KTheory and Hopf invariant one
Summer 2022
Welcome to the course page for our Summer 2022 tutorial on KTheory! This page hosts the recordings, notes, and administrative information for the class.
The class will run 9pm11pm Monday/Wednesday/Friday from June 20 to July 29. Final papers will be due before the start of the Fall semester.
Notes
Before the tutorial starts, have a read through the warmup notes entitled Have a go at the exercises, and raise any questions/discussion points on Discord.
Day 1. 
[Recording]  [Notes]  


Day 2. 
[Recording]  [Notes]  [Problem Set 1]  


Day 3. 
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Day 4. 
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Day 5. 
[Recording]  [Notes]  [Problem Set 2]  


Day 6. 
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Day 7. 
[Recording]  [Notes]  [Problem Set 3]  


Day 8. 
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Day 9. 
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Day 10. 
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Day 11. 
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Day 12. 
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Day 13. 
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Day 14. 
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Day 15. 
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Format
There will be 3 classes each week for 6 weeks, each 2 hours long. The third class of each week will typically be a problem session where we work on and discuss exercises coming from the week's content. When times get tough I might reserve (part of) the final day to put a finer point on some more complicated pieces, e.g. when we get to Bott periodicity.
References
We will try to closely follow Allen Hatcher's Vector Bundles and KTheory, freely available online. It (especially the introductory chapters) are very accessible. Hatcher's Algebraic Topology could be a useful reference, but we will try hard not to rely on basically any algebraic topology for as long as possible. There are also many other references, including Atiyah's Ktheory.
Prerequisites
Surprisingly little background is required for the most part: if you know about groups, rings, ideals, and kernels and images of homomorphisms, then you know enough algebra. If you have seen a definition of a compact metric space (or especially of a general topological space) and of continuous maps between them, then you should know enough topology. If you have seen inner products, duals, and direct sums (and maybe tensor products) of vector spaces, then you definitely know enough linear algebra. A further course in algebraic topology would open the door to more final paper ideas, but is definitely not necessary.
Final paper
Sample ideas for the final paper include: Bott periodicity: we'll see Hatcher's proof, but there are many other proofs of Bott periodicity. You could compare two different proofs.
 The Serre–Swan theorem: in particular Swan's (1962) topological version, which asserts that the category of finite rank vector bundles over a compact Hausdorff space $X$ is equivalent to the category of finitely generated projective modules over the ring $C(X)$.
 Clifford algebras exhibit a 2fold and 8fold periodicity in the complex and real cases, respectively. You could discuss the relation to Bott periodicity in Ktheory.
 Formal group laws, the correspondence between complex Ktheory and the multiplicative formal group law (and similarly for ordinary cohomology?).
 Real Ktheory (not the same as real Ktheory!) from Atiyah's Ktheory and reality.
 Spectra and the relation to Ktheory.
 The noncommutative analogue: the Ktheory of $C^*$algebras.
 $\Lambda$rings.
 For people who know about characteristic classes: Schubert calculus.
 For people who know about algebraic topology and stable homotopy groups: the complex and/or real $J$homomorphism (see e.g. Adams' ON THE GROUPS $J(X)$IV).
 Some other topic that came up in class, or your own idea! (Talk to me.)
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Plan
Week 1.Vector bundles 

Week 2.Classifying maps 

Week 3.Ktheory 

Week 4.Complex Bott periodicity 

Week 5.Hopf invariant one 

Week 6.Extras 
