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Cobordisms of fold maps and maps with prescribed number of cusps

Ekholm, Tobias
Szűcs, András
Published in Kyushu Journal of Mathematics. 2007, vol. 61, no. 2, p. 395-414
Abstract A generic smooth map of a closed $2k$-manifold into $(3k-1)$-space has a finite number of cusps ($Sigma^{1,1}$-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities ($Sigma^{1,0}$-singularities). Two fold maps are fold bordant if there are cobordisms between their source- and target manifolds with a fold map extending the two maps between the boundaries, if the two targets agree and the target cobordism can be taken as a product with a unit interval then the maps are fold cobordant. We compute the cobordism groups of fold maps of $(2k-1)$-manifolds into $(3k-2)$-space. Analogous cobordism semi-groups for arbitrary closed $(3k-2)$-dimensional target manifolds are endowed with Abelian group structures and described. Fold bordism groups in the same dimensions are described as well.
Keywords smooth manifoldsfold mapssingularitiescobordism
Note 14 pages, 1 figure
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EKHOLM, Tobias, SZŰCS, András, TERPAI, Tamas. Cobordisms of fold maps and maps with prescribed number of cusps. In: Kyushu Journal of Mathematics, 2007, vol. 61, n° 2, p. 395-414.

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Deposited on : 2010-10-12

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