Articles of topological k theory

How to compare the K-theory and singular cohomology of a classifying space

In their foundational paper “Vector bundles and homogeneous spaces,” Atiyah and Hirzebruch show, among many other things, that for $G$ a compact, connected Lie group, the K-theory of the classifying space $BG$, taken as an inverse limit of the K-theory of a family of compact approximations of $B_n G$, is isomorphic to the completion of […]

Yoneda's lemma and $K$-theory.

The K-theoretic form of Bott periodicity is that $\tilde{K}(X)\cong \tilde{K}(\Sigma^2 X)$, i.e., $\langle X, BU\times \mathbb{Z} \rangle \cong \langle \Sigma^2 X, BU\times \mathbb{Z} \rangle$. The (reduced) suspension/loopspace adjunction implies that $\langle \Sigma^2 X, BU\times \mathbb{Z}\rangle \cong \langle X,\Omega^{2} (BU\times \mathbb{Z}) \rangle$. Therefore, $\langle X, BU\times \mathbb{Z} \rangle \cong \langle X, \Omega^{2} (BU\times \mathbb{Z}) \rangle$ for […]

Proof of external product theorem using K-theory

I am using Hatcher’s K-Theory book to work through the proof of the external product theorem: $\mu:K(X) \otimes \mathbb{Z}[H]/(H-1)^2 \to K(X) \otimes K(S^2) \to K(X \times S^2)$ is an isomorphism So far I have shown that $\mu$ is surjective. I am trying to work through the inverse function $\nu$. We use the notation $[E,f]$ where […]

Prerequisite to start learning Topological K-Theory

Wanted to start learning K-theory to see if it is suitable for a undergraduate thesis (from an algebraic-topological view). For reference, I only know Hatcher’s book (mostly because I’ve read his book about algebraic-topology) But I wanted to know what are the necessary prerequisite to achieve a good level of understanding of the subject. I’ve […]

Tautological line bundle over rational projective space

Is the tautological line bundle over $\mathbb{Q}P^{n}$ a non trivial bundle? Here, $\mathbb{Q}P^{n}$ has the natural topology induced from the standard topology of $\mathbb{Q}$ as a subset of $\mathbb{R}$. What about if we change the topology by consideration of $p-$adic topology on rational numbers?

$K$-theory of smooth manifolds: continuous vs. smooth vector bundles

Suppose I have a smooth manifold $M$, and want to consider the $K$-theory $K^0(M)$. I could define this in the usual way (by taking the Grothendieck group of the monoid of equivalence classes of vector bundles) or in a ‘smooth’ way (considering only smooth vector bundles, and taking the Grothendieck group as usual). I haven’t […]

$K(\mathbb R P^n)$ from $K(\mathbb C P^k)$

EDIT: I found a brief discussion of this in Husemoller’s Fibre Bundles, chapter 16 section 12. Here to compute $\tilde K(\mathbb R P^{2n+1})$ he says to consider the map $$ \mathbb R P^{2n+1} = S^{2n+1}/\pm 1\to \mathbb C P^n = S^{2n+1}/U(1). $$ Under this map the canonical line bundle over $\mathbb C P^n$ pulls back […]

Why is stable equivalence necessary in topological K-theory?

The topological $K$-theory of a complex compact manifold $X$ is the commutative monoid $K(X)$ of isomorphism classes of complex vector bundles. Two classes $[E]$ and $[F]$ are equivalent in $K$-theory if they are stably equivalent: there exists a trivial bundle $G$ with $$ (E \oplus G) \cong (F \oplus G). $$ In what sense is […]

Topological K-theory references

I am interesting in learning about (topological) K-theory. As far as I can see there are 3 main references used: 1) Atiyah’s book: This looks to be very readable and requires minimal pre-requesities. However, the big downside is there are no exercises 2) Allan Hatcher’s online notes: If his Algebraic Topology book is any guide, […]

How to understand the Todd class?

I am reading the article “K-Theory and Elliptic Operators”(, which is about Atiyah-Singer index theorem. In page 14 the article discussed the Thom isomorphism: $$\psi:H^{k}(X)\rightarrow H^{n+k}_{c}(E)$$ and $$\phi:K(X)\rightarrow K(E)$$ with $\psi: x \rightarrow \pi^{*}x* \lambda_{E}$ and $\phi:x \rightarrow \pi^{*}x\cup \mu$. Greg further defined a correction factor $\mu(E)$ such that $$\psi(\mu(E)\cup \operatorname{ch}(x))=\operatorname{ch}(\phi(x))$$ He analyzed $\mu(E)$ by […]