# Thursday, June 03 ## Part 1 > Jeremey Hanh, MIT: something abelian about commutative ring spectra > See Tablbot 2017 notes and Richter's survey article "Commutative Ring Spectra". :::{.remark} The 2017 Talbot was on structured ring spectra, i.e. "brave new algebra", where we study $E_n\dash$ring spectra. Getting the subject off the ground requires many definitions, care of May, EKMM, Lurie, and more. ::: :::{.remark} The key objects we'll be considering: - $\SS$, the initial $E_\infty$ ring spectrum - Thom $E_\infty$ring spectra: $\MO, \MSO, \MSp, \MStr, \MU$, and the sphere fits into this pattern as framed bordism We construct *other* $E_\infty$ rings primarily to study these motivating examples. ::: :::{.example title="?"} A ring spectrum that shows up naturally when ?: $\MU$, for which \[ \pi_* \MU \cong \ZZ[x_1, x_2, \cdots] && \text{where } \abs{x_i} = 2i \] Localing at a prime $p$ splits $\MU_{(p)}$ into sum of suspensions $\BP$ where \[ \pi_* \BP = \ZZ_{(p)}[x_1, x_2, \cdots] && \text{where } \abs{x_i} = 2p^i - 2 .\] We can write \[ \BP = \MU / \gens{x_j \st j\neq p^i - 1} ,\] so $\BP$ has simpler homotopy, and $\MU$ splits into a sum of copies of $\BP$, which are easier to work with because the homotopy is more sparsely distributed. $\BP$ is a bit of a "designer" spectrum, and $\MU$ is more geometric. ::: :::{.question} Does this splitting preserve the face that $\MU$ is an $E_\infty$ ring spectrum? I.e. is $\BP$ an $E_\infty$ ring spectrum? ::: :::{.answer} No! See Lawson and Senger, who prove that $\BP$ not an $E_{2p^2 + q}\dash$ring spectrum. It turns out that understanding the $E_4$ structure is sufficient for many applications. ::: :::{.theorem title="Basterra, Mandell"} $\BP$ is an $E_4\dash$algebra retract of $\MU$. ::: :::{.remark} Some open questions: 1. Is $\BP$ an $E_5\dash\MU\dash$algebra? 2. The $E_n$ operad acts on $\BP$, so is $\BP$ a $\Disk_2\dash$algebra? In particular, can one take factorization homology against unframed surfaces? This would correspond to a trivial $S^1\dash$action. ::: :::{.remark} Studying other $E_\infty$ rings naturally leads to problems in obstruction theory. We study $\SS$ via chromatic homotopy theory. The basic strategy: - Resolve $\SS$ by other $E_\infty$ rings, namely the $K(n)\dash$local sphere $L_{K(n)} \SS$. Note that this is a Bousfield localization. This is useful precisely because of the chromatic convergence theorem, and one can build a tower whose associated graded are these local spheres. - Resolve $L_{K(n)} \SS$ for a fixed $n$ by the Hopkins-Miller $\EO\dash$theories, which are $E_\infty$ rings of the form $E_n^{hG}$ for $G$ a finite group acting on the height $n$ Morava $E\dash$theory $E_n$. These $\EO$ theories are supposed to be the basic building blocks of the $K(n)\dash$local spheres. (Check) ::: :::{.question title="A big one"} Can one generally construct the $\EO$ localizations above? ::: :::{.remark} Proved for a fixed degree/height? Check. A recent triumph of obstruction theory is that the building block $\EO$ theories have been built. This was a prominent topic in 2017 Talbot. ::: :::{.remark} How one builds $\EO$ theories: - Build $E_n$ as a homotopy commutative ring using the Landweber exact functor theorem, which is not too difficult. - Promote the $E_n$ to $E_\infty$ rings using obstruction theory. - See Robinson, Goerss-Hopkins, Lurie, Pstragowki and VanKoughnett. These allow one to make $G\dash$actions by $E_\infty$ ring maps. - Rough sketch: build an $E_\infty$ ring in $\ho\Sp$ (the homotopy category of spectra), which is already a homotopy commutative ring. Then do this in the homotopy 2-category of $\Sp$, then the homotopy 3-category of $\Sp$, and so on. This uses that an $\infty\dash$category is a sequence of $n\dash$categories. ::: :::{.question} Can one compute $\pi_* \EO$ for $\EO \da E_n^{hG}$ for various $n$ and $G$? ::: :::{.example title="?"