\input{"preamble.tex"} \let\Begin\begin \let\End\end \newcommand\wrapenv[1]{#1} \makeatletter \def\ScaleWidthIfNeeded{% \ifdim\Gin@nat@width>\linewidth \linewidth \else \Gin@nat@width \fi } \def\ScaleHeightIfNeeded{% \ifdim\Gin@nat@height>0.9\textheight 0.9\textheight \else \Gin@nat@width \fi } \makeatother \setkeys{Gin}{width=\ScaleWidthIfNeeded,height=\ScaleHeightIfNeeded,keepaspectratio}% \title{ \textbf{ Title } } \begin{document} \date{} \author{D. Zack Garza} \maketitle \newpage % Note: addsec only in KomaScript \addsec{Table of Contents} \tableofcontents \newpage \hypertarget{thursday-june-03}{% \section{Thursday, June 03}\label{thursday-june-03}} \hypertarget{part-1}{% \subsection{Part 1}\label{part-1}} \begin{quote} Jeremey Hanh, MIT: something abelian about commutative ring spectra See Tablbot 2017 notes and Richter's survey article ``Commutative Ring Spectra''. \end{quote} \begin{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. \end{remark} \begin{remark} The key objects we'll be considering: \begin{itemize} \tightlist \item \(\SS\), the initial \(E_\infty\) ring spectrum \item Thom \(E_\infty\)ring spectra: \(\MO, \MSO, \MSp, \MStr, \MU\), and the sphere fits into this pattern as framed bordism \end{itemize} We construct \emph{other} \(E_\infty\) rings primarily to study these motivating examples. \end{remark} \begin{example}[?] A ring spectrum that shows up naturally when ?: \(\MU\), for which \begin{align*} \pi_* \MU \cong \ZZ[x_1, x_2, \cdots] && \text{where } \abs{x_i} = 2i \end{align*} Localing at a prime \(p\) splits \(\MU_{(p)}\) into sum of suspensions \(\BP\) where \begin{align*} \pi_* \BP = \ZZ_{(p)}[x_1, x_2, \cdots] && \text{where } \abs{x_i} = 2p^i - 2 .\end{align*} We can write \begin{align*} \BP = \MU / \gens{x_j \st j\neq p^i - 1} ,\end{align*} 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. \end{example} \begin{question} Does this splitting preserve the face that \(\MU\) is an \(E_\infty\) ring spectrum? I.e. is \(\BP\) an \(E_\infty\) ring spectrum? \end{question} \begin{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. \end{answer} \begin{theorem}[Basterra, Mandell] \(\BP\) is an \(E_4\dash\)algebra retract of \(\MU\). \end{theorem} \begin{remark} Some open questions: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item Is \(\BP\) an \(E_5\dash\MU\dash\)algebra? \item 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. \end{enumerate} \end{remark} \begin{remark} Studying other \(E_\infty\) rings naturally leads to problems in obstruction theory. We study \(\SS\) via chromatic homotopy theory. The basic strategy: \begin{itemize} \item 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. \item 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) \end{itemize} \end{remark} \begin{question}[A big one] Can one generally construct the \(\EO\) localizations above? \end{question} \begin{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. \end{remark} \begin{remark} How one builds \(\EO\) theories: \begin{itemize} \tightlist \item Build \(E_n\) as a homotopy commutative ring using the Landweber exact functor theorem, which is not too difficult. \item Promote the \(E_n\) to \(E_\infty\) rings using obstruction theory. \begin{itemize} \tightlist \item See Robinson, Goerss-Hopkins, Lurie, Pstragowki and VanKoughnett. These allow one to make \(G\dash\)actions by \(E_\infty\) ring maps. \item 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. \end{itemize} \end{itemize} \end{remark} \begin{question} Can one compute \(\pi_* \EO\) for \(\EO \da E_n^{hG}\) for various \(n\) and \(G\)? \end{question} \begin{example}[?] The key to the Kervaire invariant one question is computing \(\pi_* E_4 ^{hC_8}\), and captures information about diffeomorphism classes of exotic spheres. \end{example} \begin{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. \end{observation} \begin{example}[?] 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 \(E_2^{hC_{24}}\). \begin{align*} E_1^{hC_2} = \KO\localize{2} &= L_{K(1)}(\ko) \\ E_2^{hC_{24}} &= L_{K(2)}(\tmf) .\end{align*} \end{example} \begin{remark} These lifts are closer to geometry than \(\EO\) theories, e.g.~there is an \(E_\infty\) ring map due to Ando-Hopkins-Rezk \begin{align*} \MString \to \tmf .