1. Introduction. Mahowald proved the following theorem [Mah77]: Theorem 1.1 (Mahowald). There is an equivalence of spectra

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1 A C 2 -EQUIVARIANT ANALOG OF MAHOWALD S THOM SPECTRUM THEOREM MARK BEHRENS AND DYLAN WILSON Abstract. We prove that the C 2 -equivariant Eilenberg-MacLane spectrum associated with the constant Mackey functor F 2 is equivalent to a Thom spectrum over Ω ρ S ρ`1. 1. Introduction Let µ be the Möbius bundle over S 1, regarded as a virtual bundle of dimension 0. The mod 2 Moore spectrum is the Thom spectrum Mp2q» ps 1 q µ. The classifying map for µ extends to a double loop map rµ : Ω 2 S 3 Ñ BO. Mahowald proved the following theorem [Mah77]: Theorem 1.1 (Mahowald). There is an equivalence of spectra pω 2 S 3 q rµ» HF 2. The bundle µ may also be regarded C 2 -equivariant virtual bundle over S 1, by endowing both S 1 and the bundle with the trivial action. Since B C2 O is an equivariant infinite loop space [Ati68], the classifying map for µ extends to an Ω ρ -loop map rµ : Ω ρ S ρ`1 Ñ B C2 O. Here, ρ is the regular representation of C 2. The purpose of this paper is to prove the following. Theorem 1.2. There is an equivalence of C 2 -spectra pω ρ S ρ`1 q rµ» HF 2. (Here, F 2 denotes the constant Mackey functor with value F 2.) Acknowledgements. Many tricks in this paper have been independently discovered by Doug Ravenel, and the first author s involvement in this project is an outgrowth of mathematical discussions with Agnés Beaudry, Prasit Bhattacharya, Dominic Culver, Doug Ravenel, and Zhouli Xu. The authors also benefited from valuable input from Mike Hill. The first author was supported by NSF grant DMS Date: August 20,

2 2 MARK BEHRENS AND DYLAN WILSON Conventions. Equivariant objects in this paper either live in Top C2, the category of C 2 -spaces, or Sp C2, the category of genuine C 2 -spectra. In both of these categories, the equivalences are those equivariant maps which induce equivalences on both C 2 -fixed points and underlying fixed points. We let H denote the Eilenberg- Maclane spectrum HF 2, with underlying spectrum H : HF 2. We use H and π C2 to denote ROpC 2 q-graded homology and homotopy groups (i.e. not the Mackey functors) of C 2 -equivariant spaces and spectra, and H and π to denote the ordinary homology and homotopy groups of non-equivariant spaces and spectra. We let σ denote the sign representation of C 2, and let ρ 1 ` σ denote the regular representation. For a representation V, SpV q denotes the unit sphere in V, and S V denotes its one point compactification, and V denotes its dimension. 2. Equivariant preliminaries Euler class. Let a denote the Euler class in π C2 σs, given geometrically by the inclusion There is a cofiber sequence S 0 ãñ S σ. (2.1) C 2` Ñ S 0 ãñ S σ so the cofiber of a is stably given by (2.2) Ca» Σ 1 σ C 2`. The transfer induces a map u : S 1 σ tr ÝÑ Σ 1 σ C 2`» Ca which serves as a Thom class for the representation σ: For X P Sp C2, we have u : S 1 Ñ Ca ^ S σ. Said differently, π C2 k pxq π kpx C2 q, π C2 V px ^ Caq π V px e q. (2.3) π C2 X ^ Ca π X e ru s. Tate square. We will let X h : F pec 2`, Xq, X Φ : X ^ ĄEC 2 denote the homotopy completion and geometric localization of X, respectively. The fixed points of X h are the homotopy fixed points of X, and the fixed points of X Φ

