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VIDEO LIBRARY |
International Seminar for Young Researchers "Algebraic, Combinatorial and Toric Topology"
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On homology of the S. Abramyan Laboratory of algebraic geometry and its applications, National Research University "Higher School of Economics" (HSE), Moscow |
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Abstract: 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 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 $$ \varOmega^{SU}\cong\mathbb Z\left[{\textstyle\frac12}\right][y_2,y_3,\ldots],\quad \deg y_i=2i. $$ 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 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 |