$C_2$-equivariant topological modular forms

Abstract

We compute the homotopy groups of the $C_2$ fixed points of equivariant topological modular forms at the prime 2 using the descent spectral sequence. We then show that as a ${\mathrm {TMF}}$-module, it is isomorphic to the tensor product of ${\mathrm {TMF}}$ with an explicit finite cell complex.

Appendices

Connective $C_2$-equivariant ${\mathrm {tmf}}$

At the prime 2, one prized property of ${\mathrm {tmf}}$ is

$$\begin{aligned} H_* {\mathrm {tmf}}= {\mathscr {A}}\square _{A(2)} {\mathbb {F}}_2, \end{aligned}$$

which lets us identify the $E_2$-page of its Adams spectral sequence as ${{\,\mathrm{Ext}\,}}_{{\mathscr {A}}(2)}({\mathbb {F}}_2, {\mathbb {F}}_2)$.

It is natural to expect $C_2$-equivariant ${\mathrm {tmf}}$ to have a similar properties, and there have been attempts to construct $C_2$-equivariant ${\mathrm {tmf}}$ along these lines (e.g. [24]). The goal of this section is not to construct $C_2$-equivariant ${\mathrm {tmf}}$, but to deduce properties of any such construction based on its homology.

Theorem A.1

Let ${\mathrm {tmf}}_{C_2} \in {\mathrm {Sp}}_{C_2}$ be a spectrum such that

$$\begin{aligned} ({H\underline{{\mathbb {F}}}}_2)_\star {\mathrm {tmf}}_{C_2} = {{\mathscr {A}}^{C_2} \square _{{\mathscr {A}}^{C_2}(2)}} {\mathbb {M}_2}\end{aligned}$$

and the $C_2$ action on the underlying spectrum $\iota {\mathrm {tmf}}_{C_2}$ is trivial. Then

$$\begin{aligned} H_* ({\mathrm {tmf}}_{C_2})^{C_2} \cong H_*({\mathrm {tmf}}\oplus {\mathrm {tmf}}\oplus {\mathrm {tmf}}\otimes L) \end{aligned}$$

as a Steenrod comodule. If $({\mathrm {tmf}}_{C_2})^{C_2}$ has a ${\mathrm {tmf}}$-module structure, then we have an isomorphism of 2-completed ${\mathrm {tmf}}$-modules.

Note that it is L that appears here, not DL, so $\varDelta ^{-1} ({\mathrm {tmf}}_{C_2})^{C_2}$ and ${\mathrm {TMF}}^{C_2}$ are duals as ${\mathrm {TMF}}$-modules (after completion).

Remark A.2

We would expect that the underlying spectrum of ${\mathrm {tmf}}_{C_2}$ is ${\mathrm {tmf}}$ with the trivial action, and that there is a natural ring map ${\mathrm {tmf}}\rightarrow ({\mathrm {tmf}}_{C_2})^{C_2}$, giving the target a ${\mathrm {tmf}}$-module structure. The proof uses weaker (albeit less natural) assumptions, which is what we state in the theorem.

Since all we know about this hypothetical ${\mathrm {tmf}}_{C_2}$ is its homology, it will be cleaner to work with a general $C_2$-spectrum whose homology is ${{\mathscr {A}}^{C_2} \square _{{\mathscr {A}}^{C_2}(n)}} {\mathbb {M}_2}$ for some n, specializing to $n = 2$ only at the very end. Nevertheless, we shall shortly see that such a spectrum can only exist for $n \le 2$, as in the non-equivariant case.

1.1 Conventions

• For consistency with existing literature, we use $(-)^{C_2}$ to denote the genuine fixed points of a genuine $C_2$-spectrum, instead of the previous $(-)^{\mathscr {B}C_2}$.

• We work with ${{\,\mathrm{RO}\,}}(C_2)$-graded homotopy groups throughout. We have an explicit identification ${{\,\mathrm{RO}\,}}(C_2) = {\mathbb {Z}}[\sigma ]/(\sigma ^2 - 1)$, and we shall write $p + q\sigma = (p + q, q)$ — the first degree is the total degree of the representation and the second degree is the number of $\sigma$’s, also called the weight. We use $\star$ to denote an ${{\,\mathrm{RO}\,}}(C_2)$-grading and $*$ for an integer grading.

• We let $\rho :S^0 \rightarrow S^\sigma$ be the natural inclusion of the representation spheres (more commonly known as a). Then $\rho \in \pi _{-1, -1}(S^0)$.

• We write $\iota :{\mathrm {Sp}}_{C_2} \rightarrow {\mathrm {Sp}}$ for the underlying spectrum functor.

1.2 $C_2$-equivariant homotopy theory

Let X be a $C_2$-spectrum. The main difficulty in analyzing $H_* X^{C_2}$ is that taking categorical fixed points is not symmetric monoidal, so there is no a priori relation between $H{\mathbb {F}}_2 \otimes X^{C_2}$ and $({H\underline{{\mathbb {F}}}}_2 \otimes X)^{C_2}$. To understand $X^{C_2}$, we consider two other functors ${\mathrm {Sp}}_{C_2} \rightarrow {\mathrm {Sp}}$ that are symmetric monoidal, namely the geometric fixed points $X^{\varPhi C_2}$ and the underlying spectrum $\iota X$. The goal of this section is to understand the effects of these constructions on ${{\,\mathrm{RO}\,}}(C_2)$-graded homotopy groups.

