Michael Atiyah’s work in algebraic topology

By Graeme Segal

Abstract

In 1960 algebraic topology was at the centre of the mathematical stage, but Michael Atiyah burst into the field and changed its focus and its language. I describe his work of the following decade and its influence, keeping to the themes of -theory and generalized cohomology to minimise the overlap with Dan Freed’s account of Atiyah’s work on index theory, which also appears in this issue.

In a number of talks and interviews (e.g., Reference Min) Michael Atiyah spoke thoughtfully about his way of doing mathematics. He explained that he had rarely set out to solve a specific problem: rather he would be intrigued by some aspect of the mathematical landscape and feel driven to find out how it worked and get to the bottom of it. His work in algebraic topology exemplifies this very well. In the 1960s, by inventing -theory and the idea of a generalized cohomology theory, he changed the complexion of algebraic topology in the world, and, even after all his later work, I would say that the perspective of algebraic topology remained central in his mathematics. In his own eyes, however, he was a more general kind of geometer, who used the ideas of algebraic topology when they were needed. He felt the main stream of algebraic topology was too inward-looking, and held himself a little apart from it; but in the second half of his career he was always eager to apply his topologist’s expertise in areas of mathematics which were new to him.

He began research in 1952 as an algebraic geometer. Though he was attracted by work in the older ‘Italian’ style, he wisely chose Hodge—the most ‘modern’ geometer in Cambridge—as his supervisor. Hodge had been deeply influenced by Lefschetz, who was at once the greatest exponent of topological ideas in algebraic geometry and the creator of much of modern algebraic topology. But Hodge’s approach to topology was different from Lefschetz’s. It focussed on the differential forms on algebraic varieties, and their integrals, combining de Rham’s theorem with Chern’s differential-geometric construction of characteristic classes. There are many parallels between Hodge’s career and Michael’s, and one of them is that Hodge’s greatest triumph—the proof that every cohomology class (with real coefficients) of a closed smooth manifold has a unique harmonic form representative—is essentially a theorem of analysis, a field in which Hodge was not an expert and needed help from others.

Until 1959 Michael’s work was in algebraic geometry, and though he expressed it in the modern language of topology, treating ruled surfaces as fibre bundles and speaking of ‘vector bundles’ rather than ‘linear systems’, his methods were algebraic. His thesis, submitted in 1955, on algebraic differential forms, displayed a command of spectral sequences and of the cohomology theory of sheaves.

One paper from this period stands out for illustrating so well his knack, throughout his career, of being the first to put his finger on an essential point. In Reference AtCl he shows that the characteristic classes of a holomorphic vector bundle on a complex manifold all come from one basic class —now called the Atiyah class—which describes the extension of sheaves (or of holomorphic vector bundles)⁠Footnote1 on

1

Here the bundle is identified with its sheaf of holomorphic sections, while is the sheaf of holomorphic 1-forms with values in , and is the sheaf of 1-jets of holomorphic sections of . We observe that the sheaf of homomorphisms from to , whose first cohomology is the home of the extension-class, can be identified with

Backlinks: Reference 1, Reference 2, Reference 3, Reference 4, Reference 5, Reference 6, Reference 7, Reference 8.

The class is the holomorphic analogue of the curvature defined in the differentiable context for a bundle with a connection. The characteristic classes of , in the Dolbeault cohomology , are obtained from as , where is a polynomial of degree invariant under conjugation by , just as the Chern–Weil description of the corresponding classes in de Rham cohomology is .

The best-known of Michael’s early papers, however, is his study Reference EllCve of the moduli space of holomorphic bundles on an elliptic curve, which foreshadows his later interest in gauge theory. It is described in Simon Donaldson’s contribution to this volume Reference Do.

