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Dans cet article, on montre comment le cobordisme d’applications et le cobordisme fibré fournissent les obstructions à des problèmes de lissage topologique de singularités avec un lieu singulier compact. On calcule dans le cas des petites dimensions les groupes de cobordisme fibré. Les résultats connus sur le cobordisme fibré ou sur son image dans le cobordisme d’application permettent le calcul d’un certain nombre de ces obstructions.

Cobordismes de plongments et produits homotopiques

O. Burlet (1971)

Commentarii mathematici Helvetici

Cobordismo di immersioni di codimensione uno

Rosa Gini (2001)

Bollettino dell'Unione Matematica Italiana

Cobordisms and Reidemeister torsions of homotopy lens spaces.

Gadgil, Siddhartha (2001)

Geometry & Topology

Cobordisms of fold maps of 2k+2-manifolds into $ℝ^{3 k + 2}$

Tamás Terpai (2008)

Banach Center Publications

We calculate the group of cobordisms of k-codimensional maps into Euclidean space with no singularities more complicated than fold for a 2k+2-dimensional source manifold in both oriented and unoriented cases.

Cochain operations defining Steenrod- $⌣_{i}$ -products in the bar construction.

Kadeishvili, T. (2003)

Georgian Mathematical Journal

Cocordance and Isotopy of Smooth Embeddings in Low Codimensions.

Allen E. Hatcher (1973)

Inventiones mathematicae

Cocycle invariants of codimension 2 embeddings of manifolds

Józef H. Przytycki, Witold Rosicki (2014)

Banach Center Publications

We consider the classical problem of a position of n-dimensional manifold Mⁿ in $ℝ^{n + 2}$ . We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of a knotting $M ⁿ \to ℝ^{n + 2}$ . In particular, we show that for any fixed quandle, quandle coloring, and shadow quandle coloring, of a diagram of Mⁿ embedded in $ℝ^{n + 2}$ we have (n+1)- and (n+2)-(co)cycle invariants (i.e. invariant under Roseman moves).

Codimension 2 fibrators that are closed under finite product.

Im, Young Ho, Kang, Mee Kwang, Woo, Ki Mun (1998)

International Journal of Mathematics and Mathematical Sciences

Codimension one foliations on solvable manifolds.

Shigenori Matsumoto (1993)

Commentarii mathematici Helvetici

Codimension one foliations without compact leaves.

Paul S. Schweitzer (1995)

Commentarii mathematici Helvetici

Codimension one minimal foliations and the fundamental groups of leaves

Tomoo Yokoyama, Takashi Tsuboi (2008)

Annales de l’institut Fourier

Let $ℱ$ be a transversely orientable transversely real-analytic codimension one minimal foliation of a paracompact manifold $M$ . We show that if the fundamental group of each leaf of $ℱ$ is isomorphic to $Z$ , then $ℱ$ is without holonomy. We also show that if $π_{2} (M) ≅ 0$ and the fundamental group of each leaf of $ℱ$ is isomorphic to $Z^{k}$ ( $k \in Z_{\geq 0}$ ), then $ℱ$ is without holonomy.

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