Equivariant Cohomology? Loring W

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WHAT IS... ? Equivariant Cohomology? Loring W. Tu

Many invariants in geometry and topology can be computed as integrals. For example, in classical b diﬀerential geometry the Gauss–Bonnet theorem states that if M is a compact, oriented surface in Euclidean 3-space with Gaussian curvature K and volume form vol, then its Euler characteristic is 1 χ(M) = K vol. 2π ZM On the other hand, if there is a continuous vector

field X with isolated zeros on a compact, oriented b manifold, the Hopf index theorem in topology computes the Euler characteristic of the manifold as the sum of the indices at the zeros of the vector Figure 1. A circle action on a sphere. field X. Putting the two theorems together, one obtains 1 zeros of the vector field are precisely the fixed (1) K vol = i (p), 2π Z X points of the action (see Figure 1). M p∈ZeroX(X) Recall that the familiar theory of singular co- where iX (p) is the index of the vector field X at homology gives a functor from the category of the zero p. This is an example of a localization topological spaces and continuous maps to the formula, for it computes a global integral in terms category of graded rings and their homomor- of local information at a finite set of points. More phisms. When the topological space has a group generally, one might ask what kind of integrals action, one would like a functor that reflects can be computed as finite sums. A natural context both the topology of the space and the action of for studying this problem is the situation in which the group. Equivariant cohomology is one such there is a group acting on the manifold with functor. isolated fixed points. In this case, one can try to The origin of equivariant cohomology is some- relate an integral over the manifold to a sum over what convoluted. In 1959 Borel defined equivariant the fixed point set. singular cohomology in the topological category Rotating the unit sphere S2 in R3 about the 1 using a construction now called the Borel construc- z-axis is an example of an action of the circle S on tion. Nine years earlier, in 1950, in two influential the sphere. It has two fixed points, the north pole articles on the cohomology of a manifold M acted and the south pole. This circle action generates on by a compact, connected Lie group G, Cartan a continuous vector field on the sphere, and the ∗ constructed a differential complex G(M), dG out of the differential forms on M and the Lie Loring W. Tu is professor of mathematics at Tufts Uni- Ω versity, Medford, MA. His email address is loring.tu@ algebra of G. Although the term “equivariant tufts.edu. cohomology” never occurs in Cartan’s papers,