Immediately after finishing his thesis, Michael spent the year 1955–56 at the Institute for Advanced Study (IAS) in Princeton. He often wrote of the enormous stimulation he got from meeting the young stars Serre, Milnor, Hirzebruch, Bott, and Singer, as well as the older mathematicians Kodaira and Spencer. The 1950s were a decade of spectacular flowering of algebraic topology, and for anyone interested in geometry it must have seemed the most exciting thing on the mathematical stage. Among the main developments—all being worked on in Princeton, and all to prove relevant to Michael’s work—were

the reformulation of the subject in the language of category theory;

the homotopy-theoretic understanding of fibre bundles and their classifying spaces and characteristic classes;

Thom’s invention of the cobordism ring , and the reduction of its calculation to homotopy theory;

the use of cohomology operations, and especially the Steenrod algebra, to calculate the cobordism ring, the development of the Adams spectral sequence, and the solution of the Hopf invariant problem;

Bott’s application of Morse theory to the topology of Lie groups, culminating in his periodicity theorem;

Milnor’s discovery of exotic smooth structures on spheres, and his work with Kervaire relating differential topology to the homotopy theory of spheres and Grassmannians.

1. The beginnings of -theory

Of the mathematicians Michael met in Princeton, Hirzebruch (who had corresponded with Todd and given the Todd classes their name) was the nearest to him in mathematical background and became his close collaborator for the next few years. A little older than Michael, he was already a considerable figure in the mathematical world. He was an expert in algebraic topology, as well as in sheaf-theoretic methods in complex analysis. He had developed a very practical calculus for encoding the multiplicative characteristic classes of vector bundles as formal power series, which enabled him to read off a formula for the signature of a compact oriented smooth manifold—or indeed for any multiplicative cobordism invariant—in terms of the Pontryagin classes of the tangent bundle. But he was most famous for his ‘higher-dimensional Riemann–Roch’ theorem, which, for a holomorphic vector bundle on a compact complex manifold , expresses the Euler number

in terms of the Chern classes of and of the tangent bundle of . (Here is the cohomology of with coefficients in the sheaf of holomorphic sections of .)

Hirzebruch’s calculus of characteristic classes

As Hirzebruch’s calculus of characteristic classes is so central in Michael’s work, I shall interpolate here a brief account of it. A characteristic class for complex vector bundles, with coefficients in a commutative ring , assigns to a bundle on a space a cohomology class . Because an -dimensional bundle can be described by a map —unique up to homotopy—from to the classifying space or Grassmannian , we can describe by a sequence of classes such that . The characteristic class is called additive if and multiplicative if .

Because the direct-sum map

induces an injection in cohomology, any characteristic class is determined by its values on sums of line bundles, and a multiplicative class is determined by the single power-series . In fact is just the subring of elements in

which are symmetric under permuting , so any choice of gives rise to the multiplicative class with , and we can rewrite this as a function of the elementary symmetric functions of , which are the Chern classes of the bundle.

A multiplicative class is stable, i.e., does not change if a trivial bundle is added to , if and only if the power-series has constant term 1. In this case we can think of as an element of the cohomology of the stabilized classifying space , and defines a ring-homomorphism .

The picture for multiplicative characteristic classes of real vector bundles is little different, as long as the bundles are orientable and we use a coefficient ring in which is invertible. We must replace by , but the upshot is just that stable multiplicative classes correspond to the even power-series

Finally, multiplicative characteristic classes in rational cohomology are essentially the same thing as rational-valued genera , i.e., ring-homomorphisms

from Thom’s ring of cobordism classes of oriented manifolds. For there is a ring-homomorphism

which associates to a closed oriented -manifold the image of its fundamental homology class under the classifying map of its tangent bundle, and Thom proved that this map becomes an isomorphism when tensored with .

The multiplicative characteristic classes that will come into this account are

the total Chern class , corresponding to the series ;

Hirzebruch’s -class for real bundles, corresponding to the series , which gives his formula for the signature;

the Todd class, corresponding to the series ;

the -class for real bundles, corresponding to the series ;

the Euler class of an oriented even-dimensional real bundle, which is multiplicative but not stable—it vanishes if the bundle has a nonvanishing section—and corresponds to the element .

