$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory

Author:

Panov Taras Evgenievich123,Chernykh George413

Affiliation:

1. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

2. Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow

3. HSE University, Moscow

4. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract

We study the $SU$-linear operations in complex cobordism and prove that they are generated by the well-known geometric operations $\partial_i$. For the theory $W$ of $c_1$-spherical bordism, we describe all $SU$-linear multiplications on $W$ and projections $MU \to W$. We also analyse complex orientations on $W$ and the corresponding formal group laws $F_W$. The relationship between the formal group laws $F_W$ and the coefficient ring $W_*$ of the $W$-theory was studied by Buchstaber in 1972. We extend his results by showing that for any $SU$-linear multiplication and orientation on $W$, the coefficients of the corresponding formal group law $F_W$ do not generate the ring $W_*$, unlike the situation with complex bordism.

Funder

Russian Foundation for Basic Research

HSE Basic Research Program

Foundation for the Development of Theoretical Physics and Mathematics BASIS

Ministry of Science and Higher Education of the Russian Federation

Publisher

Steklov Mathematical Institute

Subject

General Mathematics

Reference26 articles.

1. The methods of algebraic topology from the viewpoint of cobordism;S. P. Novikov;Izv. Akad. Nauk SSSR Ser. Mat.,1967

2. THE METHODS OF ALGEBRAIC TOPOLOGY FROM THE VIEWPOINT OF COBORDISM THEORY

3. Math. Notes;R. E. Stong,1968

4. $SU$-бордизмы: структурные результаты и геометрические представители

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