Cohomological methods in transformation groups.

*(English)*Zbl 0799.55001
Cambridge Studies in Advanced Mathematics. 32. Cambridge: Cambridge University Press. xi, 470 p. (1993).

The book under review deals with the theory of compact transformation groups and, more specially, with the study of torus and \(p\)-torus actions by means of Algebraic Topology. At the same time an introduction for the beginner and a reference for the specialist, this book has a dual presentation of results, well adapted at each category of readers. For the beginner there are statements using CW-complexes and ordinary singular cohomology. Specialists will be interested in sections written for more general spaces and using the Alexander-Spanier cohomology. A major part of this text is devoted to recent research on these transformation groups, research to which the two authors made a large contribution.

In the thirties, P. A. Smith initialized the use of algebraic-topologic methods in the study of group actions; nowadays, this domain bears the name of “P. A. Smith Theory”. Other important contributions were made by A. Adem, A. Borel, G. Bredon, W. Browder, G. Carlsson, T. Chang, P. Conner, E. Floyd, D. Gottlieb, S. Halperin, W.-Y. Hsiang, R. Oliver, D. Quillen, T. Skjelbred, J. C. Su and many others. The book takes its place in the series of synthesis works on the study of transformation groups and completes them; let us cite in particular: G. E. Bredon, Introduction to compact transformation groups (1972; Zbl 0246.57017); W.-Y. Hsiang, Cohomology theory of topological transformation groups, Ergebn. Math. Grenzgeb. 85 (1975; Zbl 0429.57011); T. tom Dieck, Transformation groups, de Gruyter Stud. Math. 8 (1987; Zbl 0611.57002)].

The algebraic invariants of a torus action \(G\) on a space \(X\) belong to an algebraic model of the Borel space \(X_ G\); the nature of this model depends on the type of invariants under consideration. Since the space \(X_ G\) is the total space of a fibration with basis the classifying space \(BG\) and fibre the space \(X\), the authors can use the perturbation theory of Hirsch-Brown to construct such a model from the tensor product of the models of the basis and of the fibre. In the case of an \(S^ 1\) action, the algebraic setting is the theory of minimal models of D. Sullivan [Publ. Math., Inst. Haut. Étud. Sci. 47(1977), 269-331 (1978; Zbl 0374.57002)]. For actions of \(G = \mathbb{Z}_ p\), the models come from cellular chain complexes and use the homotopy theory of differential modules recalled in Appendix B. Let us now describe in detail the contents of this work, which is divided in 5 Chapters and 2 Appendices.

Chapter 1 recalls the definition of a \(G\)-CW-complex \(X\), the construction of the associated Borel fibration \(X_ G\) and the equivariant cohomology of \(X: H^*_ G(X) := H^*(X_ G)\). The major part of this chapter is devoted to an algebraic Borel construction in the case of a \(p\)-torus, with an explicit description of the diagonal and of the induced module and algebra structures. It is also shown how this construction gives the cohomology of fixed point sets using evaluations. The practical use of these models is illustrated by proofs of several results of P. A. Smith.

Chapter 2 is a summary of Sullivan’s rational homotopy theory in terms of commutative graded differential algebras. This presentation contains most of the notions required for the study of rational spaces by means of algebraic objects: the algebra of PL-forms, algebraic homotopy, minimal models, models of a fibration and formality. Only the realization functor is omitted. Although this chapter does not contain any exercises, numerous examples and bibliographic citations will give the beginner ample opportunity to acquaint himself with the subject. However, the list of references has to be completed by the book of P. A. Griffiths and J. W. Morgan, Rational homotopy theory and differential forms, Prog. Math. 16 (1981; Zbl 0474.55001)].

