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Tate Blueshift and Vanishing for Real Oriented Cohomology 3

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Tate Blueshift and Vanishing for Real Oriented Cohomology 3 TATE BLUESHIFT AND VANISHING FOR REAL ORIENTED COHOMOLOGY GUCHUAN LI, VITALY LORMAN, AND J.D. QUIGLEY Abstract. We study the Tate construction for certain Real oriented cohomology theories. First, we show that after suitable completion, the Tate construction with respect to a trivial Z/2-action on height n Real Johnson-Wilson theory splits into a wedge of height n − 1 Real Johnson-Wilson theories. Our result simultaneously recovers a blueshift result of Ando-Morava-Sadofsky for (classical) Johnson-Wilson theory and extends a blueshift result of Greenlees-May for real topological K-theory to all chromatic heights. Second, we prove that the Tate construction with respect to a trivial finite group action on Real Morava K-theory vanishes, generalizing a Tate vanishing result of Greenlees and Sadofsky for (classical) Morava K-theory. In the process of proving our results, we develop the theory of completions of module spectra over Real complex cobordism, C2-equivariant chromatic Bousfield localizations, and parametrized Tate constructions. Contents 1. Introduction 1 2. Real representations and Real oriented cohomology theories 7 3. Completion and Bousfield localization 11 4. Tate constructions 15 5. Blueshift for Real oriented cohomology theories 23 6. Tate vanishing 30 References 36 1. Introduction arXiv:1910.06191v3 [math.AT] 9 Aug 2021 1.1. Motivation and main results. Real oriented homotopy theory has played an im- portant role in algebraic topology over the past few decades. Its central objects, Real ori- ented cohomology theories, are genuine C2-equivariant cohomology theories equipped with a choice of Thom class for Real vector bundles. Examples include the K-theory of Real vector bundles KR [3], Real cobordism MR [36], height n Real Johnson-Wilson theory E(n) and Real Morava K-theory K(n) [29], and certain forms of topological modular forms with level structure [22]. These cohomology theories have astounding applications, including the solution of the Kervaire invariant one problem [23], nonimmersions of real projective spaces [6][33][34], and the Atiyah-Segal completion theorem [2]. Combined with new techniques in equivariant stable homotopy theory, real oriented cohomology has led to incredible progress in stable homotopy theory [8][24][20][37]. The purpose of this paper is to study Real oriented cohomology theories via their Tate constructions. Let E be a spectrum and let Σ be a finite group acting on E. The Σ-Tate 1 2 GUCHUANLI,VITALYLORMAN,ANDJ.D.QUIGLEY construction of E, denoted EtΣ, is the cofiber of the additive norm map from the Σ- hΣ homotopy orbits EhΣ to the Σ-homotopy fixed points E [16]. The Tate construction has many interesting applications ranging from descent in chromatic homotopy theory [3][43][52] to trace methods for algebraic K-theory [21][44][51]. We focus on E(n) and K(n). The C2-spectrum E(n) has been studied extensively in work of Kitchloo, Wilson, and the second author [31][32][33], and the ER(n) := E(n)C2 - cohomology of a variety of spaces is now known, including RP ∞ [33], CP ∞ [38], BO(n) [35], and half of the Eilenberg-MacLane spaces [30]. The Tate construction of a Real oriented spectrum with respect to its natural C2-action often vanishes: Fajstrup showed [14] that KRtC2 ≃ ∗, and Hu and Kriz showed [29] that ER(n)tC2 ≃ ∗ for all n ≥ 1. In this paper, however, we are interested in the Tate construc- tion with respect to a trivial action. n+2 n−1 Theorem A (Real oriented Tate blueshift, Theorem 5.12). Let λn := 2 (2 − 1) + 1 C2 and let Σ2 act trivially on ER(n) := E(n) . (1) For n ≥ 2 there is a map of spectra (1−λn 1)j tΣ2 ∧ lim Σ − ER(n − 1) −→ (ER(n) )b i In−1 j≤i _ that becomes an isomorphism on homotopy groups after competion at In−1 or equiv- alently after K(n − 1)-localization. Here In−1 is a shift of I¯n−1 = (p, v¯1,..., v¯n−2) ⊂ E(n)⋆ into integer degrees using the invertible class y ∈ πλ+σE(n). (Seeb Section 2.4 for more on the coefficients of E(n).) b (2) For n = 1, there is an equivalence of spectra lim Σ4jER(0) ≃ ER(1)tΣ2 , i j≤i _ where ER(0) ≃ HQ and ER(1) ≃ KO(2). As n increases, the Hurewicz image of ER(n) becomes increasingly complicated. For example, ER(n) detects the first n Hopf invariant one and Kervaire invariant one elements and the first n−1κ ¯-elements in the stable homotopy groups of spheres, if they are nontrivial [37]. Theorem 5.12 therefore provides an equivalence for studying the more complicated