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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].

##### MSC:
 57R45 Singularities of differentiable mappings in differential topology 57R19 Algebraic topology on manifolds and differential topology 57R90 Other types of cobordism
##### Keywords:
avoiding ideal; cusp; fold map
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