March 2011 Notices of the AMS 423 Cartan’s complex turns out to compute the real turning every action into a free action without equivariant singular cohomology of a G-manifold changing the homotopy type of the space. The (a manifold with an action of a Lie group G), in idea is to find a contractible space EG on which much the same way that the de Rham complex the group G acts freely. Then EG × M will have the of smooth differential forms computes the real same homotopy type as M, and no matter how G singular cohomology of a manifold. Without ex- acts on M, the diagonal action of G on EG × M will plicitly stating it, Cartan provided the key step in a always be free. The homotopy quotient MG of M by proof of the equivariant de Rham theorem, before G, also called the Borel construction, is defined to equivariant cohomology was even defined! In fact, be the quotient of EG × M by the diagonal action ∗ a special case of the Borel construction was al- of G, and the equivariant cohomology HG (M) of ∗ ready present in Cartan’s earlier article (Colloque M is defined to be the cohomology H (MG) of ∗ de Topologie, C.B.R.M., Bruxelles, 1950, p. 62). Ele- the homotopy quotient MG. Here H ( ) denotes ments of Cartan’s complex are called equivariant singular cohomology with any coefficient ring. differential forms or equivariant forms. Let S(g∗) A contractible space on which a topological be the polynomial algebra on the Lie algebra g of G; group G acts freely is familiar from homotopy it is the algebra of all polynomials in linear forms theory as the total space of a universal principal on g. An equivariant form on a G-manifold M is a G-bundle π : EG → BG, of which every principal differential form ω on M with values in the poly- G-bundle is a pullback. More precisely, if P → nomial algebra S(g∗) satisfying the equivariance M is any principal G-bundle, then there is a condition: map f : M → BG, unique up to homotopy and ∗ = −1 ◦ ∈ called a classifying map of P → M, such that the ℓg ω (Ad g ) ω for all g G, ∗ bundle P is isomorphic to the pullback bundle ∗ where ℓg is the pullback by left multiplication f (EG). The base space BG of a universal bundle, by g and Ad is the adjoint representation. An uniquely defined up to homotopy equivalence, is equivariant form ω is said to be closed if it called the classifying space of the group G. The = satisfies dGω 0. classifying space BG plays a key role in equivariant What makes equivariant cohomology particu- cohomology, because it is the homotopy quotient larly useful in the computation of integrals is of a point: the equivariant integration formula of Atiyah-Bott = × = = (1984) and Berline-Vergne (1982). In case a torus ptG (EG pt)/G EG/G BG, ∗ acts on a compact, oriented manifold with isolated so that the equivariant cohomology HG (pt) of a fixed points, this formula computes the integral of point is the ordinary cohomology H∗(BG) of the a closed equivariant form as a finite sum over the classifying space BG. fixed point set. Although stated in terms of equi- It is instructive to see a universal bundle for variant cohomology, the equivariant integration the circle group. Let S2n+1 be the unit sphere formula, also called the equivariant localization in Cn+1. The circle S1 acts on Cn+1 by scalar formula in the literature, can often be used to com- multiplication. This action induces a free action of pute the integrals of ordinary differential forms. It S1 on S2n+1, and the quotient space is by definition opens up the possibility of machine computation the complex projective space CP n. Let S∞ be ∞ 2n+1 C ∞ of integrals on a manifold. the union n=0 S , and let P be the union ∞ C n n=0 P . SinceS the actions of the circle on the Equivariant Cohomology spheresS are compatible with the inclusion of one Suppose a topological group G acts continuously sphere inside the next, there is an induced action on a topological space M. A first candidate for of S1 on S∞. This action is free with quotient space equivariant cohomology might be the singular CP ∞. It is easy to see that all homotopy groups cohomology of the orbit space M/G. The example of S∞ vanish, for if a sphere Sk maps into the above of a circle G = S1 acting on M = S2 by infinite sphere S∞, then by compactness its image rotation shows that this is not a good candidate, lies in a finite-dimensional sphere S2n+1. If n is since the orbit space M/G is a closed interval, large enough, any map from Sk to S2n+1 will be a contractible space, so that its cohomology is null-homotopic. Since S∞ is a CW complex, the trivial. In this example, we lose all information vanishing of all homotopy groups implies that about the group action by passing to the quotient it is contractible. Thus the projection S∞ → CP ∞ M/G. is a universal S1-bundle and, up to homotopy A more serious deficiency of this example is equivalence, CP ∞ is the classifying space BS1 of that it is the quotient of a nonfree action. In the circle. general, a group action is said to be free if the If H∗( ) is a cohomology functor, the constant stabilizer of every point is the trivial subgroup. It map M → pt from any space M to a point induces is well known that the orbit space of a nonfree a ring homomorphism H∗(pt) → H∗(M), which action is often “not nice”—not smooth or not gives H∗(M) the structure of a module over the Hausdorff. However, the topologist has a way of ring H∗(pt).Thusthecohomologyofapointserves