Apart from these multiplicative classes, a central role is played by the Chern character , given by the symmetric function

It has the properties

Returning to our story, we can now state the Riemann–Roch formula:

Hirzebruch saw that the formula implies that the Euler number , which prima facie depends on the very rigid holomorphic structure of and , is actually a purely topological invariant of the topological space and the two complex vector bundles and . Because is an integer, while the Riemann–Roch expression for it is a polynomial with rational coefficients in the Chern classes, the theorem implies a rich array of congruences between the topologically defined characteristic numbers of the manifold. These look like strong constraints on its topology. It was natural to ask whether they hold for all smooth manifolds, or at least for a larger class than just smooth algebraic varieties.

When their active collaboration began, Atiyah and Hirzebruch were both thinking about two of the important developments of 1957–58: Grothendieck’s generalization of Hirzebruch’s theorem, and Bott’s periodicity theorem, which gave an explicit homotopy equivalence

between the stabilized classifying space of the unitary groups and its two-fold loop-space. The two results were not obviously related.

Grothendieck’s theorem concerned a proper map of smooth algebraic varieties, and it was stated in terms of Grothendieck groups. For any category in which there is a notion of short exact sequence,

the Grothendieck group of is the abelian group generated by the isomorphism classes of objects of , subject to the relations for every exact sequence. Grothendieck applied this construction to the category of coherent sheaves on a variety to define his -group , and observed that for a smooth variety over the complex numbers it is related to the cohomology of as a topological space by the Chern character map

Coherent sheaves can be pushed forward by proper maps, and Grothendieck showed that a proper map induces a homomorphism⁠Footnote2

2

It is not, however, simply the push-forward of sheaves which induces Grothendieck’s map on . Because is not an exact functor, must be defined as the alternating sum of the right-derived functors of .

closely related to the Gysin homomorphism which is the map induced by on homology regarded as a map of cohomology by using the Poincaré duality isomorphisms of both and . Grothendieck’s theorem asserts that the diagram

does not quite commute, but commutes if the Chern character maps on the left and right are multiplied by the invertible elements and of the respective cohomology rings. Hirzebruch’s Riemann–Roch theorem is contained in Grothendieck’s as the case when is a point, for a coherent sheaf on a point is simply a finite-dimensional vector space, whose Chern character is its dimension, and is the alternating sum of the cohomology groups .

The strategy of Grothendieck’s proof was to have a lasting influence on Michael’s work. If the theorem holds for maps and , then it obviously holds for . But a projective variety has an embedding in projective space, so can be factorized

where the first map is and the second is the projection. It is enough, therefore, to prove the theorem for an embedding and for the projection of a product with . In the differentiable category, because a closed manifold can be embedded in a sphere, the analogous factorization of a map of smooth manifolds is , and Michael had the idea that the natural way to push vector bundles forward under the projection is by Bott periodicity. This was a very new angle on Bott’s theorem, which in the topological world was mainly seen in the light of Milnor’s immediate recognition Reference BoM of it as the crucial fact of homotopy theory needed to prove the nonparallelizability of spheres of dimension greater than 7.

Atiyah and Hirzebruch put Grothendieck’s theorem together with Bott periodicity and obtained a differentiable Riemann–Roch theorem⁠Footnote3 Reference DiffRR which answered many of the questions about the integrality of characteristic numbers mentioned above. It also put in the foreground the relevance of a spin structure on the manifold—a ‘higher’ kind of orientability—for integrality properties, and (using the version for real vector bundles which was part of the announcement) it gave a new proof of Rokhlin’s theorem that the signature of a 4-manifold is divisible by 16 if it has a spin structure. Nevertheless, in its first incarnation the differentiable theorem was not very geometrical or illuminating. It conjectured—and, sufficiently for most applications, established—that if a map of differentiable manifolds satisfies a certain spin-orientability condition, one can associate to it a map of Grothendieck groups of vector bundles which can be calculated cohomologically by the same formula which Grothendieck found in the algebraic case. But the announcement gave no reason for the existence of —indeed it asserted discouragingly that in the differentiable case there is no analogue of the holomorphic (or algebraic) operation of pushing forward a coherent sheaf. This pessimism perhaps came from defining differently for embeddings and projections: it would have been more accurate to say that in the differentiable context there was no analogue of coherent sheaves. Understanding the situation better was to be the goal of much of Michael’s work for more than ten years, and it led him into index theory, which, as he later remarked, is “really the same thing as -theory”.