The homotopy groups of the categorical fixed points are easy to compute. Essentially by definition, we have

$$\begin{aligned} \pi _* X^{C_2} = \pi _{*, 0} X. \end{aligned}$$

The underlying spectrum and geometric fixed points are obtained by performing certain constructions in ${\mathrm {Sp}}_{C_2}$ and then applying categorical fixed points.

In the $C_2$-equivariant world, the geometric fixed points admit a particularly simple description, since $\rho ^{-1}S^0$ is a model of $\tilde{E}C_2$:

Lemma A.3

The geometric fixed points is given by

$$\begin{aligned} X^{\varPhi C_2} = (\rho ^{-1} X)^{C_2}, \end{aligned}$$

and we can write its homotopy groups as

$$\begin{aligned} \pi _* X^{\varPhi C_2} = \pi _{*, 0} (\rho ^{-1} X) = \pi _\star X / (\rho - 1), \end{aligned}$$

where in the latter formulation, an element of bidegree (s, w) in $\pi _\star X$ gives an element of degree $s - w$ in $\pi _* X^{\varPhi C_2}$.

The underlying spectrum is slightly more involved. It can be expressed as

$$\begin{aligned} \iota X = ((C_2)_+ \otimes X)^{C_2} = F((C_2)_+, X)^{C_2}, \end{aligned}$$

since $(C_2)_+$ is self-dual. To compute its homotopy groups, we use the cofiber sequence

$$\begin{aligned} (C_2)_+ \rightarrow S^0 \overset{\rho }{\rightarrow }S^\sigma . \end{aligned}$$

We can either apply $F(-, X)$ or $(-) \otimes X$ to this, and get

Lemma A.4

There are long exact sequences

$$\begin{aligned} \cdots \longrightarrow \pi _{s + 1, w + 1} X \overset{\rho }{\longrightarrow }\pi _{s, w} X \overset{\mathrm {res}}{\longrightarrow }\pi _{s, w} (C_2)_+ \otimes X \longrightarrow \pi _{s, w + 1} X \overset{\rho }{\longrightarrow }\cdots . \end{aligned}$$

and

$$\begin{aligned} \cdots \overset{\rho }{\longrightarrow }\pi _{s, w - 1} X \longrightarrow \pi _{s, w} (C_2)_+ \otimes X \overset{\mathrm {tr}}{\longrightarrow }\pi _{s, w} X \overset{\rho }{\longrightarrow }\pi _{s - 1, w - 1} X \longrightarrow \cdots . \end{aligned}$$

Moreover, $\mathrm {res} \circ \mathrm {tr}$ is $1 + \gamma$ on $\pi _* \iota X = \pi _{*, 0} (C_2)_+ \otimes X$, where $\gamma$ is the generator of $C_2$.

Corollary A.5

Suppose $\rho$ is injective on $\pi _{*, 0} X$ and $\pi _{*, 1} X$. Then

$$\begin{aligned} \pi _* \iota X = (\pi _\star X / \rho )_{*, 0} \end{aligned}$$

with $\gamma = -1$.

Proof

We first apply the first sequence with $w = 0$. By assumption, the last $\rho$ is injective, hence has trivial kernel. So $\mathrm {res}$ is surjective with kernel given by $\rho$-multiples.

Now consider the second sequence with $w = 0$. Since the second $\rho$ is injective, it has trivial kernel, so $\mathrm {tr}$ has zero image. Hence $1 + \gamma = \mathrm {res} \circ \mathrm {tr} = 0$. $\square$

1.3 The $C_2$-equivariant Steenrod algebra

We recall the computation of the $C_2$-equivariant Steenrod algebra.

Lemma A.6

([8, Proposition 6.2])

$$\begin{aligned} {\mathbb {M}_2}\equiv \pi _\star {H\underline{{\mathbb {F}}}}_2 = {\mathbb {F}}_2[\tau , \rho ] \oplus \left\{ \frac{\gamma }{\rho ^k \tau ^\ell }\mid k, \ell \ge 1\right\} \end{aligned}$$

with the obvious ring structure. The degrees of the classes are

$$\begin{aligned} |\tau | = (0, -1),\quad |\rho | = (-1, -1),\quad |\gamma | = (1, 0). \end{aligned}$$

The terms involving $\gamma$ are known as the “negative cone”. They will not play a role in the story (though we will have to verify that they indeed do not play a role).

