-bordism: structure results and geometric representatives

@article{Limonchenko2019bordismSR,
  title={-bordism: structure results and geometric representatives},
  author={Ivan Yu. Limonchenko and Taras Panov and Georgy Chernykh},
  journal={Russian Mathematical Surveys},
  year={2019},
  volume={74},
  pages={461 - 524}
}
The first part of this survey gives a modernised exposition of the structure of the special unitary bordism ring, by combining the classical geometric methods of Conner–Floyd, Wall, and Stong with the Adams– Novikov spectral sequence and formal group law techniques that emerged after the fundamental 1967 paper of Novikov. In the second part toric topology is used to describe geometric representatives in -bordism classes, including toric, quasi-toric, and Calabi–Yau manifolds. Bibliography: 56… 
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On homology of the $MSU$ spectrum
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Polynomial generators of msu*[1/2] related to classifying maps of certain formal group laws
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We study the SU -linear operations in complex cobordism and prove that they are generated by the well-known geometric operations ∂i. For the theory W of c1-spherical bordism, we describe all SU…

References

SHOWING 1-10 OF 63 REFERENCES
Complex cobordism and formal groups
This paper surveys the current state of the theory of cobordism, focusing on geometric and universal properties of complex cobordism, the Landweber-Novikov algebra, and the formal group law of…
Projective toric polynomial generators in the unitary cobordism ring
According to Milnor and Novikov's classical result, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators: , . In this paper we solve the well-known…
THE METHODS OF ALGEBRAIC TOPOLOGY FROM THE VIEWPOINT OF COBORDISM THEORY
The goal of this work is the construction of the analogue to the Adams spectral sequence in cobordism theory, calculation of the ring of cohomology operations in this theory, and also a number of…
Bordism and Cobordism
In (10), (11) Wall determined the structure of the cobordism ring introduced by Thom in (9). Among Wall's results is a certain exact sequence relating the oriented and unoriented cobordism groups.…
On toric generators in the unitary and special unitary bordism rings
We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a…
Calabi Yau Hypersurfaces and SU-Bordism
V. V. Batyrev constructed a family of Calabi–Yau hypersurfaces dual to the first Chern class in toric Fano varieties. Using this construction, we introduce a family of Calabi–Yau manifolds whose…
In Search of a Class of Representatives for SU -Cobordism Using the Witten Genus
OF DISSERTATION In Search of a Class of Representatives for SU -Cobordism Using the Witten Genus In algebraic topology, we work to classify objects. My research aims to build a better understanding…
On Thom spectra, orientability, and cobordism
Notation, Conventions and Other Preliminaries.- Spectra and (Co)homology Theories.- Phantoms.- Thom Spectra.- Orientability and Orientations.- K- and KO-Orientability.- Complex (Co)bordism.-…
Toric manifolds and complex cobordisms
Algebraic topology of manifolds defined by simple polytopes V M Bukhshtaber and T E Panov Torus actions, combinatorial topology, and homological algebra V M Buchstaber and T E Panov Hirzebruch genera…
Spaces of polytopes and cobordism of quasitoric manifolds
Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we…
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