Volume 10, issue 3 (2010)

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Multiplicative properties of Morin maps

Gábor Lippner and András Szűcs

Algebraic & Geometric Topology 10 (2010) 1437–1454
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 ${\Sigma }^{{1}_{r}}$ (or ${A}_{r}$) singular strata defines a ring homomorphism to ${\Omega }_{\ast }\otimes ℚ$, 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 ${\Sigma }^{2}$ singularity type with $ℚ$ coefficients. Then we provide a product formula for the ${\Sigma }^{2}$ singularity in $ℚ$ and the ${\Sigma }^{1,1}$ singularity in ${ℤ}_{2}$ coefficients.

Keywords
product map, Morin singularity
Mathematical Subject Classification 2000
Primary: 57R20, 57R42, 57R45
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