Geodesic
Singularities and stable homotopy groups of spheres. II
Journal of Singularities, Tome 17 (2018), pp. 28-57

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We consider compositions of immersions of n-manifolds into R^{n+2} with a projection to a hyperplane, and investigate the cobordism groups of such maps when we allow only a given finite set of (Morin) singularities. The classifying spaces for these cobordism groups allow a concrete description. The spectral sequence computing these groups can be identified with that arising from the filtration of complex projective spaces in the extraordinary homology theory formed by stable homotopy groups. The differentials of this spectral sequence describe the incidences of the different singularity strata. These incidence classes are described by elements of the stable homotopy groups of spheres and turn out to have surprising periodicity properties. The first non-zero such incidence class always belongs to the image of the J-homomorphism, which is cyclic. Combining the results of Mosher, Adams and Atiyah the element that describes the incidence can be calculated exactly.
DOI : 10.5427/jsing.2018.17b
Classification : primary57R45, secondary57R90, 55P42, 55T25 
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     author = {Andr\'as Sz\'{u}cs and Tam\'as Terpai},
     title = {Singularities and stable homotopy groups of spheres. {II}},
     journal = {Journal of Singularities},
     pages = {28--57},
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     year = {2018},
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András Szűcs; Tamás Terpai. Singularities and stable homotopy groups of spheres. II. Journal of Singularities, Tome 17 (2018), pp. 28-57. doi: 10.5427/jsing.2018.17b

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