3 A C 2-EQUIVARIANT ANALOG OF MAHOWALD S THOM SPECTRUM THEOREM 3 are the geometric fixed points of X. X is recovered from these approximations by the pullback ( Tate square ) [GM95] X X Φ X h X t where the spectrum X t is the equivariant Tate spectrum X t : px h q Φ. Note that a generalization of the argument establishing (2.2) yields an equivalence Taking a colimit, we see that we have Σ kσ 1 Cpa k q» Spkσq`. hocolim Σ kσ 1 Cpa k q» EC 2`, k hocolim S kσ» EC Ą 2. k It follows that homotopy completion and geometric localization can be reinterpreted as a-completion and a-localization: X h» X^a, X Φ» Xra 1 s. In this manner, the Tate square is equivalent to the a-arithmetic square X Xra 1 s X^a X^a ra 1 s Using (2.3), the a-bockstein spectral sequence takes the form π px e qru, as ñ π C2 px h q. The a-bockstein spectral sequence can be regarded as an ROpC 2 q-graded version of the homotopy fixed point spectral sequence (see [HM17, Lem. 4.8]). The mod 2 Eilenberg-MacLane spectrum. We have [HK01] where π C2 H F 2 ra, us F 2ra, us pa 8, u 8 q tθu u 1 σ, θ 2σ 2. The a-u divisible factor in π H is best understood from the Tate square, using π C2 H h F 2 ra, u 1 s, π C2 H Φ F 2 ra 1, us.

4 4 MARK BEHRENS AND DYLAN WILSON Actually, the second isomorphism lifts to an equivalence H ΦC2» Hra 1 us : ł Σ i H iě0 so we have H Φ X H px ΦC2 qra 1, us and, restricting the grading to trivial representations, we get (2.4) H Φ X H px ΦC2 qra 1 us. By applying π C2 V to the map H ^ X Ñ H ^ X ^ Ca we get a homomorphism (2.5) Φ e : H V pxq Ñ H V px e q. Taking geometric fixed points of a map gives a map S V Ñ H ^ X S V C 2 Ñ H ΦC2 ^ X ΦC2 Using (2.4) and passing to the quotient by the ideal generated by a 1 u, we get a homomorphism (2.6) Φ C2 : H V pxq Ñ H V C 2 px ΦC2 q. A useful lemma. Our main computational lemma is the following. Lemma 2.7. Suppose that X P Sp C2 H pxq such that and suppose that tb i u is a set of elements of (1) tφ e pb i qu is a basis of H px e q, and (2) tφ C2 pb i qu is a basis of H px ΦC2 q. Then H pxq is free over H, and tb i u is a basis. Proof. The set tb i u corresponds to a map H ^ ł S bi Ñ H ^ X. Assumption (1) implies this map is an equivalence upon applying Φ e, while assumption (2) implies this map is an equivalence upon applying Φ C2. The result follows. 3. Homology of ρ-loop spaces We spell out some specific algebraic structure carried by the equivariant homology of a ρ-loop space. A more detailed and general study of this algebraic structure will appear in [Hil].

5 A C 2-EQUIVARIANT ANALOG OF MAHOWALD S THOM SPECTRUM THEOREM 5 Products. Suppose X Ω ρ Y P Top C2 is a ρ-loop space. Then X is in particular a 1-loop space, and is therefore an equivariant H-space with product m : X ˆ X Ñ X. However, the σ-loop space structure also endows X with a twisted product related to the transfer. Namely, let S σ Ñ S σ {S 0 «C 2` ^ S 1 be the pinch map. This gives rise to a twisted product rm : N ˆΩY Ñ Ω σ Y where N ˆZ : MappC 2, Zq Z ˆ Z õ C 2 is the norm (with respect to Cartesian product). In particular, there is a map (3.1) rm : N ˆΩ 2 Y Ñ X. Upon applying fixed points to the map (3.1), we get an additive transfer (3.2) t : X e Ñ X C2. In homology, the H-space structure give rise to a product m : H V X b H W X Ñ H V `W X. Using the equivariant commutative ring spectrum structure of H [Ull13], the twisted product rm gives rise to a norm map (see [BH15, Thm. 7.2]) n : H k X e Ñ H kρ X. Dyer-Lashof operations. X has even more structure: X is an E ρ -algebra [GM17]. Specifically, regard Spρq as a C 2 ˆ Σ 2 -space where C 2 acts on ρ and Σ 2 acts antipodally. Then the E ρ -structure gives a map Spρq ˆΣ2 Xˆ2 Ñ X. Note that H is itself an E ρ -ring spectrum, because it is actually an equivariant commutative ring spectrum, so H ^ X` is an E ρ -ring in H-modules. Given x P H V pxq, represented by a map there is an induced composite H ^ Spρq` ^Σ2 S 2V x : S V Ñ H ^ X`, 1^1^x^x ÝÝÝÝÝÝÑH ^ Spρq` ^Σ2 ph ^ X`q^2 ÑH ^ H ^ X` ÑH ^ X` (where the unlabeled maps come from the E ρ -ring and H-module structure of H ^ X`). Applying π C2, we get a total power operation Ppxq : r H pspρq` ^Σ2 S 2V q Ñ H X. For the purposes of this paper we will be only concerned with the case of V kρ σ. We will need the following lemma.

