$K$-theory is one of those fields of mathematics that has always seemed too deep and not at all approachable for me. As it turns out, it has a relatively low barrier for entry but of course one could still study it their whole life and barely make a dent. There are a few flavors of $K$-theory. We usually split it into topological and algebraic $K$-theory. Ok great, well what does that mean? Hatcher defined topological $K$-theory as a cohomology theory for vector bundles and algebraic $K$-theory as a homology theory for varieties. Once again, what does that mean? Well before we get into that, first lets narrow the scope and just focuse on topological $K$-theory. Great, now all we have to do is learn about cohomology of vector bundles, should be easy enough.

I will assume we are ok with the basic theory of vector bundles, specifically complex bundles. If you don't know what that is then fret not, we really only need the definition to start talking about the objects of interest. As a refresher, maybe I will define vector bundles.

Def: A vector bundle is a triplet $\xi = (E,\pi,B)$ with

• $E$ a topological space, usually a manifold, sometimes called the "total" space $E(\xi)$
• $B$ a topological space, sometimes called the "base" space
• A continuous surjection $\pi:E \to B$ where the fibres over each $b \in B$ are isomorphic to $\mathbb{C}^n$

There are a few more details regarding local trivializations etc. but we can go over that if needed. Usually, I denote vector bundles with just $E$ and this is also a pretty standard practice. But keep in mind that its technically a triplet of two topological spaces and some kind of projection.

Ok now we can finally move on to some $K$-theory stuff. Let $X$ be the base space (a compact Hausdorff space) and let $E$ and $F$ be complex vector bundles over $X$. The addition $[E]+[F] = E \oplus F$ is well defined. Well defined enough in fact that we even get a semigroup under the direct sum. This is great, but we can do even better. The question is, how can we complete this and actually form and abelian group? Thanks to King Grothendieck, we can build and abelian group $K(X)$ from the semigroup of vector bundles. $K(X)$ is the Grothendieck group of isomorphism classes of vector bundles over $X$. This is called the first $K$-theory of $X$. Ok this is also great but how is this a group? Grothendieck defined an equivalence relation on the vector bundles that essentially gives us a "subtraction" operation, thus giving additive inverses to elements of $K(X)$.

We say two vector bundles $E$ and $F$ are stably isomorphic if there exists some $n$ such that $E \oplus \mathbb{C}^n \cong F \oplus \mathbb{C}^n$. Further, we define the cancellation law via $$ (E-E')+(F-F') = (E\oplus F)-(E' \oplus F') $$ Where $[E]-[F] = [E']-[F']$ if there exists a bundle $G$ such that $$ E\oplus F' \oplus G \cong E' \oplus F \oplus G $$ So $K(X)$ becomes a group under this cancellation law. As it turns out, $K(X)$ is even a commutative ring with unit with multiplication defined via the tensor product $$ ([E]-[F])\cdot ([G]-[H]) = [E \otimes G] + [F \otimes H] - [E\otimes H] - [F \otimes G] $$ I should mention too that the elements of $K(X)$ are not vector bundles, but rather formal differences of vector bundles $[E]-[F]$.

We usually refer to such $G$ as some kind of stabilizer. So while $E$ and $F$ may have horribly mismatched dimension and topologies, $G$ may be nice enough to get around this. If dimensions and topologies agree, then $G$ is usually just the zero or trivial bundle, but if they don't agree then we need some more sophistication encoded in $G$. Lets look at a quick example.

Let $X=S^2$, $E$ be the Hopf line bundle (you don't need to know what this is exactly, but its the usualy line bundle with twist), and $\varepsilon^n$ be the $n$-dimensional trivial line bundle. Consider the pairs $(E,\varepsilon^1)$ and $(\varepsilon^2, \varepsilon^1 \oplus \varepsilon^1)$. Recall also that $E \oplus \varepsilon^1 \cong \varepsilon^2$. For these two pairs to be stably isomorphic, we need a $G$ such that $$ E \oplus (\varepsilon^1 \oplus \varepsilon^1) \oplus G \cong \varepsilon^2 \oplus \varepsilon^1 \oplus G $$ So clearly we can't take $G$ to be the zero bundle, we need something nice yet nontrivial. If we choose $G=\varepsilon^1$ then we actually get $\varepsilon^3$ on both sides and thus the desired isomorphism. So in a nutshell, we need a $G$ that gives enough "space" and topology to undo any weird knots or homotopies in the vector bundles. Note that existence of $G$ is only guaranteed under certain conditions and there are plenty of theorems that give us more insight into when we can have such a $G$.

Now I just want to mention two important results in $K$-theory. First, consider the map $$ \mu : K(X) \otimes K(S^2) \to K(X \times S^2) $$ As it turns out, $\mu$ is actually an isopmorphism. This means $K(X) \otimes K(S^2)$ is actually isomorphic to $K(X \times S^2)$. This is suspiciously neat, but actually leads to an even cooler result.

Perhaps the most important and deep result in $K$-theory is Bott periodicity. Bott says that $$ K(S^n) \cong K(S^{n+2}) $$ for ALL nontrivial $n$. Ok how the hell does this work. Well I looked up a few different flavors of the proof and it is far from simple. There are multiple approaches to the proof using tools from different fields of math, while they are well known and documented, they are extremely deep and difficult. This makes sense however, it is very rare that such a nice theorem has a simple and clean proof. While any given proof of Bott is extremely technical, they have all been described as beautiful. For me it is one of those theorems that still just absolutely blows my mind. On the surface it seems like there's no world where this should hold, so the fact it holds for all $n$ is just ridiculous, its so cool. Its always a good idea to look for counter examples to proved theorems in order to deepen your understanding so I did exactly that. Every example I worked out just continued to blow my mind. Using $\mu$ to construct sphere isopmorphisms over chosen base spaces then finding $K$-theories for each is just a really cool loop. Ok this is neat but what about for real vector bundles? As it turns out, Bott covers that too, but we have 8-periodic isopmorphisms as opposed to 2. Specifically, $$ KO(S^n) \cong KO(S^{n+8}) $$ Here, $KO$ just means the real $K$-theory rather than the general complex case. This difference also has some deep theory and I still don't understand hardly any of it.

We always see periodicity in mathematics, trigonometry is usually where we are first exposed to it. But this is one of the most fascinating examples of periodicity I have come across to date. I hope to one day be able to understand one of the proofs of Bott periodicity but for now its beauty will continue to torment me.