# K-Theory and Hopf invariant one

Summer 2022

Welcome to the course page for our Summer 2022 tutorial on K-Theory! This page hosts the recordings, notes, and administrative information for the class.

The class will run 9pm-11pm 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 warm-up notes entitled Have a go at the exercises, and raise any questions/discussion points on Discord.

#### Day 1.

[Recording] [Notes]
Motivation, vector bundles, maps, sections, deconstruction

#### Day 2.

[Recording] [Notes] [Problem Set 1]
Trivializing open covers, restrictions, transition functions, cocycle condition, direct sums, tensor products, homs

#### Day 3.

[Recording] [Notes]
Subbundles, inner products (and existence), complements, isomorphism classes

#### Day 4.

[Recording] [Notes]
More isomorphism classes, (associated) fiber bundles, pullback bundles, invariance under homotopy

#### Day 5.

[Recording] [Notes] [Problem Set 2]
Complex vector bundles, stably isomorphic, (reduced) K-groups, ring structure, clutching functions

#### Day 6.

[Recording] [Notes]
More on clutching functions, homotopy invariance (complex case), examples, canonical line bundle $H \to \mathbb{C}\mathrm{P}^1$, statement of the fundamental product theorem

#### Day 7.

[Recording] [Notes] [Problem Set 3]
Computing a pullback, $\mathrm{GL}(\mathbb{C}^n)$ is path-connected, $\mathbb{C}\mathrm{P}^1 \cong S^2$, more clutching function calculations: $H \oplus H \cong (H \otimes H) \oplus \varepsilon_1$, the problem with infinite dimensional bundles, realification and complexification

#### Day 8.

[Recording] [Notes]
External tensor products of vector bundles, bundles over a quotient, cones and suspensions, long exact sequence of a closed subspace

#### Day 9.

[Recording] [Notes]
Quotient by a contractible subspace, operations on pointed spaces, consequences of the LES of a closed subspace, reduced external product, Bott periodicity, the six-term exact sequence

#### Day 10.

[Recording] [Notes]
K-groups of $\mathbb{R}\mathrm{P}^2$ and $\mathbb{R}\mathrm{P}^4$, reduced K-ring structure, unreduced analogues, proof of the fundamental product theorem (part one)

#### Day 11.

[Recording] [Notes]
Proof of the fundamental product theorem (part two)

#### Day 12.

[Recording] [Notes]
Division algebras and parallelizability, H-spaces, the K-ring of a product of spheres, nonexistence of H-space structures on $S^{2 n}$, the Hopf invariant

#### Day 13.

[Recording] [Notes]
Hopf invariant one maps from H-spaces, Adams operations, the splitting principle, Hopf invariant one theorem

#### Day 14.

[Recording] [Notes]
Final paper ideas, finite cell/CW complexes, $\mathbb{C}\mathrm{P}^n$ is a CW complex, $K^*(\mathbb{C}\mathrm{P}^n)$ as a group, $K^*(\mathbb{C}\mathrm{P}^n)$ as a ring

#### Day 15.

[Recording] [Notes]
Leray–Hirsch theorem for K-theory, proof of the splitting principle, projectivization/flag bundles, proof of the Leray–Hirsch theorem

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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 K-Theory, 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 K-theory.

## 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 2-fold and 8-fold periodicity in the complex and real cases, respectively. You could discuss the relation to Bott periodicity in K-theory.
• Formal group laws, the correspondence between complex K-theory and the multiplicative formal group law (and similarly for ordinary cohomology?).
• Real K-theory (not the same as real K-theory!) from Atiyah's K-theory and reality.
• Spectra and the relation to K-theory.
• The noncommutative analogue: the K-theory 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
Sections of a vector bundles and basic operations: direct sums, duals, tensor products Inner products, general fiber bundles Examples of associated bundles: sphere/disk, projective, flag, Grassmann

#### Week 2.

Classifying maps
Pullback bundles, clutching functions The universal bundle, (optional depending on student background) CW structure on Grassmannians

#### Week 3.

K-theory
The Grothendieck group, $K_0(X)$ is an abelian group, $K_0(X)$ is a ring Fundamental product theorem (following Hatcher's VBKT Section 2.2)

#### Week 4.

Complex Bott periodicity
The long exact sequence of K-groups for a closed $A \subset X$ Deducing complex Bott periodicity as a consequence of the fundamental product theorem K-theory is an extraordinary cohomology theory

#### Week 5.

Hopf invariant one
Reduction to the case of $H$-spaces, splitting principle Adams operations, Adams' theorem on Hopf invariant one

#### Week 6.

Extras
Optimistically time to talk about characteristic classes or other topics (such as the image of $J$) Though realistically I expect overflow here from previous weeks depending on student background/other interested topics