Intuitively, a vector bundle on a topological space X is a continuous family of (real or complex, finite dimensional) vector spaces parametrised by X. Unfortunately, for lack of space only few cases can be visualised like the bundle of identical copies of the real line parametrised by the circle X. This becomes easier if one accepts lines that are compressed to short open intervals as in the figure on the right. Bundles as shown count among "trivial" ones — while the Moebius strip below is a good illustration of a non-trivial vector bundle over (i.e. parametrised by) the circle.

Vector bundles form a bridge between topology and (linear) algebra. Contexts in which they naturally arise include

	differentiation say of a scalar function on the two-dimensional sphere X, which involves the so-called tangent bundle consisting of all tangent planes to X,
	in algebraic geometry the construction of scalar functions on a variety, which often relies on functions with values in a line (i.e. one-dimensional vector) bundle,
	eigenspaces of linear mappings that depend on a continuous parameter — beyond finite-dimensional linear algebra this idea has been particularly fruitful in the study of differential operators.

The first aim of the course is to define real and complex vector bundles and clarify its basic properties. A key observation will be that the vector bundles that exist over a given space X tell a lot about the topology of X — in a sense that reminds of algebraic topology: so the existence of a non-trivial vector bundle like the Moebius strip over the circle reflects the hollowness of the latter: such a bundle could not exist over the full disk or an interval. A celebrated discovery by Micheal Atiyah and Friedrich Hirzebruch in the 1960s was that the vector bundles over X indeed can be organised in an abelian group and even a ring K(X) which allows to do systematic algebraic topology in a new, and in some sense elementary, way. To explain this, and put this K-Theory in the context of algebraic topology, is the second main goal of the course.

Many questions in algebraic topology can be treated equally well using any of various possible approaches. On the other hand each particular approach has one or the application where it is more suitable than the others. A surprising application where K-theory is best pertains to so-called finite-dimensional division algebras over the real field. They include the field of complex numbers as the only true extension field, then Hamilton's quaternion algebra which has real dimension 4 and is a skew-field (no longer commutative), and the Cayley octonions which form an extension of dimension 8 where even associativity must be sacrificed. A still common feature of all these is that every non-zero element is invertible (that is the meaning of division algebra). What we will prove using K-Theory is that no other such systems of generalised numbers can exist — and that for essentially topological reasons.

The first class will be held on Tuesday 18 October. Rather than for regular Übung I will use the Friday 21 October meeting to explain a few supplements to point set topology (which were not treated in the "Einführung" course), as well as other points you may wish to know about.

This observation suggests that the framework of Rⁿ and its open subset simply is too narrow for differential calculus, and should be extended by allowing for things that only locally look like Rⁿ: these are called manifolds. 'Thing' here means a topological space X of course, and as you will easily guess it will be required that every point of X admits a so-called chart, i.e. an open neighbourhood together with a homeomorphism from the latter to some open subset of Rⁿ. While this seems straightforward enough the charts have to be chosen in quite a subtle way in order to make analytic tools available on X and turn X into a 'differential' manifold. But once this is done differential calculus suddenly becomes a more lucid notion than ever before, stressing the original idea of differentiation as linear approximation. You wonder about linearity when looking at the sphere as an example of such a differential manifold? Well the trick is to attach to it its tangent space at every point; the collection of all these is the so-called tangent bundle, and it is excellent fun to learn that linear algebra works not only in vector spaces but more generally in vector bundles.

The gist of this is that literally all analysis can be done, and often is done with advantage, on manifolds — from this point of view every local problem can be reduced to one in coordinate analysis. Such a reduction amounts to choosing a chart, and diligent choice of that chart may already contribute to solve the problem in a similar way as in linear algebra, where a problem in a vector space may be simplified by the choice of a good basis. A well-known case in point would be the alternative use of cartesian or polar coordinates on the sphere.

Examples of manifolds are manifold: taking solutions of differentiable equations is a good source, and the theory itself provides many more, in particular as two manifolds may be glued to form a new one. Algebraic geometers encounter manifolds as varieties — solution sets of polynomial equations — that are non-singular: their topological properties as manifolds are deeply related to their algebro-geometric ones (though none of this can be even touched upon in this course). Manifolds occur in group theory in the guise of Lie groups, which by definition are groups and manifolds simultaneously. Last but not least physicists usually work on manifolds rather than coordinate spaces — though often using a language that tends to hide the fact.

