Record: No

Level: Graduate

Language: Chinese

Prerequisite

Algebraic topology and algebraic geometry

Abstract

Steenrod operations are natural transformations of cohomology groups of spaces being compatible with suspensions, which play an important role especially in homotopy theory, for example, the Hopf invariant one problem and Adams spectral sequence. In this course, we introduce the Steenrod operations for both singular cohomologies and motivic cohomologies, focusing on constructions and basic properties. In particular, we prove the Adem relations and give the Hopf algebra (algebroid) structure of Steenrod algebra and its dual.

Reference

1. O. Pushin, Steenrod operations in motivic cohomology, thesis

2. G. Powell, Steenrod operations in motivic cohomology, Luminy lecture notes

J. Riou, Opérations de Steenrod motiviques, arXiv:1207.3121v1

C. Mazza, V. Voevodsky, C. Weibel, Lecture notes on motivic cohomology, American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA (2006).

A. Suslin and V. Voevodsky, Relative cycles and Chow sheaves, Annals of Mathematical Studies, vol. 143, Princeton University Press, 2000

A. Hatcher, Algebraic topology

Syllabus

1. Classical Steenrod operations

2. Classical Steenrod algebra and its dual

3. Equidimensional cycles

4. Motivic Steenrod operations

5. Motivic Steenrod algebra and its dual

Lecturer Intro

Nanjun Yang got my doctor and master degree in University of Grenoble-Alpes, advised by Jean Fasel, and bachelor degree in Beihang University. Currently he is a assistant researcher in BIMSA. His research interests are motivic cohomology and Chow-Witt ring. He proposed the theory of split Milnor-Witt motives, which applies to the computation of the Chow-Witt ring of fiber bundles. The corresponding results have been published independently on journals such as Camb. J. Math and Doc. Math.

Lecturer Email: yangnanjun@bimsa.cn

TA: Dr. Dongsheng Wu, wds2021fall@bimsa.cn