424 Notices of the AMS Volume 58, Number 3 as the coefficient ring in any cohomology theory. space, giving rise to a representation of the group For the equivariant real singular cohomology of a G on the tangent space TpM. Invariants of the circle action, the coefficient ring is representation are then invariants of the action at ∗ ∗ ∗ 1 the fixed point. For a circle action, at an isolated H (pt; R) = H pt 1 ; R = H (BS ; R) S1 S fixed point p, the tangent space T M decomposes = ∗ C ∞ R ≃ R p H ( P ; ) [u], into a direct sum Lm1 ⊕⊕ Lmn , where L is the polynomial ring generated by an element the standard representation of the circle on the C u of degree 2. For the action of a torus T = complex plane and m1,...,mn are nonzero S1 ×× S1 = (S1)ℓ, the coefficient ring is the integers. The integers m1,...,mn are called the ∗ = exponents of the circle action at the fixed point polynomial ring HT (pt; R) R[u1,...,uℓ], where each ui has degree 2. p. They are defined only up to sign, but if M is oriented, the sign of the product m1 mn is well Equivariant Integration defined by the orientation of M. = 1 ℓ Let G be a compact, connected Lie group. Over a When a torus T (S ) acts on a compact, compact, oriented G-manifold, equivariant forms oriented manifold M with isolated fixed point set can be integrated, but the values are in the coef- F, for any closed T -equivariant form ω on M, the ∗ R equivariant integration formula states that ficient ring HG (pt; ), which is generally a ring of polynomials. According to Cartan, in the case of a i∗ω (3) ω = p , circle action on a compact, oriented manifold, an Z T M pX∈F e (νp) equivariant form of degree 2n is a sum ∗ where ip ω is the restriction of the equivariantly 2 (2) ω = ω2n + ω2n−2 u + ω2n−4 u T closed form ω to a fixed point p and e (νp) is the n ++ ω0 u , equivariant Euler class of the normal bundle νp to p in M. Of course, the normal bundle to a point p 2j S1 1 where ω2j ∈ (M) is an S -invariant 2j-form in a manifold M is simply the tangent space TpM, on M. If ω is closed under the Cartan differential, Ω but formula (3) is stated in a way to allow easy then it is called an equivariantly closed extension of generalization: when F has positive-dimensional the ordinary differential form ω2n. The equivariant components, the sum over the fixed points is integral M ω is obtained by integrating each ω2j replaced by an integral over the components C of over M. IfR M has dimension 2n, then the integral the fixed point set and ν is replaced by ν , the = p C M ω2j vanishes except when j n, and one has normal bundle to the component C. In formula (3), R the degree of the form ω is not assumed to be ω = ω2n + ω2n−2 u ZM ZM ZM equal to the dimension of the manifold M, and so the left-hand side is a polynomial in u ,...,u , ++ n 1 ℓ ω0 u while the right-hand side is a sum of rational ZM expressions in u1,...,uℓ, and it is part of the = + ++ = T ω2n 0 0 ω2n. theorem that the equivariant Euler classes e (νp) ZM ZM are nonzero and that there will be cancellation on One peculiarity of equivariant integration is the right-hand side so that the sum becomes a the possibility of obtaining a nonzero answer polynomial. while integrating a form over a manifold whose Return now to a circle action with isolated dimension is not equal to the degree of the form. fixed points on a compact, oriented manifold M − For example, if M has dimension 2n 2 instead of dimension 2n. Let ω be a closed equivariant of 2n, then the integral over M of the equivariant form of degree 2n on M. Since the restriction of a 2n-form ω above is form of positive degree to a point is zero, on the right-hand side of (3) all terms in ω except ω un ω = ω2n−2 u, 0 ZM ZM restrict to zero at a fixed point p ∈ M: since for dimensional reasons all other terms n ∗ = ∗ j are zero. From this, one sees that an equivariant ip ω (ip ω2n−2j ) u integral for a circle action is in general not a real jX=0 number, but a polynomial in u. = ∗ n = n (ip ω0) u ω0(p) u .

S1 Localization The equivariant Euler class e (νp) turns out to be n What kind of information can be mined from the m1 mn u , where m1,...,mn are the exponents fixed points of an action? If a Lie group G acts of the circle action at the fixed point p. Therefore, smoothly on a manifold, then for each g ∈ G, the the equivariant integration formula for a circle action assumes the form action induces a diffeomorphism ℓg : M → M. At ∈ → ω0 a fixed point p M, the differential ℓg∗ : TpM ω = ω = (p). Z 2n Z TpM is a linear automorphism of the tangent M M pX∈F m1 mn

