Torus Action on Quaternionic Projective Plane and Related Spaces
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Torus Action on Quaternionic Projective Plane and Related Spaces Anton Ayzenberg1 Received: 3 October 2019 / Revised: 10 September 2020 / Accepted: 22 October 2020 © Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2020
Abstract For an effective action of a compact torus T on a smooth compact manifold X with nonempty finite set of fixed points, the number 21 dim X − dim T is called the complexity of the action. In this paper, we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that HP 2 /T 3 ∼ = S5 6 2 4 2 ∼ and S /T = S , for the homogeneous spaces HP = Sp(3)/(Sp(2) × Sp(1)) and S 6 = G 2 / SU(3). Here, the maximal tori of the corresponding Lie groups Sp(3) and G 2 act on the homogeneous spaces from the left. Next we consider the quaternionic analogues of smooth toric surfaces: they give a class of eight-dimensional manifolds with the action of T 3 . This class generalizes HP 2 . We prove that their orbit spaces are homeomorphic to S 5 as well. We link this result to Kuiper–Massey theorem and its generalizations studied by Arnold. Keywords Torus action · Complexity one · Quaternions · Octonions · Quasitoric manifold · Kuiper–Massey theorem Mathematics Subject Classification 55R91 · 57S15 · 57S25 · 22F30 · 57R91 · 57S17 · 20G20 · 57M60 · 20G41
1 Introduction Consider an effective action of the compact torus T k on a compact smooth manifold X = X 2n , such that the set of fixed points is finite and nonempty. The number n − k can be shown to be nonnegative; it is called the complexity of the action.
The article was prepared within the framework of the HSE University Basic Research Program.
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Anton Ayzenberg [email protected] International Laboratory of Algebraic Topology and Its Applications, Faculty of Computer Science, Higher School of Economics, Moscow, Russia
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Buchstaber and Terzic [6–8] introduced the theory of (2n, k)-manifolds to study the orbit spaces of nonfree actions of a compact torus T k on 2n-manifolds. Using this theory, they proved the homeomorphisms G 4,2 /T 3 ∼ = S 5 and F3 /T 2 ∼ = S 4 , where 4 G 4,2 is the Grassmann manifold of complex 2-planes in C , and F3 is the manifold of complete flags in C3 . The torus actions are naturally induced from the standard torus actions on C4 and C3 , respectively. In both cases of G 4,2 and F3 , the complexity of the natural torus action is equal to 1. Karshon and Tolman proved in [15] that, for Hamiltonian actions of complexity one, the orbit space is homeomorphic to a sphere provided that the weights of the tangent representation at each fixed point are in general position (see Definition 2.2). This result covers the cases of G 4,2 and F3 . If X 2n is a quasitoric manifold with the action of T n , and T n−1 ⊂ T n is a subtorus, such that the induced action of T n−1 on X 2n is in general position, then the orbit space X 2n /T n−1 is homeomorphic to a sphere [4], as well. For an action of a torus T on a space X , consider the fibration X → X ×T E T → BT and the
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