The true strength of the notion of manifold pertains to global rather than local aspects, and two particular ones I will present in the course. The first is integration: not surprisingly this inherently global notion makes good if not better sense on a manifold than on Rⁿ, and is by and large the modern version of classical vector analysis. Unlike the latter, which dwells upon a zoo of seemingly unrelated formulas in (mostly) two or three variables, the more general vector analysis on manifolds is systematic and governed by just a few succinctly stated laws, which culminate in a famous and beautiful integral formula named after Stokes.

The second global aspect treated in the course concerns differential equations — a notion which also lives naturally on manifolds rather than coordinate spaces. The majority of the vast work that has been dedicated to differential equations assumes one such equation as given, and studies its solutions — an approach virtually pressed upon us by the fact that basic natural laws in physics, but also biology and economics, invariably are differential equations, while observable phenomena correspond to the myriad solutions of these few equations.

In the course I will look at ordinary differential equations from quite a different angle, namely as a tool to solve problems in topology. A first acquaintance with topology gives the (correct) impression that there are plenty of homeomorphisms around. But it is not at all easy to construct them: while you may have worked out for yourself simple ones like a homeomorphism between a square and a disk you can only take on faith the often stated fact that every (one-handled) cup is homeomorphic to a torus! The prospect of having to construct homeomorphisms explicitly on a case by case basis is simply not very exciting. But there is a systematic way of constructing homeomorphisms — in fact diffeomorphisms — from differential equations. Let me explain the idea. Given an ordinary differential equation like f'=f² the standard view is to study the initial value problem, looking for the unique solution to a given initial condition. In geometric terms the solution is a kind of time-parametrised 'flow line' through a given starting point. We may also vary the starting point: then the collection of all flow lines becomes a more comprehensive object called a flow. If we go to the other extreme and fix a positive moment of time while varying the starting point such a flow often produces differentiable mappings — which are in fact diffeomorphisms since the flow can be reversed by a reflection in the time axis.

How can we obtain flows with prescribed properties? It turns out that fundamental results from ordinary differential equations may be re-phrased for manifolds to establish a perfect dictionary between flows and differential equations and their respective properties. Now while flows (like homeomorphisms) are near impossible to construct directly, it is easy to construct differential equations because they are intrinsically linear objects (even those which are not linear in the standard sense). The strategy to construct flows and thereby diffeomorphism thus becomes equally simple and beautiful: first translate the properties you want the flow to have into the language of differential equations, secondly construct such a differential equation (even if on an abstract level), and from that differential equation finally return to the corresponding flow.

On the students' part the course will assume, evidently, a good grasp of the first year courses and the material from the "Einführung". The latter is also roughly covered by the first 11 sections of my notes "A Topology Primer", apart from the fundamental group — whose role is one of motivation rather than a strict prerequisite.

Nevertheless the concept of this course differs from that of "A Topology Primer". We will in the first part not deal with general topological spaces at all but build very concrete ones called simplicial complexes, which one might consider topologist's LEGO models. The combinatorics of such simplicial complexes gives rise in quite a natural way to algebraic entities called homology groups, which enjoy striking invariance properties.

In the second part of the course simplicial complexes will serve as a model in order to extend these invariants to arbitrary topological spaces. Applications and furthers extensions will be discussed as time permits.

An essential concern of mine is to highlight the outstanding beauty of topology. If topology arrived too late on the scene to be the queen of mathematics, the title being taken by number theory, she certainly is a strong contender for that of Miss Mathematics.

True topology is often called algebraic because it is based on the surprising fact that topology, an inherently geometric concept, is governed by rich algebraic structures. Under the name of homology these structures can be made quite explicit, and the vast majority of results in topology, and topological results in other fields, depend in some way on calculations of homology. Euler's famous theorem on polyhedra is a first example: if v, e, and f respectively denote the number of vertices, edges, and faces of a compact polyhedron then v-e+f=2.

Homology will therefore be the central topic of the course. Among numerous possibilities to explain homology, none very simple but some very complicated, I have opted for the simplest and most explicit one, which is based on the notion of cell complexes.

Students who choose this course must be thoroughly familiar with undergraduate analysis and linear algebra. They should also have a good knowlegde of basic topological concepts commonly subsumed under the name of general or point set topology, including neighbourhoods, separation of points, connectedness, compactness, and quotient topological spaces. All these notions are covered by the two hour course "Einführung in die Topologie" I have taught the previous term, and in fact (and not surprisingly) that course essentially defines the prerequisites for the one to come. Thus you may want to check details by consulting the corresponding lecture notes (see below). These notes were specifically written with German and international students in mind who wish to join the topology course at the beginning of the winter term. The text is self-contained and if you do not, or not fully meet the requirements as stated you may find the notes suitable for preparatory self-study. I will also be happy to give further information and personal assistance, and you should not hesitate to contact me at any time if you have any doubts or queries.