Chapter 3 begins with the main result of the theory: the Borel localization theorem for \(G\)-CW-complexes of finite dimension with a finite number of orbit types:

Let \(G\) be a compact Lie group and \(k\) be a commutative ring. If \(S\) is a multiplicative subset of the center of \(H^*(BG;k)\), then the inclusion of the fixed point set \(X^ S \hookrightarrow X\) induces an isomorphism between the localizations \(S^{-1}H^*_ G(X;k) \cong S^{-1}H^*_ G(X^ S;k)\).

For a compact connected group acting with fixed points, this result is extended to rational equivariant homotopy defined in section 3. Section 5 consists of an evaluation theorem in the case of a torus action, \((S^ 1)^ m\). Many applications to the structure of fixed point sets are given; in particular one has, under suitable hypotheses, \(\dim_ kH^*(X^ G;k) \leq \dim_ k H^*(X;k)\), (Theorem 3.10.4). Note that many sections of this chapter are devoted to the cohomology of Alexander- Spanier. The generality of certain results imply an increased technicality; for instance, the non-localized version of the Hsiang fundamental theorem of fixed points consists of a statement two pages long (Theorem 3.8.7). A part from the models of rational homotopy, the proofs require numerous tools coming from commutative algebra; most of them are recalled in Appendix \(A\).

Chapter 4 contains a study of the torus rank, \(\text{rank}_ 0 X\), of a space \(X\), defined as the maximal dimension of a torus acting almost- freely on \(X\). Section 3 recalls the various upper bounds of \(\text{rank}_ 0X\) coming from the Lie algebra of rational homotopy. Section 4 is devoted to the conjecture on the torus rank attributed to S. Halperin:

If \(\text{rank}_ 0 X \geq r\) then \(\dim_ \mathbb{Q} H^*(X;\mathbb{Q}) \geq 2^ r\).

This conjecture is true for homogeneous spaces and Kähler manifolds; the main part of section 4 consists of its proof in case \(r \leq 3\).

In the next section, the exponent of Browder and Gottlieb \(e(X,G)\) is introduced for the action of a finite torus, \((\mathbb{Z}_ p)^ r\); \(e(X,G)\) is defined as the order of the cokernel of \(H^ n_ G(X;\mathbb{Z}) \to H^ n(X;\mathbb{Z})\), when \(X\) verifies \(H^ n(X;\mathbb{Z}) = \mathbb{Z}\) and \(H^ j(X;\mathbb{Z}) = 0\), \(j > n\). If \(X\) is a topological manifold, \(e(X,G)\) is equal to the order of the smallest orbit, [Browder-Gottlieb]. This definition is then extended to the Tate equivariant cohomology. Chapter 4 contains also a proof of a theorem due to Dwyer-Wilkerson which determines \(H^*_ G(X^ K;\mathbb{F}_ p)\) as a function of the structure of modules over the Steenrod algebra of a localization of \(H^*_ G(X;\mathbb{F}_ p)\).

Chapter 5 deals with the particular case of \(k\)-PD-spaces, that is to say, spaces \(X\) whose cohomology \(H^*(X;k)\) is a Poincaré algebra, not necessarily in a graded sense. The first result generalizes the case of a compact Lie group acting differentiably on a compact manifold:

If \(X\) is a \(G\)-CW-complex of finite dimension which is a \(k\)-PD-space, then all the components of fixed point sets are also \(k\)-PD-spaces, (\(G = S^ 1\) and \(k = \mathbb{Q}\), or \(G\) is a \(p\)-torus and \(k = \mathbb{F}_ p\)).

This chapter ends with the definitions of Gysin morphisms and Euler classes in the \(k\)-PD-spaces setting. In particular, they show that there exists a generalization of a Borel formula between the dimensions of a \(G\)-space \(X\), its fixed point set \(X^ G\), and the fixed point sets \(X^ K\) of subtori \(K \subset G\) of codimension one.