3

The first announcement was Hirzebruch’s Bourbaki seminar in February 1959.

In formulating their theorem, Atiyah and Hirzebruch invented the -theory’ of algebraic topology, and with it the idea of a generalized cohomology theory. Both ideas immediately took hold in algebraic topology, and it is worth reflecting on the reasons. After Grothendieck’s work it cannot have been such a step to consider the Grothendieck group of vector bundles on a compact space , and it was known that this could be identified with , the homotopy classes of maps from to . Puppe had recently published his semi-infinite exact sequence Reference P for any such functor, and Bott’s theorem immediately extended this to the doubly infinite sequence which is the defining property of a generalized cohomology theory. Furthermore, Barratt Reference B and others had—without using the words—already shown that the stable homotopy classes of maps from a varying compact space into any given fixed space do indeed form a generalized cohomology theory. So in some sense the crucial point was a matter of language: the new concepts were ideally suited to formulate the ideas then at the forefront of algebraic topology. It certainly helped that both Atiyah and Hirzebruch were superb and charismatic lecturers.

That, though, is the perspective of hindsight. In his own commentaries Michael wrote that “introducing the odd-dimensional -groups seemed at the time a daring generalization”. We should remember, too, that nothing about Grothendieck’s group suggested it could be just one component of a -graded group: its elements already included the classes of algebraic cycles of all dimensions in . Indeed the splitting of into its cohomological components as eigenspaces of the Adams operations, with the topological dimension arising as an eigenvalue, seems to me one of the enduring mysteries of the subject. We shall return to this in §3 and §7.

Of course the new theory would not have had such impact without the applications that came with it. Probably the most striking early application was made by Frank Adams, Michael’s rival from undergraduate days at Trinity College, Cambridge, who in 1961 used -theory to prove the long-conjectured theorem that the maximum number of linearly independent tangent vector fields on a sphere is the Radon–Hurwitz number . (Clifford algebras and spinors appeared here once again in connection with -theory, for is the largest number such that the Clifford algebra with generators⁠Footnote4 acts on the vector space .) Adams, with the invention of the Adams spectral sequence and the solution of the Hopf invariant problem in 1958, was already established as a leader in algebraic topology, but with hindsight it seems fair to say that the vector fields theorem was a fruit ready to be plucked in 1961. The question is to determine for which there is a cross-section of the forgetful map

4

This means that we have anticommuting skew matrices such that . If is a point of this gives us tangent vectors at .

from the Stiefel manifold of orthonormal -tuples of vectors in to the unit sphere. Ioan James Reference J had made a crucial reduction of the problem to one about the “stunted projective spaces” , and Michael had reworked this in Reference ThCpl into a convenient language of Thom complexes, and in his paper Reference At-Todd with Todd had solved the analogous but simpler problem for the unitary groups.

Michael’s paper Reference ThCpl on Thom complexes is perhaps the one where he writes most like an orthodox algebraic topologist. It is worth dwelling on because, without proving any deep new theorem, it was influential in steering the evolution of the subject towards the Atiyah perspective. The Thom space of a real vector bundle on a compact space is the one-point compactification of the space , and Reference ThCpl develops the idea of as a twisted suspension of . (If is the trivial bundle , then is the -fold suspension of .) It goes on to show that in the stable homotopy category is well-defined even when is replaced by a virtual bundle, i.e., the formal difference of two vector bundles. Most importantly, it shows that in the stable homotopy category (which is an additive category), the dual object to a closed -manifold is the Thom space , where is the tangent bundle of . If is embedded in , the dual of is therefore the -fold desuspension of the Thom space of the normal bundle . Thus , and if is orientable, the fundamental class corresponds to the Thom class .

