--- date: 2021-10-21 18:42 modification date: Friday 22nd October 2021 23:39:43 title: K-Theory aliases: ["algebraic K theory", "K theory"] --- --- - Tags - #homotopy/stable-homotopy #higher-algebra/K-theory - Refs: - A nice thesis: - [ ] Modern construction in terms of [infty-categories](Unsorted/infinity%20categories.md): - [ ] Inna's article: - [ ] Some lectures: - [ ] - [ ] - [ ] [Algebraic K Theory for schemes](Algebraic%20K%20Theory%20for%20schemes.md) - [ ] [Why study K theory](Why%20study%20K%20theory.md) - [ ] [Adams Atiyah, K Theory and the Hopf Invariant](attachments/adams1966.pdf) #resources/notes - [ ] [Markus Land, “Manifolds, algebraic K-theory, and excision”, GeoTop Opening Symposium 2nd ed.](https://www.youtube.com/watch?v=5Hsa5iRfLng&list=PLAMniZX5MiiJ39Q8B6_czpe-_qOrMtZbl&index=6) #resources/videos - [x] [Teena Gerhardt - 1/3 Algebraic K-theory and Trace Methods](https://www.youtube.com/watch?v=URXm2zIBLyU) #resources/videos - [ ] MSRI workshop on [infty-category](Unsorted/infinity%20categories.md), [factorization homology](factorization%20homology.md), and algebraic K-theory: #resources/videos #resources/workshops #projects/unstarted - [ ] Lurie's course. #projects/to-read #projects/lecture-notes - [x] Lecture 1: [Overview.](https://www.math.ias.edu/~lurie/281notes/Lecture1-Overview.pdf) - [x] Lecture 2: [The Wall Finiteness Obstruction.](https://www.math.ias.edu/~lurie/281notes/Lecture2-Wall.pdf) - [x] Lecture 3: [Whitehead Torsion: Part I.](https://www.math.ias.edu/~lurie/281notes/Lecture3-Whitehead.pdf) - [x] Lecture 4: [Whitehead Torsion: Part II.](https://www.math.ias.edu/~lurie/281notes/Lecture4-Whitehead2.pdf) - [x] Lecture 5: [Cell-Like Maps.](https://www.math.ias.edu/~lurie/281notes/Lecture5-CLike.pdf) - [x] Lecture 6: [Concordance of Polyhedra.](https://www.math.ias.edu/~lurie/281notes/Lecture6-Concordance.pdf) - [x] Lecture 7: [Higher Simple Homotopy Theory.](https://www.math.ias.edu/~lurie/281notes/Lecture7-Higher.pdf) - [x] Lecture 8: [Fibrations of Polyhedra.](https://www.math.ias.edu/~lurie/281notes/Lecture8-Fibrations.pdf) - [x] Lecture 9: [Fibrations of Nonsingular Simplicial Sets.](https://www.math.ias.edu/~lurie/281notes/Lecture9-Fibrations2.pdf) - [x] Lecture 10: [Combinatorial Models for Simple Homotopy Theory.](https://www.math.ias.edu/~lurie/281notes/Lecture10-First.pdf) - [ ] Lecture 11: [Equivalence of the Combinatorial Definition.](https://www.math.ias.edu/~lurie/281notes/Lecture11-Theorem.pdf) - [ ] Lecture 12: [Some Loose Ends.](https://www.math.ias.edu/~lurie/281notes/Lecture12-LooseEnds.pdf) - [ ] Lecture 13: [Homotopy Types vs Simple Homotopy Types.](https://www.math.ias.edu/~lurie/281notes/Lecture13-Part2.pdf) - [ ] Lecture 14: [(Lower) K-Theory of infty-Categories.](https://www.math.ias.edu/~lurie/281notes/Lecture14-Quasicategories.pdf) - [ ] Lecture 15: [The Wall Finiteness Obstruction Revisited.](https://www.math.ias.edu/~lurie/281notes/Lecture15-Wall.pdf) - [ ] Lecture 16: [Higher K-Theory of infty-Categories.](https://www.math.ias.edu/~lurie/281notes/Lecture16-Higher.pdf) - [ ] Lecture 17: [The Additivity Theorem.](https://www.math.ias.edu/~lurie/281notes/Lecture17-Additivity.pdf) - [ ] Lecture 18: [Additive K-Theory.](https://www.math.ias.edu/~lurie/281notes/Lecture18-Additive.pdf) - [ ] Lecture 19: [K-Theory