#### Algebraic and Geometric Topology 3 (2003),
paper no. 26, pages 777-789.

## The Chess conjecture

### Rustam Sadykov

**Abstract**.
We prove that the homotopy class of a Morin mapping f: P^p --> Q^q
with p-q odd contains a cusp mapping. This affirmatively solves a
strengthened version of the Chess conjecture [DS Chess, A note on the
classes [S_1^k(f)], Proc. Symp. Pure Math., 40 (1983) 221-224] and [VI
Arnol'd, VA Vasil'ev, VV Goryunov, OV Lyashenko, Dynamical systems
VI. Singularities, local and global theory, Encyclopedia of
Mathematical Sciences - Vol. 6 (Springer, Berlin, 1993)]. Also, in
view of the Saeki-Sakuma theorem [O Saeki, K Sakuma, Maps with only
Morin singularities and the Hopf invariant one problem,
Math. Proc. Camb. Phil. Soc. 124 (1998) 501-511] on the Hopf invariant
one problem and Morin mappings, this implies that a manifold P^p with
odd Euler characteristic does not admit Morin mappings into R^{2k+1}
for p > 2k not equal to 1,3 or 7.
**Keywords**.
Singularities, cusps, fold mappings, jets

**AMS subject classification**.
Primary: 57R45.
Secondary: 58A20, 58K30.

**DOI:** 10.2140/agt.2003.3.777

**E-print:** `arXiv:math.GT/0301371`

Submitted: 18 February 2003.
(Revised: 23 July 2003.)
Accepted: 19 August 2003.
Published: 24 August 2003.

Notes on file formats
Rustam Sadykov

University of Florida, Department of Mathematics

358 Little Hall, 118105, Gainesville, Fl, 32611-8105, USA

Email: sadykov@math.ufl.edu

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