On the homotopy fixed point sets of circle actions on product spaces
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Archiv der Mathematik
On the homotopy fixed point sets of circle actions on product spaces Jian Liu, Sang Xie, and Xiugui Liu
Abstract. For arbitrary S 1 -actions on SQm , SQn , and SQm × SQn , we show 1 the conditions for the tenability of the homotopy equivalence (SQm )hS × 1 1 1 (SQn )hS (SQm × SQn )hS . Here, X hS denotes the homotopy fixed point 1 set of an S -action on an space X. Mathematics Subject Classification. 55P62. Keywords. Homotopy fixed point set, Product spaces, Rational sphere, Rational homotopy equivalence.
1. Introduction. Given an action of a group G on a space X, we have the homotopy fixed point set X hG , that is, the G-equivariant mapping space mapG (EG, X). A G-action on X induces a G-action on the rational space XQ [7,8]. It is a classical fact [3] that X hG is homotopy equivalent to sec ξ, the space of sections of the corresponding Borel fibration ξ
X → XhG → BG. The homotopy fixed point set has been studied by many researchers. Goyo [5] studied the homotopy fixed point set X hG for a finite group G. The homotopy 1 fixed point set X hS was considered by Buijs, F´elix, Huerta, and Murillo in [1,2]. Recently, Hao, Liu, and Sun described algebraic models for the homotopy fixed point sets of S 3 -actions on some rational complexes [6]. From now on, we denote by G a compact connected Lie group. Given Gactions on spaces X, Y , respectively, we can endow X × Y with a G-action given by g · (x, y) = (g · x, g · y),
(x, y) ∈ X × Y.
Xiugui Liu is the corresponding author of this paper, and was supported in part by Tianjin Natural Science Foundation (Grant No. 19JCYBJC30200).
Arch. Math.
J. Liu et al.
Then we have a homeomorphism X hG × Y hG ∼ = (X × Y )hG . Given G-actions on spaces X, Y , and X × Y , we have the homotopy fixed point sets X hG , Y hG , and (X × Y )hG , respectively. The product space X hG × Y hG is not always rational homotopy equivalent to (X × Y )hG (see Section 2). For convenience, we denote the numbers of path connected components of 1 1 m hS 1 (SQ ) , (SQn )hS , and (SQm × SQn )hS by r(n), r(m), and r(m, n), respectively. In this paper, we show the following theorems. Theorem 1.1. For arbitrary S 1 -actions on SQm , SQn , and SQm × SQn , we have 1
1
(SQm )hS × (SQn )hS (SQm × SQn )hS
1
if one of the following conditions holds: (1) m, n are odd. (2) m is odd, n is even, m > n, r(n) = r(m, n) = 2. (3) m is odd, n is even, m < n, r(n) = r(m, n). Now we consider the case that m and n are even and m ≥ 2n. For arbitrary S 1 -actions on SQm , SQn , and SQm × SQn , we have the homotopy fixed point sets 1 1 1 (SQm )hS , (SQn )hS , and (SQm × SQn ))hS , respectively. Note that the model of (SQm × SQn )hS 1 must be of the form (A ⊗ Λ(x, x , y, y ), D),
deg x = m, deg y = n,
where Dx = Dy = 0 and Dx = x2 + λam/2 x + μam−n/2 y + Φ , Dy = y 2 + κan/2 y
μ, κ ∈ Q, Φ ∈ A+ ⊗ Λ≥2 y.
(1.1)
Theorem 1.2. Let the notations be as above. Then we have 1
1
(SQm )hS × (SQn )hS (SQm × SQn )hS
1
if one of the following conditions holds: (1) λ = 0, κ = 0, r(m) = 2, r(n) = 1. (2)
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