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Calculation of the avoiding ideal for \(\Sigma^{1,1}\). (English) Zbl 1169.57028
Golasiński, Marek (ed.) et al., Algebraic topology – old and new. M. M. Postnikov memorial conference, Będlewo, Poland, June 18–24, 2007. Warsaw: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-04-1/pbk). Banach Center Publications 85, 307-313 (2009).
The author carries out an investigation on the Kazarian spaces [M. E. Kazarian, The Arnold-Gelfand mathematical seminars, Boston, MA: Birkhäuser, 325–340 (1997; Zbl 0872.57034)] of the classes \( K^0 \approx BO(k)\) of immersions, \(K^{1,0}\) of locally stable maps without \(\Sigma^{1,1}\) singularities and \( K^\infty \approx BO\) of maps without any constraints on their singularities. More specifically, the author investigates the natural embeddings \(K^0 \overset{u}{\longrightarrow} K^{1,0}\overset{g}{ \longrightarrow} K^\infty\) and \(\overline{u} = g \circ u\), the induced map in cohomology \(g^*: H^*(BO, \mathbb{Z}_2) \longrightarrow H^*(K^{1,0},\mathbb{ Z}_2)\) is calculated and a generating system of its kernel is obtained, which characterizes the avoiding ideal for the singularity \(\Sigma^{1,1}\).
As a consequence the author presents the following interesting result: “The avoiding ideal for the singularity \(\Sigma^{1,1}\) consists of elements \(\sum_{I \in \mathcal{I}} w_I\) such that \(\sum_{I \in \mathcal{I}} c^S w_k^{\mid{I^+}\mid} w_{I\backslash I^+ }=0\), where \(\mathcal{I}\) contains only index sets \(I\) with max \(I >k\), \(I^+\) denotes \(\bigcup \{ J \subseteq I \mid \min J > k \}\) and \( S= \sum_{ i \in I^+}(i-k)\)”. He also presents bounds on the codimension of fold maps from real projective spaces to Euclidean space.
For the entire collection see [Zbl 1162.00013].

57R45 Singularities of differentiable mappings in differential topology
57R19 Algebraic topology on manifolds and differential topology
57R90 Other types of cobordism
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