The toral rank conjecture and variants of equivariant formality. (English. French summary) Zbl 1528.55006
This work concerns the restrictions on the topology of a space, \(M\), inherited from the existence of an almost free, non-trivial, action of a Lie group \(G\). (Recall that an action is said almost free if all isotropy groups are finite.) As a typical result in this direction, W. Y. Hsiang proved [Cohomology theory of topological transformation groups. Berlin-Heidelberg-New York: Springer-Verlag (1975; Zbl 0429.57011)] that, if \(G\) is compact and \(M\) is a compact \(G\)-manifold, then the action is almost free if, and only if, the rational equivariant cohomology of \(M\) is finite dimensional.
More precisely, the main goal of this paper is the study of a long standing conjecture of S. Halperin, called the toral rank conjecture or TRC, and which is stated as follows: if a (reasonable) space is equipped with an almost free action of a torus of dimension \(n\), then the sum of all its Betti numbers is greater than or equal to \(2^n\). This conjecture appears in [S. Halperin, Lond. Math. Soc. Lect. Note Ser. 93, 293–306 (1985; Zbl 0562.57015)] where the author proves that the TRC is true for homogeneous spaces, \(G/H\), with \(G\) connected and \(H\) closed and connected. Next, this conjecture has been the subject of numerous publications. For instance it has been verified for Kähler manifolds, some nilmanifolds, or with dimensional bounds linear in \(n\), but it is still open in general.
Here, the authors extend previous works in two directions when \(X\) is a compact Hausdorff \(T^n\)-space. – In the first one, they prove that the TRC is true if the associated Borel construction is formal in the sense of rational homotopy theory. – For the second one, their hypothesis is on the equivariant cohomology. Classical equivariant formality of an action means that the Serre spectral sequence of the Borel fibration degenerates at the \(E_{2}\)-term. Here, this notion is replaced by that of hyperformality. Let \(\varphi\colon H^*(BT^n;\mathbb Q)\to H^*_{T^n}(X;\mathbb Q)\) be the map induced by the Borel fibration \(X_{T^n}\to BT^n\). The action is said hyperformal if the kernel of \(\varphi\) is generated by a homogeneous regular sequence. M. Amann and L. Zoller prove that the TRC is true for hyperformal actions.
Apart from these two cases, they also expand the known hypotheses restricting the dimension of the spaces on which the torus operates and under which the TRC is true. The paper contains many concrete examples and a careful study of variations on the definition of hyperformality. Two appendices on minimal models for differential graded modules over a differential commutative graded algebra, or over an \(A_{\infty}\)-algebra, are also included.
More precisely, the main goal of this paper is the study of a long standing conjecture of S. Halperin, called the toral rank conjecture or TRC, and which is stated as follows: if a (reasonable) space is equipped with an almost free action of a torus of dimension \(n\), then the sum of all its Betti numbers is greater than or equal to \(2^n\). This conjecture appears in [S. Halperin, Lond. Math. Soc. Lect. Note Ser. 93, 293–306 (1985; Zbl 0562.57015)] where the author proves that the TRC is true for homogeneous spaces, \(G/H\), with \(G\) connected and \(H\) closed and connected. Next, this conjecture has been the subject of numerous publications. For instance it has been verified for Kähler manifolds, some nilmanifolds, or with dimensional bounds linear in \(n\), but it is still open in general.
Here, the authors extend previous works in two directions when \(X\) is a compact Hausdorff \(T^n\)-space. – In the first one, they prove that the TRC is true if the associated Borel construction is formal in the sense of rational homotopy theory. – For the second one, their hypothesis is on the equivariant cohomology. Classical equivariant formality of an action means that the Serre spectral sequence of the Borel fibration degenerates at the \(E_{2}\)-term. Here, this notion is replaced by that of hyperformality. Let \(\varphi\colon H^*(BT^n;\mathbb Q)\to H^*_{T^n}(X;\mathbb Q)\) be the map induced by the Borel fibration \(X_{T^n}\to BT^n\). The action is said hyperformal if the kernel of \(\varphi\) is generated by a homogeneous regular sequence. M. Amann and L. Zoller prove that the TRC is true for hyperformal actions.
Apart from these two cases, they also expand the known hypotheses restricting the dimension of the spaces on which the torus operates and under which the TRC is true. The paper contains many concrete examples and a careful study of variations on the definition of hyperformality. Two appendices on minimal models for differential graded modules over a differential commutative graded algebra, or over an \(A_{\infty}\)-algebra, are also included.
Reviewer: Daniel Tanré (Villeneuve d’Ascq)
MSC:
| 55N91 | Equivariant homology and cohomology in algebraic topology |
| 55P62 | Rational homotopy theory |
| 57S10 | Compact groups of homeomorphisms |
Keywords:
toral rank conjecture; almost free torus action; (equivariant) formality; rational homotopy theory; non-negative curvature; \(A_\infty\)-algebras; hyperformality; Borel fibrationReferences:
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