The Thom space of a real vector bundle can also be described as the mapping cone of , where is the complement of the zero-section in , and is a spherical fibration on , i.e., a bundle whose fibres are homotopy-spheres. Thus a Thom space can be defined for any spherical fibration, and its stable homotopy type depends only on the stable fibre-homotopy type of the spherical fibration. Michael pointed out the invariance of under fibre-homotopy equivalences of (I shall return to this when discussing operations in -theory below) but stopped short of discovering the Spivak normal fibration.⁠Footnote5 When a compact space is embedded in as a neighbourhood deformation retract of an open subset , the dual of in the stable homotopy category is the one-point compactification , and the complement is a spherical fibration—the Spivak fibration of —if and only if satisfies Poincaré duality. (The proof of this striking example of a global condition implying a property that looks local is almost obvious if is simply-connected, but depends on defining carefully what is meant by Poincaré duality when there is a fundamental group.)

5

This is described in Spivak’s 1964 thesis, and is attributed there to his advisor Milnor.

Generalized cohomology theories are not mentioned in Reference ThCpl, though it was submitted only two weeks before his paper Reference BordCob on bordism theories. Looking at Reference ThCpl now, I wondered why it fails to point out that it effectively contains a simple proof of the differentiable Riemann–Roch theorem, which was left in a somewhat unsatisfactory state in the preceding papers Reference DiffRR and Reference Tucs. I shall give the argument and then speculate about Michael’s lack of enthusiasm for it.

An -orientation of a rank real vector bundle , for any multiplicative cohomology theory , is a choice of a Thom class , i.e., a class which restricts to a generator of for each fibre of . (Here I have used the idea of cohomology with compact supports, which, for a locally compact space , is defined by .) The reason for the terminology is that the two generators of correspond to the orientations of . Because is a module over the ring , a Thom class defines an isomorphism from to .

When we have a map of closed manifolds , we can embed in some , and then, if is large compared with the dimension of , an arbitrarily small deformation of will be an embedding of —and hence of its tubular neighbourhood —inside the tubular neighbourhood of . Because one-point compactification is a contravariant functor for open embeddings, this means that induces a canonical map

in the stable homotopy category. If and are -oriented, then combining with the Thom isomorphisms gives us the desired map

In fact, if the Thom classes are multiplicative for direct sums of bundles, as is usually the case, then we need only the virtual bundle to be -oriented.

Finally, if we have two multiplicative theories and and a multiplicative transformation , then we get a multiplicative characteristic class with values in , defined on the class of bundles oriented for both theories, by⁠Footnote6

6

The reason for the minus sign before is that we want to be associated to the Thom isomorphism for the normal bundle of .

This gives us an abstract Riemann–Roch theorem (which Atiyah and Hirzebruch were to apply to multiplicative automorphisms of classical cohomology in Reference CohOps, cf. §7 below) asserting that the diagram

commutes when the vertical maps are multiplied by and .

The extensive work Reference BorH on characteristic classes by Borel and Hirzebruch had implicitly established that a real vector bundle is orientable for the theory only when it has a spin-structure, or, equivalently, when the Stiefel–Whitney class lifts to an integral class. Indeed their work associates a specific Thom class to a spin-structure, and it has the property that

where is the unique classical Thom class defined by the orientation of . So the abstract theorem gives a complete proof of the differentiable Riemann–Roch theorem, adequate for all the applications that had been made of it, e.g., in Michael’s paper Reference ImmEmb on immersions and embedding of manifolds.

Michael would not have liked the way this account treated -theory as just one among a class of cohomology theories. He saw -theory as a very special theory with a deep basis in analysis related to Bott periodicity. He was convinced that it was simpler and more natural than classical cohomology. A -theory class, he felt, represents a natural geometric object—a vector bundle—whereas a classical cohomology class involves an elaborate algebraic structure. He wanted to see the -theory Gysin maps, too, as natural geometric operations, which would imply, rather than follow from, the cohomological calculations of Reference BorH.

