
While general relativity and the Standard Model of particle physics are very different in many ways, they have one important thing in common: both are gauge theories. I will not attempt to explain what a gauge theory is here. I just want to recommend the following nice book on the early history of this subject:
1) Lochlainn O'Raifeartaigh, The Dawning of Gauge Theory, Princeton U. Press, Princeton, 1997.
This contains the most important early papers on the subject, translated into English, together with detailed and extremely intelligent commentary. It starts with Hermann Weyl's 1918 paper "Gravitation and Electricity", in which he proposed a unification of gravity and electromagnetism. This theory was proven wrong by Einstein in a oneparagraph remark which appears at the end of Weyl's paper  Einstein noticed it would predict atoms of variable size!  but it highlighted the common features of general relativity and Maxwell's equations, which were later generalized to obtain the modern concept of gauge theory.
It also contains Theodor Kaluza's 1921 paper "On the Unification Problem of Physics" and Oskar Klein's 1926 paper "Quantum Theory and FiveDimensional Relativity". These began the trend, currently very popular in string theory, of trying to unify forces by postulating additional dimensions of spacetime. It's interesting how gauge theory has historical roots in this seemingly more exotic notion. The original KaluzaKlein theory assumed a 5dimensional spacetime, with the extra dimension curled into a small circle. Starting with 5dimensional general relativity, and using the U(1) symmetry of the circle, they recovered 4dimensional general relativity coupled to a U(1) gauge theory  namely, Maxwell's equations. Unfortunately, their theory also predicted an unobserved spin0 particle, which was especially problematic back in the days before mesons were discovered.
I wasn't familiar with another item in this book, Wolfgang Pauli's letter to Abraham Pais entitled "MesonNucleon Interactions and Differential Geometry". This theory, "written down July 2225 1953 in order to see how it looks", postulated 2 extra dimensions in the shape of a small sphere. The letter begins, "Split a 6dimensional space into a (4+2)dimensional one." At the time, mesonnucleon interactions were believe to have an SU(2) symmetry corresponding to conservation of "isospin". Pauli obtained a theory with this symmetry group using the SU(2) symmetry of the sphere.
Apparently Pauli got a lot of his inspiration from Weyl's 1929 paper "Electron and Gravitation", also reprinted in this volume. This masterpiece did all the following things: it introduced the concept of 2component spinors (see "week109"), considered the possibility that the laws of physics violate parity and time reversal symmetry, introduced the tetrad formulation of general relativity, introduced the notion of a spinor connection, and explicitly derived electromagnetism from the gauge principle! A famously critical fellow, Pauli lambasted Weyl's ideas on parity and time reversal violation  which are now known to be correct. But even he conceded the importance of deriving Maxwell's equations from the gauge principle, saying "Here I must admit your ability in Physics". And he incorporated many of the ideas into his 1953 letter.
An allaround good read for anyone seriously interested in the history of physics! It's best if you already know some gauge theory.
Now let me continue the tour of homotopy theory I began last week. I was talking about simplices. Simplices are amphibious creatures, easily capable of living in two different worlds. On the one hand, we can think of them as topological spaces, and on the other hand, as purely algebraic gadgets: objects in the category of finite totally ordered sets, which we call Δ. This gives simplices a special role as a bridge between topology and algebra.
This week I'll begin describing how this works. Next time we'll get into some of the cool spinoffs. I'll keep up the format of listing tools one by one:
E. Geometric realization. In "week115" I talked about simplicial sets. A simplicial set is a presheaf on the category Δ. Intuitively, it's a purely combinatorial way of describing a bunch of abstract simplices glued together along their faces. We want a process that turns such things into actual topological spaces, and also a process that turns topological spaces back into simplicial sets.
Let's start with the first one. Given a simplicial set X, we can form a space X called the "geometric realization" of X by gluing spaces shaped like simplices together in the pattern given by X. Given a morphism between simplicial sets there's an obvious continuous map between their geometric realizations, so geometric realization is actually a functor
 : SimpSet → Top
from the category of simplicial sets, SimpSet, to the category of topological space, Top.
It's straightforward to fill in the details. But if we want to be slick, we can define geometric realization using the magic of adjoint functors  see below.
F. Singular Simplicial Set. The basic idea here is that given a topological space X, its "singular simplicial set" Sing(X) consists of all possible ways of mapping simplices into X. This gives a functor
Sing: Top → SimpSet.
We make this precise as follows.
By thinking of simplices as spaces in the obvious way, we can associate a space to any object of Δ, and also a continuous map to any morphism in Δ. Thus there's a functor
i: Δ → Top.
