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Torus actions on rationally elliptic manifolds. (English) Zbl 1458.55008

A simply connected topological space \(X\) is called rationally elliptic, if \(\dim_{\mathbb{Q}}H^\ast(X;{\mathbb{Q}})<\infty\) and \(\dim_{\mathbb{Q}}(\pi_\ast(X)\otimes {\mathbb{Q}})<\infty\). The authors study torus actions on rationally elliptic manifolds. They show that if a \(k\)-torus \(T^k\) acts smoothly and effectively on a smooth, closed, rationally elliptic \(n\)-dimensional manifold \(M^n\), then \(k\leq \lfloor \frac{2n}{3}\rfloor\). Moreover, \(k\leq \lfloor \frac{n}{3}\rfloor\), if the action is assumed to be almost free. The authors then consider maximal rank torus actions. They show that if \(T^k\) acts almost freely and \(k= \lfloor\frac{n}{3}\rfloor\), where \(n\geq 3\), then \(M^n\) is rationally homotopy equivalent to a product \(X\times \prod_{i=1}^{k-1}{\mathbb{S}}^3\), where \(X\) equals \({\mathbb{S}}^3\), \({\mathbb{S}}^5\) or \({\mathbb{S}}^2\times {\mathbb{S}}^3\). They also classify \(M^n\) up to rational homotopy equivalence if \(k= \lfloor\frac{2n}{3}\rfloor\), and show that there are finitely many rational homotopy types in each dimension. This differs essentially from the case of effective action of rank \(k=\lfloor\frac{2n}{3}\rfloor-1\), even in low dimensions. It would be interesting to know if the classifications can be improved to equivariant homeomorphism or diffeomorphism.
The largest integer \(r\) for which \(M^n\) admits an almost free \(T^r\)-action is called the toral rank of \(M^n\), and is denoted by \(\mathrm{rk}(M)\). The Toral Rank Conjecture, proposed by S. Halperin, claims that \(\dim H^\ast(M;{\mathbb{Q}})\geq 2^{\mathrm{rk}(M)}\). Assume \(M^n\) is a smooth, closed, rationally elliptic \(n\)-manifold with smooth, effective action of the \(k\)-torus \(T^k\), \(k\geq 1\). The authors apply their classification results to show that if \(k= \lfloor \frac{2n}{3}\rfloor\), or if \(T^k\) is of rank \(\lfloor \frac{n}{3}\rfloor\) and acts almost freely, then \(M^n\) satisfies the Toral Rank Conjecture.

MSC:

55P62 Rational homotopy theory
57R91 Equivariant algebraic topology of manifolds
57S15 Compact Lie groups of differentiable transformations
57S12 Toric topology
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