$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory
- Authors: Panov T.E.1,2,3, Chernykh G.S.1,4,2
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Affiliations:
- Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
- HSE University
- Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)
- Steklov Mathematical Institute of Russian Academy of Sciences
- Issue: Vol 87, No 4 (2023)
- Pages: 133-165
- Section: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/133920
- DOI: https://doi.org/10.4213/im9334
- ID: 133920
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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.
About the authors
Taras Evgenievich Panov
Lomonosov Moscow State University, Faculty of Mechanics and Mathematics; HSE University; Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)
Email: tpanov@mech.math.msu.su
Doctor of physico-mathematical sciences, Associate professor
George Sergeevich Chernykh
Lomonosov Moscow State University, Faculty of Mechanics and Mathematics; Steklov Mathematical Institute of Russian Academy of Sciences; HSE Universitywithout scientific degree, no status
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