For any space X we define
Sing(X): Δ → Set
by
Sing(X)() = hom(i(),X)
where the blank slot indicates how Sing(X) is waiting to eat a simplex and spit out the set of all ways of mapping it  thought of as a space!  into the space X. The blank slot also indicates how Sing(X) is waiting to eat a morphism between simplices and spit out a function between sets.
Having said what Sing does to spaces, what does it do to maps? The same formula works: for any map f: X → Y between topological spaces, we define
Sing(f)() = hom(i(),f).
It may take some headscratching to understand this, but if you work it out, you'll see it works out fine. If you feel like you are drowning under a tidal wave of objects, morphisms, categories, and functors, don't worry! Medical research has determined that people actually grown new neurons when learning category theory.
In fact, even though it might not seem like it, I'm being incredibly pedagogical and nurturing. If I were really trying to show off, I would have compressed the last couple of paragraphs into the following one line:
Sing()() = hom(i(),).
where Sing becomes a functor using the fact that for any category C there's a functor
hom: C^{op} x C → Set
where C^{op} denotes the opposite of C, that is, C with all its arrows turned around. (See "week78" for an explanation of this.)
Or I could have said this: form the composite
i x 1 hom Δ^{op} x Top > Top^{op} x Top > Setand dualize this to obtain
Sing: Top → SimpSet.These are all different ways of saying the same thing. Forming the singular simplical set of a space is not really an "inverse" to geometric realization, since if we take a simplicial set X, form its geometric realization, and then form the singular simplicial set of that, we get something much bigger than X. However, if you think about it, there's an obvious map from X into Sing(X). Similarly, if we start with a topological space X, there's an obvious map from Sing(X) down to X.
What this means is that Sing is the right adjoint of  , or in other words,   is the left adjoint of Sing. Thus if we want to be slick, we can just define geometric realization to be the left adjoint of Sing. (See "week77""week79" for an exposition of adjoint functors.)
G. Chain Complexes. Now gird yourself for some utterly unmotivated definitions! If you've taken a basic course in algebraic topology, you have probably learned about chain complexes already, and if you haven't, you probably aren't reading this anymore  so I'll just plunge in.
A "chain complex" C is a sequence of abelian groups and "boundary" homomorphisms like this:
d_{1} d_{2} d_{3} C_{0} < C_{1} < C_{2} < C_{3} < ...satisfying the magic equation
d_{i} d_{i+1} x = 0This equation says that the image of d_{i+1} is contained in the kernel of d_{i}, so we may define the "homology groups" to be the quotients
H_{i}(C) = ker(d_{i}) / im(d_{i+1})The study of this stuff is called "homological algebra". You can read about it in such magisterial tomes as:
2) Henri Cartan and Samuel Eilenberg, Homological Algebra, Princeton University Press, 1956.
or
3) Saunders Mac Lane, Homology, SpringerVerlag, Berlin, 1995.
But it you want something a bit more userfriendly, try:
4) Joseph J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.
The main reason chain complexes are interesting is that they are similar to topological spaces, but simpler. In "singular homology theory", we use a certain functor to convert topological spaces into chain complexes, thus reducing topology problems to simpler algebra problems. This is usually one of the first things people study when they study algebraic topology. In sections G. and H. below, I'll remind you how this goes.
Though singular homology is very useful, not everybody gets around to learning the deep reason why! In fact, chain complexes are really just another way of talking about a certain especially simple class of topological spaces, called "products of EilenbergMacLane spaces of abelian groups". In such spaces, topological phenomena in different dimensions interact in a particularly trivial way. Singular homology thus amounts to neglecting the subtler interactions between topology in different dimensions. This is what makes it so easy to work with  yet ultimately so limited.
Before I keep rambling on I should describe the category of chain complexes, which I'll call Chain. The objects are just chain complexes, and given two of them, say C and C', a morphism f: C → C' is a sequence of group homomorphisms
f_{i}: C_{i} → C_{i}'making the following big diagram commute:
d_{1} d_{2} d_{3} C_{0} < C_{1} < C_{2} < C_{3} < ...     f_{0} f_{1} f_{2} f_{3}     V V V V C_{0}' < C_{1}' < C_{2}' < C_{3}' < ... d_{1}' d_{2}' d_{3}'The reason Chain gets to be so much like the category Top of topological spaces is that we can define homotopies between morphisms of chain complexes by copying the definition of homotopies between continuous maps. First, there is a chain complex called I that's analogous to the unit interval. It looks like this:
d_{1} d_{2} d_{3} d_{4} Z+Z < Z < 0 < 0 < ...The only nonzero boundary homomorphism is d_{1}, which is given by
d_{1}(x) = (x,x)(Why? We take I_{1} = Z and I_{0} = Z+Z because the interval is built out of one 1dimensional thing, namely itself, and two 0dimensional things, namely its endpoints. We define d_{1} the way we do since the boundary of an oriented interval consists of two points: its initial endpoint, which is positively oriented, and its final endpoint, which is negatively oriented. This remark is bound to be obscure to anyone who hasn't already mastered the mystical analogies between algebra and topology that underlie homology theory!)
