Geodesic
Resolution of corank 1 singularities in the image of a stable smooth map to a space of higher dimension
Izvestiya. Mathematics , Tome 71 (2007) no. 2, pp. 391-437.

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We consider stable smooth maps from closed smooth manifolds to smooth manifolds of higher dimension. For maps with corank 1 singularities, we find universal linear relations between the Euler characteristics of the manifolds of singularities in their images. The calculations are based on resolving the singularities by a construction that generalizes the iteration principle from algebraic geometry. 
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V. D. Sedykh. Resolution of corank 1 singularities in the image of a stable smooth map to a space of higher dimension. Izvestiya. Mathematics , Tome 71 (2007) no. 2, pp. 391-437. http://geodesic.mathdoc.fr/item/IM2_2007_71_2_a5/

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