He developed these ideas in a series of papers in the 1960s, and in his pedagogical book Reference KTh, which completely avoids using classical cohomology. By nature he was impatient of foundational material, and these works present not so much foundations as a point of view. I shall review a number of its strands in turn:

(§2)

calculating the -theory of geometrically important spaces and showing how it reflects and illuminates the geometry;

(§3–§5)

elaborating the nature of -theory classes and their relation to the dual -homology classes, and especially the relation of the Gysin map to Bott periodicity;

(§6)

equivariant -theory;

(§7)

the natural operations in -theory;

(§8)

the -theory of real vector bundles and the role of Clifford algebras.

2. Calculations

The first systematic account Reference Tucs of -theory was written in early 1960. It defines for a finite CW-complex as the Grothendieck group of vector bundles on , and points out that the tensor product of bundles makes a commutative ring. Then is defined for as the (reduced) of the -fold suspension of . The Bott periodicity theorem can be stated as , and this is used to define the groups for all . The axioms of a cohomology theory are verified by using the Puppe sequence already mentioned. The paper continues by establishing the Atiyah–Hirzebruch spectral sequence, which shows how close -theory is to classical integral cohomology. In particular, for a compact space the Chern character of a vector bundle induces an isomorphism

Thus, as an additive group, differs from only by torsion. The spectral sequence is related to the natural decreasing filtration

where consists of the elements which vanish when pulled back to any space of dimension less than . If, for example, the cohomology has no torsion, or if vanishes when is odd, the spectral sequence shows that the associated graded group of the natural filtration of is

The next topic in Reference Tucs is the Gysin map induced by a map of compact smooth manifolds which satisfies a spin-orientability condition. The treatment of this essential ingredient in the differentiable Riemann–Roch theorem of Reference DiffRR, however, is still confessedly provisional and unsatisfactory, and was soon to be superseded by the viewpoint of Reference ThCpl discussed above.

After this foundational material the paper turns to its main objective, the calculation of the -theory of some geometrically interesting spaces. The foundational part is written very much in the language of the homotopy theory of the time, and is less directly geometrical than Michael would have made it a few years later: it does not mention that every vector bundle on a compact space has a complementary bundle such that is trivial, and it does not use the difference-bundle description of a relative class in , which I shall discuss in the next section.

The first space considered in Reference Tucs is the classifying space of a compact connected Lie group . The cohomology of had been studied throughout the 1950s, most intensively by Borel and Hirzebruch Reference BorH. The results were complicated: even for it is not so easy to describe the integral cohomology ring explicitly, and the result for was unknown until Quillen’s work Reference Q1. For -theory the situation is very different. Every finite-dimensional complex representation of defines a vector bundle on , and the assignment induces a ring-homomorphism from the representation-ring , the Grothendieck group of the category of finite-dimensional representations, i.e., the free abelian group generated by the classes of irreducible representations. It turns out that is close to being an isomorphism: this is easy to prove when is connected, because if is a maximal torus in , then and restrict injectively into the explicitly known groups and , the image in each case being the subgroup invariant under the conjugation-action of the Weyl group of on . The only problem is that the space is not compact, and so one cannot expect every element of its -theory to be represented by the difference between two finite-dimensional vector bundles. For a noncompact space the correct definition of is as , but when this is equivalent to another plausible definition as the inverse-limit of a system , where is any expanding family of compact subspaces whose union is . The paper uses the inverse-limit definition, which makes it obvious that is complete in the topology induced by its natural filtration, and this motivates the theorem that

where is the completion of the representation ring in a topology which we shall return to in §7. The other -group vanishes.

From the paper turns to the homogeneous spaces , where is a connected and simply-connected compact Lie group and is a subgroup of maximal rank, i.e., one which contains a maximal torus of . This is the class of spaces typified by the projective spaces, Grassmannians, and flag manifolds long familiar in algebraic geometry. They are complex projective algebraic varieties with natural decompositions into Schubert cells, each of which is a complex affine space. Thus their integral cohomology is all in even dimensions, and is a free abelian group with a canonical basis.

Every finite-dimensional complex representation of defines a holomorphic vector bundle on , and the bundles so arising are called homogeneous. This gives us a ring-homomorphism

which corresponds to the map induced in -theory by the classifying map