6 6 MARK BEHRENS AND DYLAN WILSON Lemma 3.3. We have the following identification of the C 2 -fixed point space of the extended power: pspρq` ^Σ2 S 2pkρ σq q C2 «S 2k 1 _ S 2k. Proof. The extended power can be identified with the Thom complex of the equivariant vector bundle Spρq ˆΣ2 R 2pkρ σq Ñ Spρq{Σ 2. The fixed points is the Thom complex of the fixed point bundle. Thinking of Spρq as the unit circle in C, with C 2 acting by conjugation, the fixed points of the base are given by rspρq{σ 2 s C2 tr1s, risu. The bundle has fiber R 2pkρ σq over r1s, and because Σ 2 acts with the antipodal action mixed with the interchange action, the fiber over ris is given by R ρpkρ σq R p2k 1qρ. The result follows. Proposition 3.4. We have rh Spρq` ^Σ2 S 2pkρ σq H te 2kρ σ 1, e 2kρ σ u. Proof. Theorem 2.15 of [Wil17] implies there is a cofiber sequence S 2kρ 2σ Ñ Spρq` ^Σ2 S 2pkρ σq Ñ S 2kρ 1. There are two possibilities for the long exact sequence in H : either (a) the connecting homomorphism sends ι 2kρ 1 to zero, or (b) the connecting homomorphism sends it to θι 2kρ 2σ. Only possibility (b) is compatible with Lemma 3.3 from geometric fixed point considerations. The result follows. Thus we get a pair of Dyer-Lashof operations given by the formulas Q kρ : H kρ σ X Ñ H 2kρ σ X, Q kρ 1 : H kρ σ X Ñ H 2kρ σ 1 X Q kρ pxq : Ppxqpe 2kρ σ q, Q kρ 1 pxq : Ppxqpe 2kρ σ 1 q. Remark 3.5. If X is actually an equivariant infinite loop space, then H X has an action by equivariant Dyer-Lashof operations [Wil17], and these operations agree with those defined in that paper.

7 A C 2-EQUIVARIANT ANALOG OF MAHOWALD S THOM SPECTRUM THEOREM 7 Compatibility with fixed points. The compatibility of all this structure with the maps Φ e and Φ C2 of (2.5) and (2.6) is summarized as follows. Products: Note that X e is an E 2 -algebra, and X C2 maps Φ e and Φ C2 are algebra homomorphisms. is an E 1 -algebra. The Norms: The following diagram commutes: H k X e H k X C2 t Φ C 2 n H kρ X Φ e Sq H 2k X e Here t is the transfer (3.2) and Sq is the squaring map. Dyer-Lashof operations: The following diagrams commute, where ɛ 0, 1: H kρ σ X Q kρ ɛ H 2kρ σ ɛ X Φ e Φ e H 2k 1 X e Q 2k ɛ H 4k 1 ɛ X e H kρ σ X ΦC 2 H k X C2 Q kρ Sq H 2kρ σ X Φ C 2 H 2k X C2 4. Homology of Ω ρ S ρ`1 Theorem 4.1. There is an additive isomorphism (of H -modules) H Ω ρ S ρ`1 H b Ert 0, t 1,...s b P re 1, e 2,...s with t i 2 i ρ σ, e i p2 i 1qρ. Proof. Note that we have H Ω 2 S 3 F 2 rx 1, x 2,...s with x i 2 i 1. Here x 1 is the fundamental class ι 1, and x i : Q 2i Q 2i 1 Q 2 x 1.