March 2011 Notices of the AMS 425 In this formula, ω2n is an ordinary diﬀerential The formalism of equivariant cohomology car- form of degree 2n on M, ω is an equivariantly ries over from singular cohomology to other

closed extension of ω2n, and ω0 is the coefficient cohomology theories such as K-theory, Chow of the un term in ω as in (2). rings, and quantum cohomology. There are similar localization formulas that compare the equivari- Applications ant functor of a G-space to that of the fixed In general, an integral of an ordinary differential point set of G or of some subgroup of G (for form on a compact, oriented manifold can be example, Segal 1968 and Atiyah-Segal 1968). In the fifty years since its inception, equivariant computed as a finite sum using the equivariant cohomology has found applications in topology, integration formula if the manifold has a torus differential geometry, symplectic geometry, alge- action with isolated fixed points and the form braic geometry, K-theory, representation theory, has an equivariantly closed extension. These con- and combinatorics, among other fields, and is ditions are not as restrictive as they seem, since currently a vibrant area of research. many problems come naturally with the action of a compact Lie group, and one can always restrict Acknowledgments the action to that of a maximal torus. It makes This article is based on a talk given at the National sense to restrict to a maximal torus, instead of Center for Theoretical Sciences, National Tsing any torus in the group, because the larger the Hua University, Taiwan. The author gratefully torus, the smaller the fixed point set, and hence acknowledges helpful discussions with Alberto the easier the computation. Arabia, Aaron W. Brown, Jeffrey D. Carlson, George As for the question of whether a form has Leger, and Winnie Li during the preparation of an equivariantly closed extension, in fact a large this article, as well as the support of the Tufts collection of forms automatically do. These in- Faculty Research Award Committee in 2007–2008, clude characteristic classes of vector bundles on the Université Paris 7–Diderot in 2009–2010, and a manifold. If a vector bundle has a group action the American Institute of Mathematics and the compatible with the group action on the mani- National Science Foundation in 2010. fold, then the equivariant characteristic classes of the vector bundle will be equivariantly closed References extensions of its ordinary characteristic classes. [1] R. Bott, An introduction to equivariant cohomol- A manifold on which every closed form has ogy, in Quantum Field Theory: Perspective and an equivariantly closed extension is said to be Prospective (C. DeWitt-Morette and J.-B. Zuber, equivariantly formal. Equivariantly formal man- eds.) Kluwer Academic Publishers, Netherlands, ifolds include all manifolds whose cohomology 1999, pp. 35–57. [2] V. Guillemin and S. Sternberg, Supersymme- vanishes in odd degrees. In particular, a homoge- try and Equivariant Cohomlogy, Springer, Berlin, neous space G/H, where G is a compact Lie group 1999. and H is a closed subgroup of maximal rank, is [3] M. Vergne, Applications of equivariant coho- equivariantly formal. mology, in the Proceedings of the International The equivariant integration formula is a power- Congress of Mathematicians (Madrid, August ful tool for computing integrals on a manifold. If 2006), Vol. I, European Math. Soc., Zürich, 2007, a geometric problem with an underlying torus ac- pp. 635–664. tion can be formulated in terms of integrals, then there is a good chance that the formula applies. For example, it has been applied to show that the stationary phase approximation formula is exact for a symplectic action (Atiyah-Bott 1984), to calculate the number of rational curves in a quintic threefold (Kontsevich 1995, Ellingsrud-Strømme 1996), to calculate the characteristic numbers of a compact homogeneous space (Tu 2010), and to derive the Gysin formula for a fiber bundle with homogenous space fibers (Tu preprint 2011). In the special case in which the vector field X is generated by a circle action, the Gauss-Bonnet-Hopf formula (1) is a consequence of the equivariant integration formula. Equivariant cohomology has also helped to elucidate the work of Witten on supersymmetry, Morse theory, and Hamiltonian actions (Atiyah-Bott 1984, Jeffrey-Kirwan 1995).

426 Notices of the AMS Volume 58, Number 3