- Tags: - #subjects/homotopy/stable #higher-algebra/K-theory - Refs: - [ ] Inna’s article: https://www.ams.org/journals/notices/201907/rnoti-p1034.pdf - [ ] Some lectures: https://web.ma.utexas.edu/users/dafr/M392C-2015/Notes/lecture3.pdf - [ ] https://faculty.tcu.edu/richardson/Seminars/QuillenKtheory.pdf - [ ] https://www.dpmms.cam.ac.uk/~jes98/K-TheoryWeb.pdf - [ ] Algebraic K Theory for schemes - [ ] Why study K theory - [ ] Adams Atiyah, K Theory and the Hopf Invariant - Links: - THH - transfers, norms, wrong-way maps - equivariant K theory - chromatic - How to construct K theory - The difference between algebraic and topological K theory - The connection between K theory and projective modules - Topological cyclic homology is related and more computable.

K-Theory

Categorical K Theory

• Defining \({\mathsf{K}}_0\) for \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\): File:Pasted image 20211103190932.png
  • Where it shows up naturally in algebraic topology: the Wall finiteness obstruction. The finiteness obstruction \(w(X)\) is zero if and only if \(X\) has the homotopy type of a finite CW complex. File:Pasted image 20211103191051.png
• \({\mathsf{K}}_1\) shows up in defining Whitehead torsion: File:Pasted image 20211103191524.png

Algebraic K Theory

• It’s like a homology theory on \(\mathsf{CRing}\).
• File:Pasted image 20211105131146.png
• File:Pasted image 20211105131542.png

File:Pasted image 20211108230838.png

See Borel regulator, Lichtenbaum-Quillen conjectures, Zeta function

K(Z)

In toplogy

Examples

File:Pasted image 20211105131303.png File:Pasted image 20211105131409.png

Complex K Theory