Lemma A.7

([8, Theorem 6.41])

$$\begin{aligned} {\mathscr {A}}^{C_2} \equiv \pi _\star ({H\underline{{\mathbb {F}}}}_2 \otimes {H\underline{{\mathbb {F}}}}_2) = {\mathbb {M}_2}[\tau _0, \tau _1, \ldots , \xi _1, \xi _2 \ldots ]/(\tau _i^2 = \rho \tau _{i + 1} + {\bar{\tau }} \xi _{i + 1}), \end{aligned}$$

where ${\bar{\tau }} = \tau + \rho \tau _0$ and

$$\begin{aligned} |\tau _n| = (2^{n + 1} - 1, 2^n - 1),\quad |\xi _n| = (2^{n + 1} - 2, 2^n - 1). \end{aligned}$$

We have

$$\begin{aligned} \eta _R(\rho ) = \rho ,\quad \eta _R(\tau ) = \bar{\tau } = \tau + \rho \tau _0 \end{aligned}$$

and

$$\begin{aligned} \varDelta \xi _k = \sum \xi _{k - i}^{2^i} \otimes \xi _i,\quad \varDelta \tau _k = \sum _{i = 0}^k \xi _{k - i}^{2^i} \otimes \tau _i + \tau _k \otimes 1. \end{aligned}$$

One observes that Corollary A.5 applies to ${\mathbb {M}_2}$ and ${\mathscr {A}}^{C_2}$, so we can read off the homotopy groups of the underlying spectra of ${H\underline{{\mathbb {F}}}}_2$ and ${H\underline{{\mathbb {F}}}}_2 \otimes {H\underline{{\mathbb {F}}}}_2$. This tells us $\iota {H\underline{{\mathbb {F}}}}_2 = H{\mathbb {F}}_2$ and $\pi _* \iota ({H\underline{{\mathbb {F}}}}_2 \otimes {H\underline{{\mathbb {F}}}}_2) \cong {\mathscr {A}}$, as expected. The identification of the latter with ${\mathscr {A}}$ is canonical for the following unimaginative reason:

Lemma A.8

Let $\zeta _1, \zeta _2, \ldots \in {\mathscr {A}}$ be such that $|\zeta _i| = 2^i - 1$ and

$$\begin{aligned} \varDelta \zeta _k = \sum \zeta _{k - i}^{2^i} \otimes \zeta _i. \end{aligned}$$

If $\zeta _1 \not = 0$, then $\zeta _i = \xi _i$.

Proof

We prove this by induction on i. It is clear for $i = 1$, since there is a unique element of degree 1. Then if it is true for $k < i$, then $\zeta _i - \xi _i$ is primitive. But the only primitive elements of ${\mathscr {A}}$ are of the form $\xi _1^{2^k}$, which is not in the bidegree. So this is zero, and $\zeta _i = \xi _i$. $\square$

1.4 The homology of geometric fixed points

Since $(-)^{\varPhi C_2}$ is symmetric monoidal, in order to deduce $H_* X^{\varPhi C_2}$ from $({H\underline{{\mathbb {F}}}}_2)_\star X$, it suffices to understand ${H\underline{{\mathbb {F}}}}_2^{\varPhi C_2}$.

Lemma A.9

We have ${H\underline{{\mathbb {F}}}}_2^{\varPhi C_2} = H{\mathbb {F}}_2 [\tau ]$ as an $\mathbb {E}_1$-ring, where $|\tau | = 1$.

Truncation gives us an $\mathbb {E}_\infty$ map ${H\underline{{\mathbb {F}}}}_2^{\varPhi C_2} \rightarrow H{\mathbb {F}}_2$, which we can understand as quotienting out $\tau$.

Proof

The inclusion of the categorical fixed points gives ${H\underline{{\mathbb {F}}}}_2^{\varPhi C_2}$ the structure of an $\mathbb {E}_\infty$-$H{\mathbb {F}}_2$-algebra. Since $H{\mathbb {F}}_2[\tau ]$ is the free $\mathbb {E}_1$-ring on one generator over $H{\mathbb {F}}_2$, it suffices to observe that $\pi _* {H\underline{{\mathbb {F}}}}_2^{\varPhi C_2} = {\mathbb {F}}_2[\tau ]$ by Lemma A.3. $\square$

Corollary A.10

If $X \in {\mathrm {Sp}}_{C_2}$, then

$$\begin{aligned} H_* X^{\varPhi C_2} \cong (({H\underline{{\mathbb {F}}}}_2)_\star X)/(\rho - 1, \tau ). \end{aligned}$$

If X is a homotopy ring, then this is an isomorphism of rings.

We are, of course, not only interested in the homology groups as groups, but as Steenrod modules. This involves understanding the comparison ${H\underline{{\mathbb {F}}}}_2^{\varPhi C_2} \otimes {H\underline{{\mathbb {F}}}}_2^{\varPhi C_2} \rightarrow H{\mathbb {F}}_2 \otimes H{\mathbb {F}}_2$ given by squaring the projection ${H\underline{{\mathbb {F}}}}_2^{\varPhi C_2} \rightarrow H{\mathbb {F}}_2$.

Corollary A.11

The map $\pi _* ({H\underline{{\mathbb {F}}}}_2^{\varPhi C_2} \otimes {H\underline{{\mathbb {F}}}}_2^{\varPhi C_2}) \rightarrow \pi _* (H{\mathbb {F}}_2 \otimes H{\mathbb {F}}_2)$ sends $\xi _i \mapsto \xi _i$ and kills $\tau _0, \tau$.

Proof

After setting $\rho = 1$, we can express $\tau _{i + 1}$ in terms of $\tau _i^2$ and $\xi _{i + 1}$, so ${\mathscr {A}}^{C_2}/(\rho - 1)$ is generated polynomially over ${\mathbb {F}}_2[\tau ]$ by $\tau _0, \xi _1, \xi _2, \ldots$ with no relations. We know that $\eta _L (\tau )$ and $\eta _R (\tau )$ get mapped to zero, and these are $\tau$ and $\tau + \tau _0$ respectively. So $\tau$ and $\tau _0$ get killed.