To conclude, the book contains a lot of precise statements, references and remarks which will be appreciated by the specialist, while the numerous exercises and examples will help the beginner in his first contact with his theory. It should be noted that despite the dual purpose in the presentation both kinds of readers can easily find their way. However, the beginner has to get a good grasp of Borel constructions, Sullivan’s homotopy theory and various tools of commutative algebra before he can start to study Chapter 3 and the subsequent ones.

In the thirties, P. A. Smith initialized the use of algebraic-topologic methods in the study of group actions; nowadays, this domain bears the name of “P. A. Smith Theory”. Other important contributions were made by A. Adem, A. Borel, G. Bredon, W. Browder, G. Carlsson, T. Chang, P. Conner, E. Floyd, D. Gottlieb, S. Halperin, W.-Y. Hsiang, R. Oliver, D. Quillen, T. Skjelbred, J. C. Su and many others. The book takes its place in the series of synthesis works on the study of transformation groups and completes them; let us cite in particular: G. E. Bredon, Introduction to compact transformation groups (1972; Zbl 0246.57017); W.-Y. Hsiang, Cohomology theory of topological transformation groups, Ergebn. Math. Grenzgeb. 85 (1975; Zbl 0429.57011); T. tom Dieck, Transformation groups, de Gruyter Stud. Math. 8 (1987; Zbl 0611.57002)].

The algebraic invariants of a torus action \(G\) on a space \(X\) belong to an algebraic model of the Borel space \(X_ G\); the nature of this model depends on the type of invariants under consideration. Since the space \(X_ G\) is the total space of a fibration with basis the classifying space \(BG\) and fibre the space \(X\), the authors can use the perturbation theory of Hirsch-Brown to construct such a model from the tensor product of the models of the basis and of the fibre. In the case of an \(S^ 1\) action, the algebraic setting is the theory of minimal models of D. Sullivan [Publ. Math., Inst. Haut. Étud. Sci. 47(1977), 269-331 (1978; Zbl 0374.57002)]. For actions of \(G = \mathbb{Z}_ p\), the models come from cellular chain complexes and use the homotopy theory of differential modules recalled in Appendix B. Let us now describe in detail the contents of this work, which is divided in 5 Chapters and 2 Appendices.

Chapter 1 recalls the definition of a \(G\)-CW-complex \(X\), the construction of the associated Borel fibration \(X_ G\) and the equivariant cohomology of \(X: H^*_ G(X) := H^*(X_ G)\). The major part of this chapter is devoted to an algebraic Borel construction in the case of a \(p\)-torus, with an explicit description of the diagonal and of the induced module and algebra structures. It is also shown how this construction gives the cohomology of fixed point sets using evaluations. The practical use of these models is illustrated by proofs of several results of P. A. Smith.

Chapter 2 is a summary of Sullivan’s rational homotopy theory in terms of commutative graded differential algebras. This presentation contains most of the notions required for the study of rational spaces by means of algebraic objects: the algebra of PL-forms, algebraic homotopy, minimal models, models of a fibration and formality. Only the realization functor is omitted. Although this chapter does not contain any exercises, numerous examples and bibliographic citations will give the beginner ample opportunity to acquaint himself with the subject. However, the list of references has to be completed by the book of P. A. Griffiths and J. W. Morgan, Rational homotopy theory and differential forms, Prog. Math. 16 (1981; Zbl 0474.55001)].

Chapter 3 begins with the main result of the theory: the Borel localization theorem for \(G\)-CW-complexes of finite dimension with a finite number of orbit types:

Let \(G\) be a compact Lie group and \(k\) be a commutative ring. If \(S\) is a multiplicative subset of the center of \(H^*(BG;k)\), then the inclusion of the fixed point set \(X^ S \hookrightarrow X\) induces an isomorphism between the localizations \(S^{-1}H^*_ G(X;k) \cong S^{-1}H^*_ G(X^ S;k)\).