} The key to the Kervaire invariant one question is computing $\pi_* E_4 ^{hC_8}$, and captures information about diffeomorphism classes of exotic spheres. ::: :::{.observation} In practice, these $\EO$ theories (which are all $K(n)\dash$local) seem to be $K(n)\dash$localizations of nice connective ring spectra. ::: :::{.example title="?"} At the prime $p=2$, $E_1^{hC_2} = \KO\localize{2}$ is the localization of $\KO$ at 2, and turns out to be equal to $L_{K(1)}(k_0)$: It turns out that $E_2^{hC_{24}} = L_{K(2)} (\tmf)$, and so $\ko$ and $\tmf$ are "connective $E_\infty$ lifts" of $E_1^{hC_2}$ and $E_2^{hC_{24}}$. \[ E_1^{hC_2} = \KO\localize{2} &= L_{K(1)}(\ko) \\ E_2^{hC_{24}} &= L_{K(2)}(\tmf) .\] ::: :::{.remark} These lifts are closer to geometry than $\EO$ theories, e.g. there is an $E_\infty$ ring map due to Ando-Hopkins-Rezk \[ \MString \to \tmf .\] > Find comments See how $\MSpin$ splits as \todo[inline]{Missed} so one might expect that $\MString$ is \todo[inline]{??? Missed} ::: :::{.remark} An observation due to Hi-Kriz and Hill-Hopkins-Ravenel: using sparsity, the easiest way to compute $\pi_* \EO$ is to compute $\pi_* \eo$ where $\eo$ is a good connective lift of $\EO$. Note that $\pi_* \tmf$ is finitely-generated in each degree, and it's useful to work with something "small" in computations, for example if you're trying to rule out differentials in spectral sequences. A nice way to organize the computations of $\pi_* \EO$ is to understand them via lifts with better finiteness properties. ::: :::{.question} Can we make connective $\eo$ theories highly structured with $L_{K(n)} \eo = \EO = E_n^{hG}$ for various and $n$ and $G$? ::: > See example in comment :::{.remark} $\tmf$ is a connective $\EO$ theory, so how is $\tmf\localize{2}$ built? For full details, see the tmf book or Lurie's "Elliptic Cohomology". A sketch: $\tmf\localize{2} = \tau_{\geq 0} L_2 \tmf\localize{2}$ where the latter is built out of a finite resolution involving the following three terms: - $L_{K(2)} \tmf$ - $L_{K(1)} \tmf$ - $L_{\QQ} \tmf$ Note that $L_2$ is the second stage of the chromatic tower. The basic strategy: take the monochromatic pieces above, which are relatively easy to make and work with, and find a way to glue them together. ::: :::{.remark} An idea: one can try to make $\eo$ as a connective cover of some $L_n\dash$local object. See Lawson, Berhrens-Lawson and $\TAF$ (topological automorphic forms). This worked very well for $\tmf$, and there is currently partial progress at height 3. We're not yet able to construct a connective version of $E_4^{hC_8}$, which was needed in Kervaire Invariant One. Note that all techniques used here seem to work equally well for building $E_\infty$ as $E_n$ rings for finite $n$. > See comments ::: :::{.remark} An alternate idea that let Hill-Hopkins-Ravenel solve Kervaire Invariant One, and recently developed by Beaudry-Hill-Shi-Zeng, constructs a connective version of $E_4^{hC_8}$. However, with this construction, it's less clear how much structure there is on the object. They use the following procedure: - Put a $C_8$ action on $\MU^{\tensor 4}$ by viewing this as a norm $\NN_{C_2}^{C_8} \MU_\RR$ \todo[inline]{Check $\RR$ notation..