As a spectrum, ${H\underline{{\mathbb {F}}}}_2^{\varPhi C_2}$ splits as a sum of $H{\mathbb {F}}_2$’s, we know that the map is surjective, and thus $\xi _1$ must hit $\xi _1$ since it is the only element left in the degree. Finally, the images of the $\xi _i$ satisfy the conditions of Lemma A.8, so must be equal to $\xi _i$. $\square$

Corollary A.12

If $X \in {\mathrm {Sp}}_{C_2}$, then the coaction on $H_* X^{\varPhi C_2}$ is given by taking the $C_2$-equivariant coaction

$$\begin{aligned} ({H\underline{{\mathbb {F}}}}_2)_\star X \rightarrow {\mathscr {A}}^{C_2} \otimes _{{\mathbb {M}_2}} ({H\underline{{\mathbb {F}}}}_2)_\star X \end{aligned}$$

and quotienting out by $(\rho - 1, \tau )$ on the left and $(\rho - 1, \tau \otimes 1, \tau _0 \otimes 1)$ on the right.

1.5 Spectra whose homology is ${{\mathscr {A}}^{C_2} \square _{{\mathscr {A}}^{C_2}(n)}} {\mathbb {M}_2}$

We now introduce the standard quotient coalgebras of ${\mathscr {A}}^{C_2}$:

Definition A.13

We define

$$\begin{aligned} {\mathscr {A}}^{C_2}(n) = {\mathscr {A}}^{C_2}/(\xi _i^{2^{n - i + 1}}, \tau _{n + 1}, \tau _{n + 2}, \ldots ). \end{aligned}$$

If $n < 0$, we set ${\mathscr {A}}^{C_2}(n) = {\mathbb {M}_2}$. Then we have

$$\begin{aligned} {{\mathscr {A}}^{C_2} \square _{{\mathscr {A}}^{C_2}(n)}} {\mathbb {M}_2}= {\mathbb {M}_2}[{\bar{\xi }}_1^{2^n}, \ldots , {\bar{\xi }}_n^2, {\bar{\xi }}_{n + 1}, {\bar{\xi }}_{n + 2}, \ldots , \bar{\tau }_{n + 1}, \bar{\tau }_{n + 2}, \ldots ]. \end{aligned}$$

As usual, the bar denotes the Hopf algebroid antipode.

Definition A.14

We let $Y_n \in {\mathrm {Sp}}_{C_2}$ be any spectrum such that

$$\begin{aligned} ({H\underline{{\mathbb {F}}}}_2)_\star Y_n = {\mathscr {A}}^{C_2} \square _{{\mathscr {A}}^{C_2}(n)} {\mathbb {M}_2}\end{aligned}$$

and the $C_2$ action on the underlying spectrum is trivial. If $n \ge m$, a map $f:Y_n \rightarrow Y_m$ is admissible if it induces the natural injection in homology.

It is helpful to note that

Lemma A.15

There is a unique admissible map $Y_n \rightarrow {H\underline{{\mathbb {F}}}}_2$.

Since we understand the situation of ${H\underline{{\mathbb {F}}}}_2$ completely, we get to learn about $Y_n$ through comparison.

Proof

Since $H_\star Y_n$ is free over ${\mathbb {M}_2}$, the universal coefficients theorem tells us such maps are classified by ${\mathbb {M}_2}$-module maps $H_\star Y_n \rightarrow {\mathbb {M}_2}$, which in turn is in bijection with ${\mathscr {A}}^{C_2}$-comodule maps $H_\star Y_n \rightarrow {\mathscr {A}}^{C_2}$. $\square$

Lemma A.16

$H_* \iota Y_n = {\mathscr {A}}\square _{{\mathscr {A}}(n)} {\mathbb {F}}_2$ with trivial $C_2$ action as a Steenrod comodule, and admissible maps induce the natural inclusion.

This implies that $Y_n$ cannot exist for $n > 2$.

Proof

Since $\iota$ is symmetric monoidal, we have $H_* \iota Y_n = \pi _* \iota ({H\underline{{\mathbb {F}}}}_2 \otimes Y_n)$.

Observe that in $({H\underline{{\mathbb {F}}}}_2)_\star Y_n$, the $\rho$-torsion term of smallest weight is $\frac{\gamma }{\rho \tau }$ with weight 2. So $\rho$ is injective on $({H\underline{{\mathbb {F}}}}_2)_{*, 0} Y_n$ and $({H\underline{{\mathbb {F}}}}_2)_{*, 1} Y_n$, and Corollary A.5 applies. This tells us $H_* \iota Y_n = (({H\underline{{\mathbb {F}}}}_2)_\star Y_n / \rho )_{*, 0}$. It is then easy to check that the ring map ${\mathscr {A}}\square _{{\mathscr {A}}(n)} {\mathbb {F}}_2 \rightarrow H_* \iota Y_n$ sending ${\bar{\xi }}_i^k$ to $(\tau ^{2^{i - 1} - 1}\bar{\tau }_{i - 1})^k$ is an isomorphism. Since the ring is 2-torsion, we have $\gamma = -1 = 1$.