8 8 MARK BEHRENS AND DYLAN WILSON Define t 0 P H 1 Ω ρ S ρ`1 to be the fundamental class, and define the other generators e i and t i by Consider the product e i : npx i q, with ɛ i P t0, 1u and k i ě 0. We compute t i : Q 2iρ Q 2i 1ρ Q ρ t 0. t ɛ e k : t ɛ0 0 tɛ1 1 ek1 1 ek2 2 P H pω ρ S ρ`1 q Φ e pt ɛ e k q x 2k1`ɛ0 1 x 2k2`ɛ1 2. Mapping out of the cofiber sequence (2.1) gives a fiber sequence ΩN ˆΩS ρ`1 Ñ Ω ρ S ρ`1 Ñ ΩS ρ`1 ÝÑ N ˆΩS ρ`1. Upon taking fixed points we get a fiber sequence In particular there is an equivalence and we have Ω 2 S 3 t ÝÑ pω ρ S ρ`1 q C2 Ñ ΩS 2 null ÝÝÑ ΩS 3 pω ρ S ρ`1 q C2» ΩS 2 ˆ Ω 2 S 3. H pω ρ S ρ`1 q C2 P rys b P rtpx 1 q, tpx 2 q,...s where y is the image of the fundamental class under the map It follows that Thus the set S 1 Ñ pω ρ S ρ`1 q C2. Φ C2 pt ɛ e k q y ɛ0`2ɛ1`4ɛ2` tpx 1 q k1 tpx 2 q k2. tt ɛ e k u Ă H X satisfies the hypotheses of Lemma 2.7, and the result follows. 5. The equivariant Mahowald theorem In order to prove Theorem 1.2 we will need to establish a Thom isomorphism H pω ρ S ρ`1 q rµ H Ω ρ S ρ`1. We will do so in two steps. Recall that an E 0 -algebra is just a spectrum X equipped with a map S 0 Ñ X. Let Free E ρ : Alg E0 psp C2 q Ñ Alg Eρ psp C2 q denote a homotopical left adjoint to the forgetful functor. An explicit model for this functor is the homotopy pushout of E ρ -algebras: Free Eρ ps 0 q Free Eρ pxq S 0 Free E ρ pxq

9 A C 2-EQUIVARIANT ANALOG OF MAHOWALD S THOM SPECTRUM THEOREM 9 We will need the following theorem. Theorem 5.1. Let f : X Ñ B C2 O classify a virtual bundle of dimension zero and denote by f : Ω ρ Σ ρ X Ñ B C2 O the associated Ω ρ -map. Then there is a canonical equivalence of E ρ -algebras in Sp C2 Free E ρ px f q pω ρ Σ ρ Xq f. Proof. Combine the equivariant approximation theorem [GM17, RS00] with Theorem IX.7.1 and Remark X.6.4 of [LMSM86]. Remark 5.2. The non-equivariant version of Theorem 5.1 was first observed by Mark Mahowald, and then proven by Lewis. A nice modern account in the nonequivariant setting via universal properties can be found in [AB14]. Proposition 5.3. There is a Thom isomorphism H pω ρ S ρ`1 q rµ H Ω ρ S ρ`1. Proof. Let Free E ρ,h : Alg E0 pmod H q Ñ Alg Eρ pmod H q denote a homotopical left adjoint to the forgetful functor. Along with the previous theorem, we will need two facts: (1) H ^ p q : Sp C2 Ñ Mod H is symmetric monoidal. (2) There is a Thom isomorphism H ^ ps 1 q µ H ^ S 1`. The proposition is now proved by the following string of equivalences: H ^ `Ω ρ Σ ρ S 1 µ H ^ Free Eρ `ps 1 q µ by Theorem 5.1 Free E ρ,h `H ^ ps 1 q µ by (1) Free E ρ,h `H ^ S 1` by (2) H ^ Free E ρ `S1` by (1) H ^ Ω ρ Σ ρ S 1`. Proof of Theorem 1.2. The Thom class is represented by a map pω ρ S ρ`1 q rµ Ñ H. We wish to show this map is an isomorphism on H. The homology of H is the C 2 -equivariant Steenrod algebra, computed in [HK01] to be with H H H rτ 0, τ 1,, ξ 1, ξ 2, s{pτ 2 i τ i 2 i ρ σ, ξ i p2 i 1qρ. pu ` aτ 0 qξ i`1 ` aτ i`1 q It suffices to show it is surjective, since the two homologies are abstractly isomorphic and of finite type. Observe that the composite Mp2q» ps 1 q µ Ñ pω ρ S ρ`1 q rµ Ñ H

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