact manifolds of a theorem of Moser (6) and a corollary of that theorem. The theorem is that if M is a compact connected oriented manifold and if w and t are C°° volume forms on M such that JMu> = Jmt then there is a C°° diffeomorphism M such that tp*r = w. The corollary is that if : M -> M is a diffeomorphism such that

91 citations

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12 Nov 2009TL;DR: In this paper, the Brouwer degree is defined for regular values of smooth mappings between smooth oriented manifolds of the same dimension, as the sum of signs of the Jacobians over the inverse image set, extended to nonregular values through de Rham's approach based upon differential forms, before giving the extension to continuous maps.

Abstract: The literature devoted to degree theory and its applications is abundant, but the richness of the topics is such that it is not surprising to see regularly the publication of new books in this area. The emphasis of the present one is on Brouwer degree considered from the viewpoint of differential topology, and the applications have essentially a topological flavour. The book starts with an interesting chapter devoted to the history of the concept of degree, inspired by and completing H.-W. Siegberg's article [Amer. Math. Monthly 88 (1981), no. 2, 125–139], and which is, as justly observed by the authors, `biased by their personal opinions and preferences'. After a second chapter recalling the definition and basic properties of manifolds and their mappings, the degree is defined first for regular values of smooth mappings between smooth oriented manifolds of the same dimension, as the sum of signs of the Jacobians over the inverse image set. The notion is extended to nonregular values through de Rham's approach based upon differential forms, before giving the extension to continuous maps. An interesting application is given to the Hopf invariant before more classical ones to the Jordan separation theorem and Brouwer fixed point theorems on a ball and on a sphere. Chapter IV develops the Brouwer degree for continuous mappings of the closure of a bounded open set of a Euclidean space into this space, in a now-classical analytical way. Chapter V is somewhat less standard, by providing a proof of Hopf's result that the degree is the only homotopy invariant for spheres. This chapter ends with a study of gradient vector fields and Hopf fibrations. The book contains a series of interesting exercises and problems, a list of names of mathematicians cited, historical references, a bibliography restricted to some twenty books, a list of symbols and an index. It is an interesting contribution to the literature, trying to give `the simplest possible presentation at the lowest technical cost'.

90 citations