The comodule structure follows from a similar calculation. $\square$

We can similarly compute

Lemma A.17

$$\begin{aligned} H_* Y_n^{\varPhi C_2} = {\mathbb {F}}_2[\bar{\tau }_{n + 1}, {\bar{\xi }}_1^{2^n}, \ldots , {\bar{\xi }}_n^2, {\bar{\xi }}_{n + 1}, \ldots ] \end{aligned}$$

with the coactions of the ${\bar{\xi }}_i$ as usual and

$$\begin{aligned} \psi (\bar{\tau }_{n + 1}) = 1 \otimes \bar{\tau }_{n + 1}. \end{aligned}$$

An admissible map $Y_n \rightarrow Y_m$ acts as the “identity” on the ${\bar{\xi }}_i^{2^n}$’s and sends $\bar{\tau }_{n + 1}$ to $\bar{\tau }_{m + 1}^{2^{n - m}}$.

Our goal is to understand the homology of the categorical fixed points $Y_n^{C_2}$. To do so, we make use of the commutative diagram

where $\widetilde{Y_n}$ is defined to be the cofiber of $(Y_n)_{h C_2} \rightarrow \iota Y_n$.

To compute $H_* Y_n^{C_2}$, we have to understand the effect of $Y_n^{\varPhi C_2} \rightarrow \varSigma (Y_n)_{h C_2}$ on homology. This factors through $\widetilde{Y_n}$, and the strategy is to understand each factor separately. By Theorem B.1, we can identify the homology of the sequence

$$\begin{aligned} \iota Y_n \rightarrow \widetilde{Y_n} \rightarrow \varSigma (Y_n)_{h C_2} \end{aligned}$$

with the natural maps

$$\begin{aligned} {\mathscr {A}}\square _{{\mathscr {A}}(n)} {\mathbb {F}}_2 \rightarrow {\mathscr {A}}\square _{{\mathscr {A}}(n)} H_* \varSigma {\mathbb {RP}}^\infty _{-1} \rightarrow {\mathscr {A}}\square _{{\mathscr {A}}(n)} H_* \varSigma {\mathbb {RP}}^\infty , \end{aligned}$$

which is a short exact sequence.

As for the map $Y_n^{\varPhi C_2} \rightarrow \widetilde{Y_n}$, we know it is an equivalence in the case $n = -1$. In particular, it is injective in homology. Since $H_* Y_n^{\varPhi C_2}$ and $H_* \widetilde{Y_n}$ inject into the $n = -1$ counterparts, we know that $H_* Y_n^{\varPhi C_2} \rightarrow H_* \widetilde{Y_n}$ is injective as well. It remains to compute the image.

At this moment it is slightly more convenient to consider cohomology. We follow some constructions from [10].

For any N, we have a map $\varSigma {\mathbb {RP}}^\infty _{-N} \rightarrow \varSigma {\mathbb {RP}}^\infty _{-1}$ which presents $H^* \varSigma {\mathbb {RP}}^\infty _{-1}$ as a submodule of $H^* \varSigma {\mathbb {RP}}^\infty _{-N}$. Let $M_{n, N}^* \subseteq H^* \varSigma {\mathbb {RP}}^\infty _{-N}$ be the ${\mathscr {A}}(n)^*$-submodule generated under ${\mathscr {A}}(n)^*$ by elements of negative degree, and let $L_n^* = M_{n, N}^* \cap H^* \varSigma {\mathbb {RP}}^\infty _{-1}$ for sufficiently large N (where it no longer depends on N). Let $G_n^* = H^* \varSigma {\mathbb {RP}}^\infty _{-1} / L_n^*$. We then have a short exact sequence of ${\mathscr {A}}(n)$-comodules.

$$\begin{aligned} 0 \rightarrow G_n \rightarrow H_* \varSigma {\mathbb {RP}}^\infty _{-1} \rightarrow L_n \rightarrow 0. \end{aligned}$$

Lemma A.18

$L_n^*$ is trivial in degree 0.

Proof

In the notation of [10, Page 1], we need to know that $x^{-1}$ is not in the image of the ${{\,\mathrm{Sq}\,}}^k$’s. We have

$$\begin{aligned} {{\,\mathrm{Sq}\,}}^k x^{-k - 1} = \frac{(-k - 1)(-k - 2) \cdots (-2k)}{1 \cdot 2 \cdot \cdots \cdot k} x^{-1} = (-1)^{?} \left( {\begin{array}{c}2k\\ k\end{array}}\right) x^{-1} = 0. \end{aligned}$$

$\square$

Lemma A.19

There are no non-zero ${\mathscr {A}}$-comodule maps $H_* Y_n^{\varPhi C_2} \rightarrow {\mathscr {A}}\square _{{\mathscr {A}}(n)} L_n$.

Proof

By adjunction, this is equivalent to showing that there are no non-zero ${\mathscr {A}}(n)$-comodule maps $H_* Y_n^{\varPhi C_2} \rightarrow L_n$, or equivalently, no ${\mathscr {A}}(n)^*$-module maps $L_n^* \rightarrow H^* Y_n^{\varPhi C_2}$.