There is a way to define a "tensor product" C x C' of chain complexes C and C', which is analogous to the product of topological spaces. And there are morphisms
i,j: C → I x Canalogous to the two maps from a space into its product with the unit interval:
i, j: X → [0,1] x X i(x) = (0,x), j(x) = (1,x)Using these analogies we can define a "chain homotopy" between chain complex morphisms f,g: C → C' in a way that's completely analogous to a homotopy between maps. Namely, it's a morphism F: I x C → C' for which the composite
i F C > I x C > C'equals f, and the composite
j F C > I x C > C'equals g. Here we are using the basic principle of category theory: when you've got a good idea, write it out using commutative diagrams and then generalize the bejeezus out of it!
The nice thing about all this is that a morphism of chain complexes f: C → C' gives rise to homomorphisms of homology groups,
H_{n}(f): H_{n}(C) → H_{n}(C').In fact, we've got a functor
H_{n}: Chain → Ab.And even better, if f: C → C' and g: C → C' are chain homotopic, then H_{n}(f) and H_{n}(g) are equal. So we say: "homology is homotopyinvariant".
H. The Chain Complex of a Simplicial Abelian Group. Now let me explain a cool way of getting chain complexes, which goes a long way towards explaining why they're important. Recall from item D. in "week115" that a simplicial abelian group is a contravariant functor C: Δ → Ab. In particular, it gives us an abelian group C_{n} for each object n of Δ, and also "face" homomorphisms
partial_{0}, ...., partial_{n1}: C_{n} → C_{n1}coming from all the ways the simplex with (n1) vertices can be a face of the simplex with n vertices. We can thus can make C into a chain complex by defining d_{n}: C_{n} → C_{n1} as follows:
d_{n} = sum (1)^{i} partial_{i}The thing to check is that
d_{n} d_{n+1} x = 0The alternating signs make everything cancel out! In the immortal words of the physicist John Wheeler, "the boundary of a boundary is zero".
Unsurprisingly, this gives a functor from simplicial abelian groups to chain complexes. Let's call it
Ch: SimpAb → Chain
More surprisingly, this is an equivalence of categories! I leave you to show this  if you give up, look at May's book cited in section C. of "week115". What this means is that simplicial abelian groups are just another way of thinking about chain complexes... or vice versa. Thus, if I were being ultrasophisticated, I could have skipped the chain complexes and talked only about simplicial abelian groups! This would have saved time, but more people know about chain complexes, so I wanted to mention them.
I. Singular Homology. Okay, now that we have lots of nice shiny machines, let's hook them up and see what happens! Take the "singular simplicial set" functor:
Sing: Top → SimpSet,
the "free simplicial abelian group on a simplicial set" functor:
L: SimpSet → SimpAb,
and the "chain complex of a simplicial abelian group" functor:
Ch: SimpAb → Chain,
and compose them! We get the "singular chain complex" functor
C: Top → Chain
that takes a topological space and distills a chain complex out of it. We can then take the homology groups of our chain complex and get the "singular homology" of our space. Better yet, the functor C: Top → Chain takes homotopies between maps and sends them to homotopies between morphisms of chain complexes! It follows that homotopic maps between spaces give the same homomorphisms between the singular homology groups of these spaces. Thus homotopyequivalent spaces will have isomorphic homology groups... so we have gotten our hands on a nice tool for studying spaces up to homotopy equivalence.
Now that we've got our hands on singular homology, we could easily spend a long time using it to solve all sorts of interesting problems. I won't go into that here; you can read about it in all sorts of textbooks, like:
5) Marvin J. Greenberg, John R. Harper, Algebraic Topology: A First Course, Benjamin/Cummings, Reading, Massachusetts, 1981.
or
6) William S. Massey, Singular Homology Theory, SpringerVerlag, New York, 1980.
which uses cubes rather than simplices.
What I'm trying to emphasize here is that singular homology is a composite of functors that are interesting in their own right. I'll explore their uses a bit more deeply next time.
