An introduction to equivariant cohomology. (English) Zbl 0970.55003

DeWitt-Morette, CĂ©cile (ed.) et al., Quantum field theory: perspective and prospective. Proceedings of the NATO ASI, held in Les Houches, France, June 15-26, 1998. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 530, 35-56 (1999).
In the introduction, the action of a group \(G\) on a space \(M\) is defined and the author shows cases when the situation can be conveniently described by quotienting. Typically \(G\) is a compact (Lie) group, \(M\) is a smooth manifold and the action is free.
The next section presents a way of repairing the situation when the action is not free. One considers the diagonal action of \(G\) on \(M\times EG\) where \(EG\) is contractible and on which the action of \(G\) is free. (The author describes an example where \(EG\) may be infinite dimensional). The resulting diagonal action of \(G\) is free on a space of the same homotopy type as \(M\). The cohomology of the resulting quotient \(M_G\) is the equivariant cohomology of \(M:H^*_G(M)\). It is also seen that \(M_G\) has the same homotopy type as \(M/G\) in the case of a free action.
Section three explains the de Rham model of equivariant cohomology when \(G\) is a connected compact Lie group. One way of getting the differential forms on the base space of the projection \(\pi:M_G\to BG\) \((BG\) is fibered by \(M)\) is by a procedure of integration along the fibers (Cartan, Weil). Concretely, the model for \(EG\) is \(\Lambda^*({\mathfrak g}^*)\otimes S({\mathfrak g}^*)\) where \({\mathfrak g}\) is the Lie algebra of \(G\). \(S({\mathfrak g}^*)\) is the symmetric algebra of \({\mathfrak g}^*\). (The author writes that it enters “vitally” in the construction but it would have been nice to explain why). The “basic forms” on \(M_G\) are those annihilated by \(i_X\) and \({\mathcal L}_X\) where \(X\) is a vector field tangent to the fibers and they can be identified with polynomials on \({\mathfrak g}\) invariant under the adjoint action of \(G\) on \({\mathfrak g}\). The case \(G=S^1\) is detailed and one sees that a modified differential (the Cartan differential) has to be introduced so that compensating terms in the “basic forms” will satisfy some constraints in order that the forms are closed. In the symplectic case, the simplest correction term is the momentum map.
The last section shows how much of the “equivariant cohomological information” is retained by restricting to the fixed points of the action. First, in the case of a locally free action of \(S^1\) on \(M\), one sees that \(H^*_{S^1}(M)\) is a torsion module over \(\mathbb{R}[u]\) so that the cohomology is trivial after localization (i.e. tensoring by \(\mathbb{R} [u,u^{-1}])\). In the same vein, a result of Borel says that in the case of an \(S^1\) semifree action, after localization \(H^*_{S^1} (M)= H^*_{S^1} (F)\) where \(F\) is the fixed point set. Next, the author shifts slightly the focus to what happens when integrating along the fibers (cf. Cartan-Weil of the previous section) and he states localization results of his and Atiyah where the computation of an integral of a form on \(M\) can be reduced to its computation on \(F\). (A proof is sketched at the end when \(F\) is made of isolated fixed points. It amounts to using the previous results of the locally free action by studying what happens when blowing up the fixed points). From this one gets the theorem of Duistermaat-Heckmann when \(M\) is a compact symplectic manifold, \(F\) is made of isolated fixed points and the action admits a moment map. It can be seen as a rigorous justification of the stationary phase expansion of the physicists. Note the impact of the moment map on such an expansion (!).
There are several typographical mistakes, 24 references and a picture of the author (which resembles poorly the one from the coverpage of the Notices of the American Mathematical Society – April 2001).
This text is directed to physicists and has excellent didactical qualities; the focus is on an intuitive understanding of the mathematics leading to a justification of formulas arising in physics. Moreover, the author indicates that equivariant cohomology could be felt in previous mathematical works but it would have been interesting to know more about the mechanisms of its genesis.
For the entire collection see [Zbl 0935.00053].


55N91 Equivariant homology and cohomology in algebraic topology
55-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology
53D20 Momentum maps; symplectic reduction
58A12 de Rham theory in global analysis