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Equivariant formality of corank-one isotropy actions and products of rational spheres

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Abstract

We completely characterize the pairs of connected Lie groups \(G > K\) such that \({{\,\textrm{rk }\,}}G - {{\,\textrm{rk }\,}}K = 1\) and the isotropy action of K on G/K is equivariantly formal. The analysis requires us to correct and extend an existing partial classification of homogeneous quotients G/K with the rational homotopy type of a product of an odd- and an even-dimensional sphere.

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Notes

  1. N.B. This terminology differs from Onishchik’s usage [37, p. 207], in which the corank is the rational dimension of the cokernel of the map .

  2. See, e.g. Adams [1, Thm. 5.13(vii), notation from Defs. 4.13, 4.38].

  3. This can be taken to be the Killing form if G is semisimple and otherwise may be taken as the direct sum of the Killing form and the negative of an arbitrary inner product on the center \(\mathfrak z(\mathfrak g)\).

  4. If we make the identifications \(H^2(BT;\mathbb R) \cong \mathfrak {t}^\vee \) and \(H^2(BS;\mathbb R) \cong \mathfrak {s}^\vee \), then \({\widetilde{e}}\) can be identified with \(\alpha \), which is another way of showing it is \(W_L\)-invariant.

  5. In this case the traditional terminology is that G is locally isomorphic to \(K_1 \times K_2\), a sensible wording since this means the Lie algebras are isomorphic.

  6. Almost effective is standard, but we prefer to make the analogy with geometric group theory, where G is said to be virtually of some class C if it admits a finite-index subgroup of class C.

  7. At least one direction survives even when H is only be a topological group acting continuously on a topological space X, and in fact holds with arbitrary coefficients. As \(EH \times E\overline{H}\) is contractible and a principal H-bundle under the diagonal action, we can use it in the Borel construction

    With this identification, the map \(X_H \longrightarrow X_{\overline{H}}\) induced by projects out the BK-coordinate, and \(X \longrightarrow X_{\overline{H}}\) factors through \(X \longrightarrow X_H\), so if the former induces a surjection in cohomology, then so does the latter.

  8. On the other hand, if \(\widehat{G}\), G, and H are connected, \(\widehat{H}\) is connected too.

  9. A result of this kind for \(G/H = S^{\textrm{odd}}\) is already proved by Montgomery–Samelson [34, §§4–7].

  10. Kramer also classifies the irreducible pairs such that G/H has the integral cohomology of a product of spheres [30, pp. x–xi]—in fact, this is his main goal of the first part of his monograph—although he is missing some cases implied by Kamerich’s classification, which was not accessible to him at the time [personal communication].

  11. He coins the charming term auletic for a pair leading to a space of the form \((K_1/H_1) \times (K_2/H_2)\), from the ancient Greek (aulos), a wind instrument with two pipes, one keyed by each hand. An online search for Kamerich finds in later life he was a teacher of mathematics as well as a longtime bassoonist for the Arnhem Symphony Orchestra and a member of a local Renaissance music ensemble. A memorial for him in 2017 was titled “His life was music.”

  12. A proof not involving knowledge of \(G_2\) is possible by computing that the cohomology of the Wu manifold \(W = \textrm{SU}(3)/\textrm{SO}(3)\) is the sum of \(H^0(W) \cong \mathbb Z\cong H^5(W)\) and \(H^3(W) \cong \mathbb Z/2 \cong H^2(W;\mathbb Z/2)\) then taking the homotopy fiber F of a representing map \(W \longrightarrow K(\mathbb Z/2,2)\) to kill \(\pi _2(W) = \mathbb Z/2\) and finding the order of \(H^4(F) \cong H_3(F) \cong \pi _3(F) \cong \pi _3(W)\) through the Serre spectral sequence of \(K(\mathbb Z/2,1) \rightarrow F \rightarrow W\).

  13. Note that we must have \(n, m\geqslant 2\) due to the simple-connectivity hypothesis.

  14. This argument, as written, would incorrectly rule out \(\textrm{SU}(3)\), on considering the long exact homotopy sequence of \(H \rightarrow G \rightarrow G/H\), but this is due to the claim that \(\pi _4(S^2) \cong \mathbb Z\) (probably a typo), which does not affect his argument since he only needs that this group is nonzero; of course \(\pi _4(S^2) \cong \mathbb Z/2\).

  15. He does not consider the remaining case, which one can however show has \(H^8 \cong \mathbb Z/3\). In fact, the pair also appears to have torsion, namely \(H^8 \cong \mathbb Z/91\).

  16. The two copies of \(\textrm{U}(1) < \textrm{Sp}(1)\) in \(G_2\) meet in \(\{\pm 1\}\), but the diagonal \(\Delta \textrm{U}(1)\) with one coordinate in does not meet the \(\textrm{U}(1)\), so this really is a direct product.

  17. Isotropy-formality in the case H is of rank one was already characterized in one of the authors’ earlier works [16], where isotropy-formal pairs \((G,H)\) were classified.

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Acknowledgements

The first author thanks Federico Pasini for help with a program evaluating degree multisets for rational sphere products G/H, Linus Kramer for verifying his analysis did not consider the cases \(S^{\textrm{even}}\times S^{\textrm{odd}}\) with \(\textrm{even} \geqslant \textrm{odd}\) and discussing the cases in which our results differed, Oliver Goertsches for comments and suggestions on an early draft, and Jason DeVito for useful conversations and literature references on homogeneous spaces, especially the dissertation of Kamerich [26] and the argument in Discussion 3.23(f). The second author thanks Mychelle Parker for letting us use her Maple code to study various subalgebras of simple Lie algebras, and thanks the Fundamental Research Funds for the Central Universities of China (2020MS040, 2023MS078) for their support. Both authors would additionally like to thank the anonymous referee for a careful and insightful reading resulting in many corrections and clarifications.

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    Jeffrey D. Carlson

  2. North China Electric Power University, Beijing, China

    Chen He

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Carlson, J.D., He, C. Equivariant formality of corank-one isotropy actions and products of rational spheres. Math. Z. 305, 21 (2023). https://doi.org/10.1007/s00209-023-03319-1

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