of Ring Spectra.](https://www.math.ias.edu/~lurie/281notes/Lecture19-Rings.pdf) - [ ] Lecture 20: [Lower K-Groups of Ring Spectra.](https://www.math.ias.edu/~lurie/281notes/Lecture20-Lower.pdf) - [ ] Lecture 21: [The Algebraic K-Theory of Spaces.](https://www.math.ias.edu/~lurie/281notes/Lecture21-ATheory.pdf) - [ ] Lecture 22: [Constructible Sheaves.](https://www.math.ias.edu/~lurie/281notes/Lecture22-Constructible.pdf) - [ ] Lecture 23: [Universal Local Acyclicity.](https://www.math.ias.edu/~lurie/281notes/Lecture23-ULA.pdf) - [ ] Lecture 24: [The Assembly Map.](https://www.math.ias.edu/~lurie/281notes/Lecture24-Assembly.pdf) - [ ] Lecture 25: [The Assembly Map II.](https://www.math.ias.edu/~lurie/281notes/Lecture25-Assembly2.pdf) - [ ] Lecture 26: [The Assembly Map III.](https://www.math.ias.edu/~lurie/281notes/Lecture26-Assembly3.pdf) - [ ] Lecture 27: [Higher Torsion.](https://www.math.ias.edu/~lurie/281notes/Lecture27-HigherTorsion.pdf) - [ ] Lecture 28: [Another Assembly Map.](https://www.math.ias.edu/~lurie/281notes/Lecture28-Proof1.pdf) - [ ] Lecture 29: [Another Assembly Map II.](https://www.math.ias.edu/~lurie/281notes/Lecture29-Proof2.pdf) - [ ] Lecture 30: [The Whitehead Space.](https://www.math.ias.edu/~lurie/281notes/Lecture30-StructureSpace.pdf) - [ ] Lecture 31: [The Whitehead Space II.](https://www.math.ias.edu/~lurie/281notes/Lecture31-Whitehead.pdf) - [ ] Lecture 32: [Proof of the Main Theorem.](https://www.math.ias.edu/~lurie/281notes/Lecture32-Mainstep.pdf) - [ ] Lecture 33: [Digression: Review of Microbundles](https://www.math.ias.edu/~lurie/281notes/Lecture33-Microbundles.pdf) - [ ] Lecture 34: [Overview of Part 3](https://www.math.ias.edu/~lurie/281notes/Lecture34-Part3.pdf) - [ ] Lecture 35: [The Setup](https://www.math.ias.edu/~lurie/281notes/Lecture35-Setup.pdf) - [ ] Lecture 36: [The Combinatorial Step (Part I)](https://www.math.ias.edu/~lurie/281notes/Lecture36-Combinatorics1.pdf) - [ ] Lecture 37: [The Combinatorial Step (Part II)](https://www.math.ias.edu/~lurie/281notes/Lecture37-Combinatorics2.pdf) - [ ] Lecture 38: [Thickenings of a Point](https://www.math.ias.edu/~lurie/281notes/Lecture38-ManifoldsI.pdf) - [ ] The Q construction: - [ ] THE LOCALISATION THEOREM FOR THE $K$-THEORY OF STABLE $\infty$-CATEGORIES (2022): - Links: - [HH](Unsorted/HH.md) - [transfers](transfers), [norms](norms), [wrong-way maps](wrong-way%20maps) - [equivariant K theory](equivariant%20K%20theory) - [chromatic](Unsorted/chromatic%20homotopy%20theory.md) - [How to construct K theory](How%20to%20construct%20K%20theory) - [The difference between algebraic and topological K theory](The%20difference%20between%20algebraic%20and%20topological%20K%20theory) - [The connection between K theory and projective modules](The%20connection%20between%20K%20theory%20and%20projective%20modules) - [Topological cyclic homology](Topological%20cyclic%20homology.md) is related and more computable. - [Quillen K theory](Unsorted/Quillen%20K%20theory.md) - [examples of K theory rings](Unsorted/examples%20of%20K%20theory%20rings.md) - [Talbot Talk 1](Unsorted/Talbot%20Talk%201.md) - [K theory in AG](K%20theory%20in%20AG.md) - [devissage](Unsorted/devissage.md) - [Milnor K theory](Milnor%20K%20theory) - [transfers](transfers.md) - [computational properties of K