Geodesic
Fibration of Classifying Spaces in the Cobordism Theory of Singular Maps
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Singularities and applications, Tome 267 (2009), pp. 280-287.

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In the cobordism theory of singular smooth maps there exist classifying spaces (analogues of Thom spectra) depending on the set of allowed singularity types. The so-called “key fibration” introduced by A. Szűcs connects these classifying spaces for different sets of allowed singularities. Here we prove the existence of such a fibration using a new, more simple and general argument than that of Szűcs. This makes it possible to extend the range of applications to some negative codimension maps. 
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T. Terpai. Fibration of Classifying Spaces in the Cobordism Theory of Singular Maps. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Singularities and applications, Tome 267 (2009), pp. 280-287. http://geodesic.mathdoc.fr/item/TM_2009_267_a21/

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