Hurewicz image of ER(n) using the less complicated Hurewicz image of ER(n − 1). The n = 1 case of Theorem A was proven by Greenlees and May in [16] using the Atiyah-Segal completion theorem. Other blueshift phenomena have been observed at low chromatic heights in [4][11][12]. Further applications include the tensor triangulated geometry of equivariant stable homotopy theory [7][5] and redshift for Mahowald invariants [41][53]. The Tate construction also has a curious effect on Real Morava K-theory. The following result is related to global Tate vanishing and ambidexterity in K(n)-local stable homotopy theory [25][27] as well as the finite generation of Morava K-theory of classifying spaces of finite groups [49]. Theorem B (Real oriented Tate vanishing, Theorem 6.14). Let G be a finite group and let G act trivially on KR(n). Then KR(n)tG ≃ ∗. Theorems A and B are real oriented counterparts of classical complex oriented Tate blueshift and vanishing results. In [1], Ando, Morava, and Sadofsky showed that if the n-th TATE BLUESHIFT AND VANISHING FOR REAL ORIENTED COHOMOLOGY 3 (classical) Johnson-Wilson theory E(n) is equipped with a trivial Σ2-action, then there is a map of spectra 2j tΣ2 ∧ (1) lim Σ E(n − 1) → (E(n) )I 1 i n− j≤i _ which induces an isomorphism on homotopy groups after completion at In−1, or equiva- lently after Bousfield localization with respect to the (n − 1)-st Morava K-theory K(n − 1). Analogously, Greenlees and Sadofsky showed [18] that (2) K(n)tG ≃ ∗ for the trivial action of any finite group G on the n-th (classical) Morava K-theory K(n). Theorem B follows from (2) using the cofiber sequence [32] y ΣλKR(n) → KR(n) → K(n) paired with a subtle analysis of the Real Morava K-theory of inverse limits. Surprisingly, Theorem A does not follow directly from its complex oriented counterpart, and moreover, the techniques of [1] do not apply to ER(n) as it is not a complex-oriented theory. We obtain Theorem A from the following C2-equivariant enrichment of the blueshift result of Ando-Morava-Sadofsky. Theorem C (Parametrized blueshift, Theorem 5.11). Let Σ2 act trivially on E(n). There is a map of genuine C2-spectra ρj tC2 Σ2 ∧ lim Σ E(n − 1) → (E(n) )I 1 i n− j≤i _ which induces an isomorphism on C2-equivariant homotopy groups after completion at In−1 = (p, v¯1,..., v¯n−2) or equivalently after K(n − 1)-localization. The construction on the right-hand side of Theorem C is the parametrized Tate construc- tion [46][48] which will be discussed below. The C2-equivariant map above recovers (1) by taking underlying spectra. The proof of Theorem A from Theorem C is not immediate: the C2-fixed points of the parametrized Tate construction are not generally the ordinary Tate construction of the C2-fixed points. To circumvent this issue, we construct a comparison map using certain homotopy limit models for trivial Tate constructions (discussed below) and special C2-equivariant properties of E(n). We then use ER(n)-orientability results from [31][35] to show that the comparison map is an equivalence of C2-spectra. Theorem D (Tate comparison, Theorem 5.8). The map E(n)tΣ2 → E(n)tC2 Σ2 is a C2-equivariant weak equivalence. Theorem D allows us to identify the C2-fixed points of the right-hand side of Theorem C with (a completion of) the classical Σ2-Tate construction of ER(n). Theorem A then follows by an explicit calculation of the C2-fixed points of the left-hand side. 4 GUCHUANLI,VITALYLORMAN,ANDJ.D.QUIGLEY 1.2. Techniques. To understand why classical techniques cannot be applied for ER(n), we sketch the construction of the map (1) and the proof that it is a K(n − 1)-local equivalence from [1]. If E is a complex oriented cohomology theory equipped with a trivial Σ2-action, then there is an isomorphism tΣ2 ∗ (3) π−∗(E ) =∼ E ((x))/[2](x) with |x| = 2, where [2](x) is the 2-series of the formal group law associated to E. By examining [2](x) for E(n), one produces an isomorphism tΣ2 ∧ ∼ ∧ (4) π∗(E(n) )In−1 = E(n − 1)∗((x))In−1 . Next, one must produce the map (1). The map is constructed using the formula for trivial Σ2-Tate constructions [16] tΣ2 ∞ (5) X ≃ lim(ΣX ∧ P−i), i ∞ ∞ where P−i = T h(−iγ → RP ). The map (1) is obtained as the inverse limit of maps from bounded below wedges to bounded below stunted projective spectra produced using special properties of MU-module spectra. Finally, one identifies In−1-completion with K(n − 1)- localization using the equivalence −1 ∧ (6) LK(j)M ≃ (vj M)Ij−1 which holds for any MU-module spectrum M. The technical core of this work is concerned with proving Real oriented analogs of (3)- (6).
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