Lorenzo Mantovani

Topological K-Theory was developed in the Sixties thanks to the common effort of A. Grothendieck, M. Atiyah, F. Hirzebruch and many others. The theory gives one possible answer to the following vague problem.

Associating with a topological space $X$ an algebraic invariant $K(X)$,
in such a way that, at the same time, we can:
manipulate the invariant $K(X)$ in a sufficiently easy and intuitive way,
and reconstruct some properties of $X$ from $K(X)$.

There exist many different algebraic invariants that can be considered a solution to this problem, but we will deal only with the so called Topological K-Theory, that we will denote by $K(X)$. More concretely $K(X)$ is a commutative ring whose construction and properties will we explained during the course. At this stage, without entering in the details of the matter, we can say that in order to actually construct and study $K(X)$ we first have to study the notion of topological vector bundle on $X$

A (complex) topological vector bundle on a topological space $X$ is, roughly speaking, a continuous map $p:E\rightarrow X$ such that

• Every point $x\in X$ has an open neighborhood $x \in U_x \subseteq X$ where there exists a homeomorphism $\phi_{U_x}: p^{-1}(U_x) \rightarrow U_x\times \mathbb C^n $
• Given $x,y \in X$, if $W:=U_x\cap U_y \not = \emptyset$, then the composition $\phi_{U_x}\circ \phi_{U_y}^{-1}: W\times \mathbb C^n \rightarrow W\times \mathbb C^n$ has the form $(p,v)\mapsto (p,M(p)v)$, where $ M(-): W \rightarrow \mathrm{GL}_n(\mathbb C)$ is some continuous map.

In simpler words, a topological vector bundle is a continuous map that locally is the projection from a cylinder to its base. There is an analogous notion of real topological vector bundle, with the same definition, where $\mathbb C$ is replaced by $\mathbb R$. Examples of real vector bundles $E$ are the cylinder $\mathbb S^1\times \mathbb R$ and the Möbius band (what is $p$ in these examples?)

Vector bundles will be the first object of study of the course. We will study how to construct them, how they behave with respect to continuous maps, and we will develop a bit of linear algebra in order to be able to manipulate them in an efficient way (after all a complex vector space $V$ is just a vector bundle on the one point space $V \rightarrow \ast$).

At this point we will turn to K-theory. The ring K(X) of complex topological K-theory of the space $X$ is constructed from the collection of vector bundles on $X$ by treating vector bundles as symbols that can be added, and multiplied as if they were numbers. We will compute some elementary example and prove the so called Bott periodicity theorem, which describes $K(\mathbb S^2)$. Basing on this, we will prove a number of results describing how $K(-)$ behaves under a number of manipulations, e.g. the extremely useful splitting principle, and prove that $K(-)$ satisfies the Eilenberg-Steenrod axioms. This will open the possibility of computing a lot of explicit examples, and we will do it.

At this point we will introduce the lambda operations (basically the external powers) and we will deduce from them the existence of the so called Adams operations. Adams operations are simply ring homomorphisms $\psi^i: K(X) \rightarrow K(X)$ that satisfy a very short list of simple axioms. The main point here is that their existence strongly restricts the behavior of $K(X)$.

At last we will turn to applications. The main application we will consider is to the so called Hopf invariant one problem. This is too technical to explain now, but roughly the problem asks to determine if there are continuous maps $\mathbb S^{2n-1} \rightarrow \mathbb S^n$ satisfying some special requirement. We will give a very elementary proof using Adams operations. We will see how some classical problems about the (non-)existence of division algebras structures on $\mathbb R^n$ (e.g. the ring of complex numbers for $n=2$, quaternions for $n=4$, octonions for $n=8$) and the (non-)existence of group structures on $\mathbb S^{n-1}$ are intimately related with the Hopf invariant one problem.

In the first part: vector bundles on topological spaces and "continuous" algebra. Some examples coming form manifolds.

In the second part: Topological K-theory, Bott periodicity.

In the third part: K-theory as a generalized cohomology theory

In the fourth part: Operations and their classification

In the fifth part: Applications to the problem of classifying real division algebras and the problem of parallelizability of the spheres $\mathbb S^n$. Reduction to the so called Hopf invariant one problem . Solution of the Hopf invariant one problem via Adams operations in K-theory.

04.11.2019 - Notions of categories, functors and natural transformations with some examples. A reference I personally like very much is the following lecture notes by P. Shapira. Section 3 of Chapter 1 (pages 14-20) contain even a bit more of what I tried to explain. The notion of cohomology theory on pairs of topological spaces, and of reduced cohomology theory on pointed CW-complexes. We have also stated that each notion determines and is determined by the other. The proof of this fact is not in the scope of our course. You can find a reference for this in the book of P. May. More precisely you can give a look at pages 137-138 for the notion of cohomology theory on pairs, and to pages 145-147 for the notion of reduced cohomology on pointed spaces.

06.11.2019 - Topological preliminaries: the category of compact spaces $\mathbf{Cpt}$, of compact pairs $\mathbf{Cpt^2}$, and of pointed compact spaces $\mathbf{Cpt}_\bullet$. Some constructions in these categories: push-outs, quotient by a subspace, the wedge sum, the smash product, reduced suspensions. We have defined the reduced $K$-theory $\widetilde K(X)$ of a pointed compact space $(X,x)$ by setting \[\widetilde K(X):=\operatorname{Ker}(K(X) \rightarrow K(x)),\] and the $K$-theory of a compact pair $K(X,A)$ by setting \[K(X,A):=\widetilde K (X/A),\] where $X/A$ is pointed by the canonical choice $\ast=A/A \subseteq X/A$. We have started to show the exactness property (we called it $(\tilde ci)$ in class): given a compact pair (X,A) and a point $x \in A$, the inclusion \( i: (A,x) \rightarrow (X,x)\) and the projection $(X,x)\rightarrow (X/A,A/A)$ induce a short exact sequence \[ \widetilde K(X/A) \rightarrow \widetilde K(X)\rightarrow \widetilde K(A) \]

Next time - Some more topological preliminaries: cones, cylinders, suspensions. Definition of $K^{-n}(X,A)$ for $n \in \mathbb N$. Finishing up the exactness and the other axioms of reduced cohomology.

The main references for this course are the following books. What we cover during the course is a strict subset of the content of such books, but we often go into some more details and computations.

• M. Atiyah, K-Theory, Benjamin, 1967.
• A. Hatcher, Vector bundles and K-Theory, Link.

For the more intrepid reader I suggest

• M. Karoubi, K-Theory - an introduction, Springer, 1978.

Another good reference for the material covered in the course is contained in

• M. Aguilar, S. Gitler, C. Prieto, Algebraic Topology from a Homotopical Viewpoint, Chapters 8-10, Springer, 2002.
• Wirthmüller, Notes on vector bundles and $K$-Theory