} The norm here gives a way of boosting a $C_2$ action on one tensor factor to a $C_8$ action on 4 tensor factors. - Quotient by some elements, possibly losing structure, to obtain a quotient $Q$. The connective version of $E_4^{hC_8}$ is $Q^{C_8}$. - It'd be interesting to know how much structure is lost here! ::: :::{.question} Some natural questions that arise here: - What group actions (with various amounts of structure, e.g. $E_\infty$) act on tensor powers $\MU^{\tensor m}$? - What structure exists on quotients of such tensor powers? ::: :::{.remark} Note that these quotients can be destructive when it comes to maintaining $E_n$ ring structures. Answering these amounts to building structured models for these connective spectra. Understanding these two questions would allow computing fixed points of certain Morava $E\dash$theories? Big question: do these tensor products admit $G\dash$actions beyond those which come from norms? ::: :::{.remark} Getting at these $\EO$ would be huge! The HHR construction is spectacular but somehow only works at the prime $2$ and for cyclic groups?? ::: ## Part 2 :::{.remark} Recall that $\BP$ is an $E_4$ ring spectrum with $\pi* \BP \cong \ZZ_{(p)}[x_1, x_2, \cdots]$. What structure exists on quotients of $\BP$? Any progress here would lead to many natural next questions, e.g. adding in group actions. There has been recent progress, some of which is ripe for generalization -- e.g. how much can be made equivariant? ::: :::{.example title="?"} Take \[ \BP \gens{ n }\da \BP / \gens{ v_n , v_{n+1}, \cdots } \implies \pi_*\BP \gens{ n } \cong \ZZ_{(p)}[x_1, x_2, \cdots, x_n] .\] ::: :::{.theorem title="Baker-Jeanneret"} For any choice of indecomposable generators $v_{n+1}, v_{n+2}, \cdots$, $\BP\gens{ n }$ is an $E_1\dash \BP$ algebra. ::: :::{.theorem title="H-Wilson"} There exists a specific choice of generators $v_{n+1}, v_{n+2}$ is an $E_3\dash\BP$ algebra. ::: :::{.remark} It'd be exciting to try to take this result and use it in the equivariant setting. We'll try to discuss a bit how this theorem is proved. ::: :::{.proposition title="?"} If $x\in \pi_{2\ell} \BP$ is any class in $\pi_* \BP$, then $\BP / \gens{ x }$ is an $E_1\dash\BP$ algebra. ::: :::{.remark} This says you can freely mod out by any generator and still obtain an $E_1$ structure. ::: :::{.proof title="?"} Let $S^0[a_{2\ell}] = S^0 \oplus S^{2\ell} \oplus S^{4\ell} \oplus \cdots$ denote the free $E_1$ ring on $S^{2\ell}$. There is an $E_1$ ring map \[ \psi: S^0[a_{2\ell}] \to \BP ,\] which hits $x$, and \[ \BP / \gens{ x } = \BP \tensor_{S^0[a_{2\ell}] } S^0 \] where we use $\eps: S^0[a_{2\ell}] \to S^0$. It suffices to prove the following lemma: :::{.lemma title="?"} $S^0[a_{2\ell}]$ and $\psi$ can be made $E_2$. ::: :::{.proof title="of lemma, in the case $\ell=1$"} Consider \[ S^0[a_2] = S^0 \oplus S^2 \oplus S^4 \oplus \cdots = \Free_{E_1}(S^4) .\] This turns out to be equal to $\Sigma_+^\infty \Loop S^3$ using ??? (see comment). Now use $S^4 = \Loop\HP^\infty$, so this is $\Sigma_+^\infty \Omega^2 \HP^\infty$, which has an $E_2$ ring structure. There is a filtration \[ S^2 \to S^4\cong \HP^1 \to \HP^2 \to \HP^3 \to \cdot \to \HP^\infty ,\] which yields a filtration \[ \Suspendpinf \Loop^2 S^4 = \Suspendpinf \Loop^2 \HP^1 \to \Suspendpinf \Loop^2 \HP^2 \to \cdots .