Now note that in $L_n^*$ all elements have positive degree. Moreover, since ${\mathscr {A}}(n)^*$ is generated by elements of degree at most $2^n$, the module $L_n^*$ is generated as an ${\mathscr {A}}(n)^*$-module by elements of degree less than $2^n - 1$. On the other hand, the elements of lowest degree in $H_* Y_n^{\varPhi C_2}$ are in degree 0 and $2^n$ (given by 1 and ${\bar{\xi }}_1^{2^n}$). Thus, any ${\mathscr {A}}(n)^*$-module map $L_n \rightarrow H^* Y_n^{\varPhi C_2}$ must kill the generators, hence must be zero. $\square$

Corollary A.20

The map $H_* Y_n^{\varPhi C_2} \rightarrow H_* \widetilde{Y_n}$ factors through ${\mathscr {A}}\square _{{\mathscr {A}}(n)} G_n$.

Corollary A.21

$H_* Y_n^{\varPhi C_2} \rightarrow {\mathscr {A}}\square _{{\mathscr {A}}(n)} G_n$ is an isomorphism.

Proof

Since $H_* Y_n^{\varPhi C_2} \rightarrow H_* \widetilde{Y_n}$ is injective, so must this map. By [10, Lemma 1.3] both sides are abstractly isomorphic as graded groups. Since they are finite dimensional in each degree, it must be an isomorphism. $\square$

Since $H_* \widetilde{Y_n} \rightarrow H_* \varSigma (Y_n)_{h C_2}$ is surjective in homology with kernel ${\mathscr {A}}\square _{{\mathscr {A}}(n)} {\mathbb {F}}_2$, we learn that:

Corollary A.22

There is a short exact sequence of comodules

$$\begin{aligned} 0 \rightarrow {\mathscr {A}}\square _{{\mathscr {A}}(n)} L_n[-1] \rightarrow H_* Y_n^{C_2} \rightarrow {\mathscr {A}}\square _{{\mathscr {A}}(n)} {\mathbb {F}}_2 \rightarrow 0. \end{aligned}$$

These $L_n$ can be explicitly computed.

Example A.23

$L_0[-1] = 0$. $L_1[-1] = {\mathbb {F}}_2$. $L_2[-1] = {\mathbb {F}}_2 \oplus H_* L$.

Specializing to the $n = 2$ case,

Lemma A.24

When $n = 2$, the short exact sequence

$$\begin{aligned} 0 \rightarrow {\mathscr {A}}\square _{{\mathscr {A}}(2)} L_2[-1] \rightarrow H_* Y_2^{C_2} \rightarrow {\mathscr {A}}\square _{{\mathscr {A}}(2)} {\mathbb {F}}_2 \rightarrow 0. \end{aligned}$$

splits non-canonically.

Proof

It is classified by an element in

$$\begin{aligned} {{\,\mathrm{Ext}\,}}_{{\mathscr {A}}}^{1, 0} ({\mathscr {A}}\square _{{\mathscr {A}}(2)} {\mathbb {F}}_2, {\mathscr {A}}\square _{{\mathscr {A}}(2)} L_2[-1]) = {{\,\mathrm{Ext}\,}}_{{\mathscr {A}}(2)} ({\mathscr {A}}\square _{{\mathscr {A}}(2)} {\mathbb {F}}_2, L_2[-1]). \end{aligned}$$

Thus, we have to equivalently show that any short exact sequence of ${\mathscr {A}}(2)^*$-modules

$$\begin{aligned} 0 \rightarrow {\mathscr {A}}^*/\!/{\mathscr {A}}(2)^* \rightarrow Z \rightarrow L_2[-1]^* \rightarrow 0 \end{aligned}$$

splits. Let $x_i \in L_2[-1]^*$ be the unique element in degree i, if exists.

Let $z_0 \in Z$ be any lift of $x_0 \in L_2[-1]^*$ and $z_1$ the unique lift of $x_1$. Then we can attempt to produce a splitting by mapping $x_0$ to $z_0$ and $x_1$ to $z_1$. Since the lowest non-zero degrees of ${\mathscr {A}}^*/\!/{\mathscr {A}}(2)^*$ are 0, 8, 12, the only obstruction is if we cannot map $x_8$ to ${{\,\mathrm{Sq}\,}}^4 {{\,\mathrm{Sq}\,}}^2 {{\,\mathrm{Sq}\,}}^1 z_1$. This occurs if and only if ${{\,\mathrm{Sq}\,}}^4 {{\,\mathrm{Sq}\,}}^4 {{\,\mathrm{Sq}\,}}^2 {{\,\mathrm{Sq}\,}}^1 z_1 \not = 0$. But ${{\,\mathrm{Sq}\,}}^4 {{\,\mathrm{Sq}\,}}^4 {{\,\mathrm{Sq}\,}}^2 {{\,\mathrm{Sq}\,}}^1 = {{\,\mathrm{Sq}\,}}^7 {{\,\mathrm{Sq}\,}}^3 {{\,\mathrm{Sq}\,}}^1$, and ${{\,\mathrm{Sq}\,}}^3 {{\,\mathrm{Sq}\,}}^1 z_1$ has degree 5, where Z is trivial. So this cannot happen. $\square$

Corollary A.25

We have

$$\begin{aligned} H_* Y_2^{C_2} = {\mathscr {A}}\square _{{\mathscr {A}}(2)} ({\mathbb {F}}_2 \oplus {\mathbb {F}}_2 \oplus H_* L). \end{aligned}$$

