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Space and Spaces Space and spaces Graeme Segal The idea of space is central in the way we think. We organize our percep- tions in physical space, we think of time as a one-dimensional continuum, and we carry spatial notions over to any number of our conceptual constructs. Nevertheless, in some sense space is `our' technology, wonderfully evolved for dealing with our experience of the physical world. It is probably only an approximation to reality. Long ago I happened to be looking at an account of quantum field the- ory written `for the general reader' by Freeman Dyson, one of my favorite writers about science. I came upon a passage where he said that there are `well-understood mathematical reasons' why when we quantize a wave-like physical system such as the electromagnetic field the result is best described in terms of particles. \Unfortunately," he added, \the reason for this can- not be explained in non-mathematical terms." I felt rather let down, for I don't like the idea of mathematics as an arcane mystery where even the ba- sic ideas can only be explained to initiates. I also wondered exactly which well-understood mathematical reasons Dyson had in mind. I have thought about the question a lot since then, but haven't come up with anything that would be of much use to non-mathematicians; I leave it as a challenge. It seems to me a challenge even to give a clear mathematical account. I suppose that Dyson was thinking of the traditional statement that a free field can be regarded as a system of independent harmonic oscillators | which mystified me so much when I first encountered it in physics lectures as an undergradu- ate | but I believe that to understand the essential point involves thinking carefully about how we use the concept of space. That is the aim of this talk. One of the triumphs of mathematics is the creation of the real numbers as a model of a one-dimensional continuum. It perfectly encodes all our intuitions of how such a thing should be1. Another great achievement | if 1Once when I was young and innocent I tried to explain to an eminent philosopher 1 not quite on the same level | is the concept of a topological space, which captures a more general intuitive idea. Primarily, a space is a set with a notion of proximity, which tells us when a point of the the space is moving continuously as a function of time, and tells us when a real-valued function on the space is continuous. Beyond that, it has the feature that different regions in the space can in some sense be studied independently. The formal definition of a topological space does not involve the real numbers, but, as you see, my account of the intuition very much does. But spaces also have a non-local aspect. The crudest way to classify topological spaces is by their homotopy-type. If we start with the category of topological spaces and identify maps when they are homotopic | when one can be deformed continuously to another | then we get the homotopy category. In this category homotopy-equivalent spaces become isomorphic, so, effectively, the homotopy category is very much smaller than the category of topological spaces. It is the receptacle for the non-local properties of a space: it records how the space is connected-up globally | what we still need to know when we know everything about the local structure near every point of the space. We have very powerful intuitions about homotopy-types, at least in low dimensions. No-one ever doubted that a real polynomial equation of odd degree has a real root, long before there was any framework to prove the intermediate-value theorem. In the same way, the winding-number proof that the complex numbers are algebraically closed | the two-dimensional analogue of the intermediate-value theorem | carries immediate conviction2. The non-local nature of the homotopy category gives it a life of its own. Though in principle its objects are sets, we often come upon well-defined objects in the homotopy category for which there is no specially singled-out topological space which represents them. The London tube map is obvi- ously well-defined up to homotopy, but what, exactly, is the underlying set? There are many more extreme examples: in mathematical logic and com- in my Oxford college why mathematicans are no longer puzzled about how Achilles can overtake a tortoise, and the resulting explosion has made me wary of transgressing cultural boundaries. 2Our intuition is of course patchy and unreliable: one is rightly certain that if one puts a disc of cloth down on a smaller circular plate so that the edge of the cloth runs twice around the boundary of the plate then the cloth must cover the plate completely, but it is less immediately evident that performing the act with ordinary cloth in ordinary space is impossible. 2 puter science it can be useful to think of a `proof' as a `path' | defined up to homotopy | from one proposition to another. Evidently these paths are not maps from an interval of the real numbers into a definite topological space. One place where the independent life of the homotopy category is impor- tant is in algebraic geometry. An algebraic variety over the complex numbers is a space, but not just any space: when a space is defined by algebraic equa- tions its homotopy-type is very strongly constrained, and has an elaborate additional structure. For the moment I'd just like to point out that the homotopy-type of a variety plays a mysterious role in unexpected contexts. I first became interested in mathematics through physics, and I was incurious about number-theoretical questions like Fermat's last theorem. I felt more interested when the theorem was restated to me as the fact that the real plane curve with equation xn + yn = 1 somehow gets from (1,0) to (0,1) without going through any points with rational coordinates. But I was gripped when I heard of Mordell's conjecture, proved by Faltings in 1983, which gives a criterion for any plane curve with a rational equation to have at most finitely many rational points. The criterion is in terms of the homotopy-type of the surface formed by the complex points of the curve: if the surface has genus greater than 1, i.e. has more than one `handle', then there are at most finitely many rational points. What can the global topology of the complex variety | or even of the subspace of points with real coordinates | have to do with the rational points? An unromantic answer is that a property of the algebraic equations happens to control both; but the homotopy-type of the surface leaps out at one, while the genus is far from being a salient feature of the algebraic description of the curve. More magical still is the case of algebraic varieties defined over a finite field by equations with integer coefficients. According to the Weil conjec- tures, proved by Grothendieck and Deligne, the number of points of such a variety is again related to the homotopy type of the associated complex variety. The essential idea of the Weil conjectures is that an algebraic va- riety over a finite field, though it is not in any ordinary sense a topological space, nevertheless has a homotopy-type which is defined by its algebraic structure. Grothendieck's construction of this homotopy-type began from the observation that to know the homotopy-type of an ordinary space one does not need to know the space as a point-set: all one needs to know is the way its contractible open subsets are fitted together combinatorially. There is a simple construction | making essential use of the real numbers | which 3 associates a space3 jCj, and hence a homotopy-type, to any category C. The partially-ordered set UX of contractible open subsets of a space X is a cate- gory, with just the inclusions as morphisms. Surprisingly, the homotopy-type of jUX j is precisely the homotopy-type of the space X. What really makes the homotopy type of a category significant is that it depends on the cat- egory only up to equivalence of categories: the category of \all" finite sets and bijections has the same homotopy type as the countable subcategory whose objects are just the sequence of particular sets f1; 2; : : : ; ng for n ≥ 0, and whose morphisms are the symmetric groups. Grothendieck defined a category of `generalized open subsets' for an arbitrary algebraic variety, and (oversimplifying slightly) the homotopy-type of the variety is defined as that of this category. When we prove the fundamental theorem of algebra by homotopy theory we are applying our intuitions to our best model of physical space. But often we use the language of spaces and geometry more like an analogy. We say, for instance, that a module is projective if it is `locally free', or speak of an ‘infinitesimal deformation' of an algebraic variety over an arbitrary field; in these cases there is no genuine space around4 of the sort to which our intuitions apply. These analogies are nonetheless very powerful tools. It is something of a surprise, therefore, that the homotopy-types we assign to alge- braic varieties over arbitrary fields are those of genuine spaces, even though the points of the spaces are not those of the variety. Over the last half- century the assignment of homotopy-types to groups, rings, and all manner of other algebraic objects, has become steadily more pervasive in mathemat- ics, since its beginnings when it was called `homological algebra'.
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