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A generalization of Thom’s transversality theorem

Lukáš Vokřínek (2008)

Archivum Mathematicum

We prove a generalization of Thom’s transversality theorem. It gives conditions under which the jet map f * | Y : Y J r ( D , M ) J r ( D , N ) is generically (for f : M N ) transverse to a submanifold Z J r ( D , N ) . We apply this to study transversality properties of a restriction of a fixed map g : M P to the preimage ( j s f ) - 1 ( A ) of a submanifold A J s ( M , N ) in terms of transversality properties of the original map f . Our main result is that for a reasonable class of submanifolds A and a generic map f the restriction g | ( j s f ) - 1 ( A ) is also generic. We also present an example of A where the...

A regularity lemma for functions of several variables.

Jean L. Journé (1988)

Revista Matemática Iberoamericana

We shall prove the following Theorem. Let Fs and Fu be two continuous transverse foliations with uniformly smooth leaves, of some manifold. If f is uniformly smooth along the leaves of Fs and Fu, then f is smooth.

Compositions of equi-dimensional fold maps

Yoshihiro Hirato, Masamichi Takase (2012)

Fundamenta Mathematicae

According to Ando's theorem, the oriented bordism group of fold maps of n-manifolds into n-space is isomorphic to the stable n-stem. Among such fold maps we define two geometric operations corresponding to the composition and to the Toda bracket in the stable stem through Ando's isomorphism. By using these operations we explicitly construct several fold maps with convenient properties, including a fold map which represents the generator of the stable 6-stem.

Fibrations sur le cercle et surfaces complexes

Anne Pichon (2001)

Annales de l’institut Fourier

Nous donnons des conditions nécessaires et suffisantes pour qu’une variété de dimension 3 se réalise comme bord d’une famille dégénérée de courbes complexes, et pour qu’un entrelacs dans une 3-variété se réalise comme bord d’un germe de fonction analytique en un point d’une surface complexe normale. Ces résultats s’appuient sur une étude des objets topologiques fournis par de telles fonctions holomorphes : soit M une variété de Waldhausen et soit L une union finie, éventuellement vide, de fibres...

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