Singularities and stable homotopy groups of spheres. II

András Szűcs and Tamás Terpai

Journal of Singularities
volume 17 (2018), 28-57

Received: 16 June 2015. Received in revised form: 10 February 2018

DOI: 10.5427/jsing.2018.17b

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Abstract:

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.

Keywords:

Global singularity theory, cobordisms of singular maps, stable homotopy theory

2010 Mathematical Subject Classification:

primary 57R45, secondary 57R90, 55P42, 55T25

Author(s) information:

András Szűcs	Tamás Terpai
Department of Analysis	Mathematical Institute
Eötvös Loránd University	Hungarian Academy of Sciences
Budapest, Pázmány P. sétény I/C	Budapest, Reáltanoda u. 13-15
H-1117 Hungary	H-1053 Hungary
email: szucs@cs.elte.hu	email: terpai@renyi.mta.hu