Multiplicative properties of Morin maps

Lippner, Gábor; Szűcs, András

doi:10.2140/agt.2010.10.1437

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Multiplicative properties of Morin maps

Gábor Lippner and András Szűcs

Algebraic & Geometric Topology 10 (2010) 1437–1454

DOI: 10.2140/agt.2010.10.1437

Abstract

In the first part of the paper we construct a ring structure on the rational cobordism classes of Morin maps (ie smooth generic maps of corank 1). We show that associating to a Morin map its $Σ^{1_{r}}$ (or $A_{r}$ ) singular strata defines a ring homomorphism to $Ω_{*} \otimes ℚ$ , the rational oriented cobordism ring. This is proved by analyzing the multiple-point sets of a product immersion. Using these homomorphisms we compute the ring of Morin maps.

In the second part of the paper we give a new method to find the oriented Thom polynomial of the $Σ^{2}$ singularity type with $ℚ$ coefficients. Then we provide a product formula for the $Σ^{2}$ singularity in $ℚ$ and the $Σ^{1, 1}$ singularity in $ℤ_{2}$ coefficients.

Keywords

product map, Morin singularity

Mathematical Subject Classification 2000

Primary: 57R20, 57R42, 57R45

References

Publication

Received: 8 August 2008

Revised: 19 May 2010

Accepted: 20 May 2010

Published: 1 July 2010

Authors

Gábor Lippner
Department of Mathematics Harvard University One Oxford Street Cambridge 02138 United States

András Szűcs
Department of Analysis Eotvos University Pazmany Peter setany 1/c Budapest 1117 Hungary

Volume 10, issue 3 (2010)