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International seminar for young researchers "Algebraic, combinatorial and toric topology"
17 декабря 2020 г. 11:50–12:30, online

On homology of the $MSU$ spectrum

S. Abramyan

Laboratory of algebraic geometry and its applications, National Research University "Higher School of Economics" (HSE), Moscow
MP4 172.6 Mb

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Аннотация: The theory of bordism and cobordism was actively developed in the 1950–1960s. Most leading topologists of the time have contributed to this development. The idea of bordism was first explicitly formulated by Pontryagin who related the theory of framed bordism to the stable homotopy groups of spheres using the concept of transversality. Key results of bordism theory were obtained in the works of Rokhlin, Thom, Novikov, Wall, Averbuch, Milnor, Atiyah.
The paper of Adams gave an opportunity to enter the new stage of developing bordism theory. It culminated in the calculation of the complex (or unitary) bordism ring $\varOmega^U$ in the works of Milnor and Novikov. The ring $\varOmega^U$ was shown to be isomorphic to a graded integral polynomial ring $\mathbb{Z}[a_i\colon i\ge1]$ on infinitely many generators, with one generator in every even degree, $\deg a_i=2i$. This result has since found numerous applications in algebraic topology and beyond.
In Novikov's 1967 work a brand new approach to cobordism and stable homotopy theory was proposed, based on application of the Adams–Novikov spectral sequence and formal group laws techniques. This approach was further developed in the context of bordism of manifolds with singularities in the works of Mironov, Botvinnik and Vershinin. The Adams-Novikov spectral sequence has also become the main computational tool for stable homotopy groups of spheres.
As an illustration of his approach, Novikov outlined a complete description of the additive torsion and the multiplicative structure of the SU-bordism ring $\varOmega^{SU}$, which provided a systematic view on earlier geometric calculations with this ring. A modernised exposition of this description is given in the survey paper of Chernykh, Limonchenko and Panov; it includes the geometric results by Wall, Conner-Floyd and Stong, the calculations with the Adams-Novikov spectral sequence, and the details of the arguments missing in Novikov's work. A full description of the SU-bordism ring $\varOmega^{SU}$ relies substantially on the calculation of $\varOmega^{SU}$ with 2 inverted, namely on proving the ring isomorphism
$$ \varOmega^{SU}\cong\mathbb Z[{\textstyle\frac12}][y_2,y_3,\ldots],\quad \deg y_i=2i. $$
This result first appeared in Novikov's work with only a sketch of the proof, stating that it can be proved using Adams' spectral sequence in a way similar to Novikov's calculation of the complex bordism ring $\varOmega^U$. Although the result has been considered as known since the 1960s, its full proof has been missing in the literature, and also not included in the survey.
The main goal of the talk is to give a complete proof of the isomorphism above using the original methods of the Adams spectral sequence. While filling in details in Novikov's sketch we faced technical problems that seemed to be unknown before. For example, the comodule structure of $H_{\bullet}(MSU; \mathbb F_p)$ over the dual Steenrod algebra $\mathcal{A}_p^*$ with odd prime $p$ has not been sufficiently detailed in the literature. The latter computation is one of the main results of the talk. We also discuss results on the structure of the Hurewicz map and the divisibility of characteristic numbers of $SU$-manifolds.

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