For a compact connected group acting with fixed points, this result is extended to rational equivariant homotopy defined in section 3. Section 5 consists of an evaluation theorem in the case of a torus action, \((S^ 1)^ m\). Many applications to the structure of fixed point sets are given; in particular one has, under suitable hypotheses, \(\dim_ kH^*(X^ G;k) \leq \dim_ k H^*(X;k)\), (Theorem 3.10.4). Note that many sections of this chapter are devoted to the cohomology of Alexander- Spanier. The generality of certain results imply an increased technicality; for instance, the non-localized version of the Hsiang fundamental theorem of fixed points consists of a statement two pages long (Theorem 3.8.7). A part from the models of rational homotopy, the proofs require numerous tools coming from commutative algebra; most of them are recalled in Appendix \(A\).

Chapter 4 contains a study of the torus rank, \(\text{rank}_ 0 X\), of a space \(X\), defined as the maximal dimension of a torus acting almost- freely on \(X\). Section 3 recalls the various upper bounds of \(\text{rank}_ 0X\) coming from the Lie algebra of rational homotopy. Section 4 is devoted to the conjecture on the torus rank attributed to S. Halperin:

If \(\text{rank}_ 0 X \geq r\) then \(\dim_ \mathbb{Q} H^*(X;\mathbb{Q}) \geq 2^ r\).

This conjecture is true for homogeneous spaces and Kähler manifolds; the main part of section 4 consists of its proof in case \(r \leq 3\).

In the next section, the exponent of Browder and Gottlieb \(e(X,G)\) is introduced for the action of a finite torus, \((\mathbb{Z}_ p)^ r\); \(e(X,G)\) is defined as the order of the cokernel of \(H^ n_ G(X;\mathbb{Z}) \to H^ n(X;\mathbb{Z})\), when \(X\) verifies \(H^ n(X;\mathbb{Z}) = \mathbb{Z}\) and \(H^ j(X;\mathbb{Z}) = 0\), \(j > n\). If \(X\) is a topological manifold, \(e(X,G)\) is equal to the order of the smallest orbit, [Browder-Gottlieb]. This definition is then extended to the Tate equivariant cohomology. Chapter 4 contains also a proof of a theorem due to Dwyer-Wilkerson which determines \(H^*_ G(X^ K;\mathbb{F}_ p)\) as a function of the structure of modules over the Steenrod algebra of a localization of \(H^*_ G(X;\mathbb{F}_ p)\).

Chapter 5 deals with the particular case of \(k\)-PD-spaces, that is to say, spaces \(X\) whose cohomology \(H^*(X;k)\) is a Poincaré algebra, not necessarily in a graded sense. The first result generalizes the case of a compact Lie group acting differentiably on a compact manifold:

If \(X\) is a \(G\)-CW-complex of finite dimension which is a \(k\)-PD-space, then all the components of fixed point sets are also \(k\)-PD-spaces, (\(G = S^ 1\) and \(k = \mathbb{Q}\), or \(G\) is a \(p\)-torus and \(k = \mathbb{F}_ p\)).

This chapter ends with the definitions of Gysin morphisms and Euler classes in the \(k\)-PD-spaces setting. In particular, they show that there exists a generalization of a Borel formula between the dimensions of a \(G\)-space \(X\), its fixed point set \(X^ G\), and the fixed point sets \(X^ K\) of subtori \(K \subset G\) of codimension one.

To conclude, the book contains a lot of precise statements, references and remarks which will be appreciated by the specialist, while the numerous exercises and examples will help the beginner in his first contact with his theory. It should be noted that despite the dual purpose in the presentation both kinds of readers can easily find their way. However, the beginner has to get a good grasp of Borel constructions, Sullivan’s homotopy theory and various tools of commutative algebra before he can start to study Chapter 3 and the subsequent ones.

Reviewer: D.Tanré (Villeneuve d’Ascq)

##### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

55P91 | Equivariant homotopy theory in algebraic topology |

57R91 | Equivariant algebraic topology of manifolds |

55N91 | Equivariant homology and cohomology in algebraic topology |