theory](computational%20properties%20of%20K%20theory.md) - [Waldhausen S construction](Unsorted/Waldhausen%20S%20construction.md) --- # K-Theory # Motivations The [Lichtenbaum-Quillen conjectures](Lichtenbaum-Quillen%20conjectures) related to [etale cohomology](Unsorted/l-adic%20cohomology.md), [[Kummer-Vandiver conjecture]], relations to [THH](Unsorted/THH.md) and [TC](Unsorted/Topological%20cyclic%20homology.md). ![](attachments/Pasted%20image%2020220527234514.png) # Notes - Algebraic K is analogous to $\KU$, complex [topological K theory](Unsorted/topological%20K%20theory.md), or $\KO$, real topological K-theory. - The [Grothendieck-Witt](Unsorted/quadratic%20form.md) group is analogous to $\KO$? ![](attachments/Pasted%20image%2020220425094133.png) ![](attachments/Pasted%20image%2020220521163500.png) ![](attachments/Pasted%20image%2020220527233346.png) ![](attachments/Pasted%20image%2020220527233440.png) ![](attachments/Pasted%20image%2020220527233450.png) # Construction ![](attachments/Pasted%20image%2020220425095202.png) See [Infinite loop space machine](Unsorted/Infinite%20loop%20space.md). ![](attachments/Pasted%20image%2020220415125059.png) ![](attachments/Pasted%20image%2020220415125115.png) # Categorical K Theory - Defining $\K_0$ for $\rmod$: ![Pasted image 20211103190932.png](Pasted%20image%2020211103190932.png) ![](attachments/Pasted%20image%2020220420102353.png) - Where it shows up naturally in algebraic topology: the [Wall finiteness obstruction](Wall%20finiteness%20obstruction). The finiteness obstruction $w(X)$ is zero if and only if $X$ has the homotopy type of a finite CW complex. ![Pasted image 20211103191051.png](Pasted%20image%2020211103191051.png) - $\K_1$ shows up in defining [Whitehead torsion](Whitehead%20torsion.md): ![Pasted image 20211103191524.png](Pasted%20image%2020211103191524.png) # Algebraic K Theory - It's like a homology theory on $\CRing$. - ![Pasted image 20211105131146.png](Pasted%20image%2020211105131146.png) - ![Pasted image 20211105131542.png](Pasted%20image%2020211105131542.png) ![Pasted image 20211108230838.png](Pasted%20image%2020211108230838.png) ![](attachments/Pasted%20image%2020220209192205.png) See [Borel regulator](Borel%20regulator.md), [Lichtenbaum-Quillen conjectures](Lichtenbaum-Quillen%20conjectures), [Zeta function](Unsorted/Riemann%20Zeta.md). ![](attachments/Pasted%20image%2020220415202658.png) ![](attachments/Pasted%20image%2020220415202738.png) ## K(Z) ![](attachments/Pasted%20image%2020220209211517.png) ## In toplogy ![](attachments/Pasted%20image%2020220209211421.png) ## Examples ![Pasted image 20211105131303.png](Pasted%20image%2020211105131303.png) ![Pasted image 20211105131409.png](Pasted%20image%2020211105131409.png) # Complex K Theory ![](attachments/Pasted%20image%2020220209184315.png) # For stacks See [stacks MOC](Unsorted/stacks%20MOC.md): ![](attachments/Pasted%20image%2020220319213903.png) # Misc - $X(R) \da ( \rmod^\free)^\cong \iso \disjoint_n \BGL_n(R)$ # Gaps Build a [simplicial set](Unsorted/simplicial%20set.md): ![](attachments/Pasted%20image%2020220420102612.png) ![](attachments/Pasted%20image%2020220420102718.png) # Quillen's construction ![](attachments/Pasted%20image%2020220420103307.png) # Misc See [syntomic cohomology](Unsorted/syntomic.md): ![](attachments/Pasted%20image%2020220515001557.png)