\] One can try to produce maps out of each filtered pieced: \begin{tikzcd} {\Sigma_+^\infty \Omega^2 S^4} &&&& \BP \\ \\ {\Sigma_+^\infty \Omega^2 \HP^2} \\ \\ {\Sigma_+^\infty \Omega^2 \HP^3} \\ \vdots \\ {\Sigma_+^\infty \Omega^2 S^3} \arrow[from=5-1, to=6-1] \arrow[from=6-1, to=7-1] \arrow[from=1-1, to=1-5] \arrow[from=1-1, to=3-1] \arrow[from=3-1, to=5-1] \arrow[curve={height=30pt}, dashed, from=3-1, to=1-5] \arrow[curve={height=24pt}, dashed, from=5-1, to=1-5] \arrow[curve={height=30pt}, dashed, from=7-1, to=1-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) At each stage, the obstruction to lifting is a map out of a free $E_2$ algebra on an odd degree class in $\pi_* \BP$, which is concentrated in even degrees. > Find comment. ::: ::: :::{.remark} So these are free as $E_1$ rings, and are secretly $E_2$ rings (although not free $E_2$ rings) which have a simple presentation that makes them easy to map into objects with even-degree homotopy. > Find comment on first obstruction. ::: :::{.remark} Proving a relativey easy proof that introduces a new technique used to show that $\BP \gens{ n }$ can be made $E_3$. ::: :::{.theorem title="?"} Connective $K(n)$ exists as an $E_1\dash\SS$ algebra. ::: :::{.remark} We hve $\pi_* K(n) = \FF_p[v_n]$ where $\abs{v_n} = 2p^n - 2$ and $K(n) = \BP / \gens{ ? }$. There is a Postnikov tower: \begin{tikzcd} &&&& {k(n)} \\ &&&& \vdots \\ \\ {v_n^2} & {\Sigma^{4p^n-4} \FF_p} &&& {\tau_{\leq 4p^n-4} k(n)} \\ \\ {v_n} & {\Sigma^{2p^n-2} \FF_p} &&& {\tau_{\leq 2p^n-2} k(n)} \\ \\ &&&& {\FF_p} && {\Sigma^{2p^n-2}\FF_p} && {\Sigma^{4p^n}} \arrow[from=1-5, to=2-5] \arrow[from=2-5, to=4-5] \arrow["{Q_n}", from=8-7, to=8-9] \arrow["{Q_n}", from=8-5, to=8-7] \arrow[from=6-5, to=8-5] \arrow[from=4-5, to=6-5] \arrow[from=4-2, to=4-5] \arrow[from=6-2, to=6-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMTEsWzEsMywiXFxTaWdtYV57NHBebi00fSBcXEZGX3AiXSxbMSw1LCJcXFNpZ21hXnsycF5uLTJ9IFxcRkZfcCJdLFs0LDUsIlxcdGF1X3tcXGxlcSAycF5uLTJ9IGsobikiXSxbNCwzLCJcXHRhdV97XFxsZXEgNHBebi00fSBrKG4pIl0sWzQsMCwiayhuKSJdLFs0LDEsIlxcdmRvdHMiXSxbNCw3LCJcXEZGX3AiXSxbNiw3LCJcXFNpZ21hXnsycF5uLTJ9XFxGRl9wIl0sWzgsNywiXFxTaWdtYV57NHBebn0iXSxbMCwzLCJ2X25eMiJdLFswLDUsInZfbiJdLFs0LDVdLFs1LDNdLFs3LDgsIlFfbiJdLFs2LDcsIlFfbiJdLFsyLDZdLFszLDJdLFswLDNdLFsxLDJdXQ==) - To build $\tau_{2p^n-2} k(n)$, one just needs to identify $Q_n \in \pi_0 \Hom(\FF_p, \Sigma^{2p^n - 1} \FF_p)$. So one needs to identify $Q_n \in \pi_* \Hom(\FF_p, \FF_p)$, the $\mathrm{mod} p$ Steenrod algebra. - To build $\tau_{\leq 4p^n - 4}k(n)$, one needs to check that $Q_n^2 = 0$ in the $\mod p$ Steenrod algebra, which is an Adem relation. Note that understanding $\pi_* \Hom(\FF_p, \FF_p)$ as a group lets on build $\tau_{\leq 2p^n - 2}k(n)$., but the next stage requires knowing this is a *ring* along with the Adem relation. Since $\Hom(\FF_p, \FF_p)$ is an $E_1$ ring, and understanding this ring structure would allow building $k(n)$ completely as a spectrum. Here $\Hom(\FF_p, \FF_p)$ parameterizes all 2-stage Postnikov towers in the sense that its homotopy groups record this data. ::: :::{.remark} How to build $k(n)$ as an $E_1$ ring instead of a spectrum: - Write down the object parameterizing two-stage Postnikov towers in the category of $E_1$ rings. This is well-known to be the **$E_1\dash$ center** $\mathcal{Z}_{E_1}(\FF_p)$, also known as $\THC(\FF_p)$, the topological Hochschild cohomology of $\FF_p$. This is known to be an $E_2$ ring and if one understands its $E_2$ structure well, one learns that $E_2$ rings are more complicated than 2-stage Postnikov towers. ::: :::{.remark} Bokstedt proved that $\pi_* \THC(\FF_p)$ is concentrated in even degrees. Thus given any class $x_{2\ell}\in \pi_{2\ell} \THC(\FF_p)$ parameterizing some 2-stage $E_1$ ring, by the previous theorem there is an $E_2$ ring map \[ S^0[a_{2\ell}] \to \THC(\FF_p) .\] :::