Corollary A.26

If $Y_2^{C_2}$ admits the structure of a ${\mathrm {tmf}}$-module, then upon 2-completion, we have an isomorphism

$$\begin{aligned} Y_2^{C_2} = {\mathrm {tmf}}\oplus {\mathrm {tmf}}\oplus {\mathrm {tmf}}\otimes L. \end{aligned}$$

Proof

The Adams spectral sequence has two $h_0$-towers at 0, and this lets us map in two copies of ${\mathrm {tmf}}$. After quotienting them out we are left with ${{\,\mathrm{Ext}\,}}_{{\mathscr {A}}(2)}({\mathbb {F}}_2, H_* L)$, which is shown in Fig. 19.

Fig. 19

${{\,\mathrm{Ext}\,}}_{{\mathscr {A}}(2)}({\mathbb {F}}_2, H_* L)$

Let $w_1$ be the class in degree 1. To map in L, we need to show that

$$\begin{aligned} 2w_1 = 0, \quad \langle \eta , 2, w_1\rangle = 0,\quad \langle \nu , \eta , 2, w_1\rangle = 0. \end{aligned}$$

These obstructions live in $\pi _1, \pi _3$ and $\pi _7$ respectively. The only non-trivial case is the last case, but the only possible class lives in the indeterminacy. So we get a map $L \rightarrow Y_2^{C_2}$, which lifts to a map ${\mathrm {tmf}}\otimes L \rightarrow Y_2^{C_2}$ via the ${\mathrm {tmf}}$-module structure. This map induces an isomorphism on homology groups, hence an equivalence on 2-completion. $\square$

Stunted projective spaces

The goal of this appendix is to prove the following folklore theorem we used in Theorem 4.1:

Theorem B.1

Let X be a genuine $C_2$-spectrum whose $C_2$ action on the underlying spectrum $\iota X$ is trivial. Then the cofiber of the composition

$$\begin{aligned} \iota X \otimes {\mathbb {RP}}^n_+ \rightarrow \iota X \otimes {\mathbb {RP}}^\infty _+ = X_{h C_2} \overset{\mathrm {Nm}}{\rightarrow }X^{h C_2} = (\iota X)^{{\mathbb {RP}}^\infty _+} \rightarrow (\iota X)^{{\mathbb {RP}}^m_+} \end{aligned}$$

is $\iota X \otimes \varSigma P_{-m - 1}^n$, where $P_{-m}^n$ is the stunted projective space.

We should think of $\iota X \otimes {\mathbb {RP}}^n_+$ as the colimit of the constant functor on $\iota X$ under ${\mathbb {RP}}^n$. In general, we can understand this colimit as follows:

Lemma B.2

Let X be any genuine $C_2$-spectrum, and restrict it to a diagram on ${\mathbb {RP}}^n$ under the inclusion ${\mathbb {RP}}^n \hookrightarrow BC_2 = {\mathbb {RP}}^\infty$. Then

$$\begin{aligned} {{\,\mathrm{colim}\,}}_{{\mathbb {RP}}^n} X \simeq (S((n + 1)\sigma )_+ \otimes X)_{h C_2} \simeq (S((n + 1)\sigma )_+ \otimes X)^{C_2}, \end{aligned}$$

where if $V \in {{\,\mathrm{RO}\,}}(C_2)$, then S(V) is the unit sphere of V.

Note that since $C_2$ acts freely on $S((n + 1) \sigma )$, this homotopy quotient agrees with the literal quotient under point set models.

Proof

The idea of the first isomorphism is that we have a 2-sheeted universal cover $S((n + 1)\sigma ) \rightarrow {\mathbb {RP}}^n$ with deck transformation group $C_2$. To take the colimit over ${\mathbb {RP}}^n$, we can take the colimit over $S((n + 1)\sigma )$, and then take the colimit over the residual $C_2$ action. The restriction of X to $S((n + 1) \sigma )$ is trivial since the following diagram commutes

so the colimit over $S((n + 1) \sigma )$ is simply given by tensoring with $S((n + 1) \sigma )_+$.

To prove this formally, we apply [16, Corollary 4.2.3.10], where we choose $\mathscr {J} = BC_2$, $K = {\mathbb {RP}}^n$ and F to be the universal cover $S((n + 1)\sigma ) \rightarrow {\mathbb {RP}}^n$ with $\mathscr {J}$ acting via deck transformations.

The second equivalence is the Adams isomorphism (see e.g. [25, Theorem 8.4]). $\square$

We recall some basic properties of stunted projective spaces. Let $\phi (n)$ be the number of integers j congruent to 0, 1, 2 or 4 mod 8 such that $0 < j \le n$.

Theorem B.3

([4, Theorem V.2.14]) There is a (unique) collection of spectra $\{P_m^n\}_{n \ge m}$ with the property that

• 1. If $m > 0$, then $P^n_m = \varSigma ^\infty {\mathbb {RP}}^n / {\mathbb {RP}}^{m - 1}$ using the standard inclusion.

• 2. If $r \equiv n \pmod {2^{\phi (k)}}$, then $P_n^{n + k} \cong \varSigma ^{n - r} P^{r + k}_r$.

Further, we have

• 3. If we restrict the $BC_2$ action on $S^{\sigma }$ to ${\mathbb {RP}}^m \subseteq {\mathbb {RP}}^\infty = BC_2$, then

  $$\begin{aligned} {{\,\mathrm{colim}\,}}_{{\mathbb {RP}}^m} S^{n\sigma } \simeq P_n^{n + m} \end{aligned}$$

• 4. $D P_m^n \simeq \varSigma P_{-n-1}^{-m-1}$.

Corollary B.4

There is a natural isomorphism

$$\begin{aligned} \lim _{{\mathbb {RP}}^m} S^{n \sigma } \simeq \varSigma P_{n - m + 1}^{n - 1}. \end{aligned}$$

Proof

$$\begin{aligned} \lim _{{\mathbb {RP}}^m} S^{n \sigma } \simeq D {{\,\mathrm{colim}\,}}_{{\mathbb {RP}}^m} S^{-n \sigma } \simeq D P_{-n}^{m - n} \simeq \varSigma P_{n - m - 1}^{n - 1}. \end{aligned}$$

$\square$

Corollary B.5

Let X be a spectrum with trivial $C_2$ action. Then

$$\begin{aligned} (S^{n \sigma } \otimes X)^{h C_2} \simeq \lim _N (\varSigma P^{n - 1}_{-N} \otimes X). \end{aligned}$$

Proof

We write X as a filtered colimit of finite spectra, $X = {{\,\mathrm{colim}\,}}_\alpha X^{(\alpha )}$. Then we have

$$\begin{aligned} \begin{aligned} (S^{n \sigma } \otimes X)^{h C_2}&= \left( S^{n \sigma } \otimes {{\,\mathrm{colim}\,}}_\alpha X^{(\alpha )}\right) ^{h C_2} \\&= \left( {{\,\mathrm{colim}\,}}_\alpha \left( S^{n \sigma } \otimes X^{(\alpha )}\right) \right) ^{h C_2} \\&= \lim _{{\mathbb {RP}}^\infty } {{\,\mathrm{colim}\,}}_\alpha \left( S^{n \sigma } \otimes X^{(\alpha )}\right) \\&= \lim _m \lim _{{\mathbb {RP}}^m} {{\,\mathrm{colim}\,}}_\alpha \left( S^{n \sigma } \otimes X^{(\alpha )}\right) \\&= \lim _m {{\,\mathrm{colim}\,}}_\alpha \lim _{{\mathbb {RP}}^m} \left( S^{n \sigma } \otimes X^{(\alpha )}\right) \\&= \lim _m {{\,\mathrm{colim}\,}}_\alpha \left( \varSigma P_{n - m - 1}^{n - 1} \otimes X^{(\alpha )}\right) \\&= \lim _m \left( \varSigma P_{n - m - 1}^{n - 1} \otimes X\right) , \end{aligned} \end{aligned}$$

where we use that finite limits commute with filtered colimits and tensoring with finite spectra. $\square$

We now recall the construction of the norm map. Let X be a genuine $C_2$-spectrum. By Lemma B.2, we have an isomorphism

$$\begin{aligned} {{\,\mathrm{colim}\,}}_{{\mathbb {RP}}^n} X = (S((n + 1)\sigma )_+ \otimes X)^{C_2}. \end{aligned}$$

Taking the colimit as $n \rightarrow \infty$ gives

$$\begin{aligned} X_{hC_2} = (S(\infty \sigma )_+ \otimes X)^{C_2}, \end{aligned}$$

where $S(\infty \sigma ) = {{\,\mathrm{colim}\,}}_n S(n \sigma )$. The norm map $X_{h C_2} \rightarrow X^{C_2}$ is then induced by the projection $S(\infty \sigma )_+ \rightarrow S^0$. By replacing X with the cofree version $X^{(EC_2)_+}$, we obtain the $X^{h C_2}$-valued version of the norm map.

Lemma B.6

Let X be a genuine $C_2$-spectrum whose $C_2$ action on the underlying spectrum $\iota X$ is trivial. Then the cofiber of the composition

$$\begin{aligned} \iota X \otimes {\mathbb {RP}}^n_+ \rightarrow \iota X \otimes {\mathbb {RP}}^\infty _+ = X_{h C_2} \overset{\mathrm {Nm}}{\rightarrow }X^{h C_2} = (\iota X)^{{\mathbb {RP}}^\infty _+} \end{aligned}$$

is $\lim _N (\iota X \otimes \varSigma P_{-N}^n)$.

Proof

This composition is obtained by taking the fixed points of

$$\begin{aligned} S((n + 1) \sigma )_+ \otimes X^{(EC_2)_{+}} \rightarrow X^{(EC_2)_+}. \end{aligned}$$

The cofiber of this map is $S^{(n+1) \sigma } \otimes X^{(EC_2)_+} \cong (S^{(n + 1)\sigma } \otimes X)^{(EC_2)_+}$. So this follows from Corollary B.5. $\square$

We can now prove Theorem B.1.

Proof of Theorem B.1

This follows from the commutative diagram

$\square$

Sage script

This appendix contains the sage script used to perform the computer calculations. The actual computations are at the end of the script and the comments indicate the lemmas they prove.