\end{align*} \begin{quote} Find comments \end{quote} See how \(\MSpin\) splits as \todo[inline]{Missed} so one might expect that \(\MString\) is \todo[inline]{??? Missed} \end{remark} \begin{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. \end{remark} \begin{question} Can we make connective \(\eo\) theories highly structured with \(L_{K(n)} \eo = \EO = E_n^{hG}\) for various and \(n\) and \(G\)? \end{question} \begin{quote} See example in comment \end{quote} \begin{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: \begin{itemize} \tightlist \item \(L_{K(2)} \tmf\) \item \(L_{K(1)} \tmf\) \item \(L_{\QQ} \tmf\) \end{itemize} 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. \end{remark} \begin{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 \(n\). \begin{quote} See comments \end{quote} \end{remark} \begin{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: \begin{itemize} \item 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. \item Quotient by some elements, possibly losing structure, to obtain a quotient \(Q\). The connective version of \(E_4^{hC_8}\) is \(Q^{C_8}\). \begin{itemize} \tightlist \item It'd be interesting to know how much structure is lost here! \end{itemize} \end{itemize} \end{remark} \begin{question} Some natural questions that arise here: \begin{itemize} \tightlist \item What group actions (with various amounts of structure, e.g.~\(E_\infty\)) act on tensor powers \(\MU^{\tensor m}\)? \item What structure exists on quotients of such tensor powers? \end{itemize} \end{question} \begin{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? \end{remark} \begin{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?? \end{remark} \hypertarget{part-2}{% \subsection{Part 2}\label{part-2}} \begin{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? \end{remark} \begin{example}[?] Take \begin{align*} \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] .\end{align*} \end{example} \begin{theorem}[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. \end{theorem} \begin{theorem}[H-Wilson] There exists a specific choice of generators \(v_{n+1}, v_{n+2}\) is an \(E_3\dash\BP\) algebra. \end{theorem} \begin{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. \end{remark} \begin{proposition}[?] If \(x\in \pi_{2\ell} \BP\) is any class in \(\pi_* \BP\), then \(\BP / \gens{ x }\) is an \(E_1\dash\BP\) algebra. \end{proposition} \begin{remark} This says you can freely mod out by any generator and still obtain an \(E_1\) structure. \end{remark} \begin{proof}[?] 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 \begin{align*} \psi: S^0[a_{2\ell}] \to \BP ,\end{align*} which hits \(x\), and \begin{align*} \BP / \gens{ x } = \BP \tensor_{S^0[a_{2\ell}] } S^0 \end{align*} where we use \(\eps: S^0[a_{2\ell}] \to S^0\). It suffices to prove the following lemma: \begin{lemma}[?] \(S^0[a_{2\ell}]\) and \(\psi\) can be made \(E_2\). \end{lemma} \begin{proof}[of lemma, in the case $\ell=1$] Consider \begin{align*} S^0[a_2] = S^0 \oplus S^2 \oplus S^4 \oplus \cdots = \Free_{E_1}(S^4) .\end{align*} 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 \begin{align*} S^2 \to S^4\cong \HP^1 \to \HP^2 \to \HP^3 \to \cdot \to \HP^\infty ,\end{align*} which yields a filtration \begin{align*} \Suspendpinf \Loop^2 S^4 = \Suspendpinf \Loop^2 \HP^1 \to \Suspendpinf \Loop^2 \HP^2 \to \cdots .\end{align*} One can try to produce maps out of each filtered pieced: \begin{center} \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} \end{center} \begin{quote} \href{https://q.uiver.app/?q=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}{Link to Diagram} \end{quote} 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. \begin{quote} Find comment. \end{quote} \end{proof} \end{proof} \begin{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. \begin{quote} Find comment on first obstruction. \end{quote} \end{remark} \begin{remark} Proving a relativey easy proof that introduces a new technique used to show that \(\BP \gens{ n }\) can be made \(E_3\). \end{remark} \begin{theorem}[?] Connective \(K(n)\) exists as an \(E_1\dash\SS\) algebra. \end{theorem} \begin{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{center} \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} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsMTEsWzEsMywiXFxTaWdtYV57NHBebi00fSBcXEZGX3AiXSxbMSw1LCJcXFNpZ21hXnsycF5uLTJ9IFxcRkZfcCJdLFs0LDUsIlxcdGF1X3tcXGxlcSAycF5uLTJ9IGsobikiXSxbNCwzLCJcXHRhdV97XFxsZXEgNHBebi00fSBrKG4pIl0sWzQsMCwiayhuKSJdLFs0LDEsIlxcdmRvdHMiXSxbNCw3LCJcXEZGX3AiXSxbNiw3LCJcXFNpZ21hXnsycF5uLTJ9XFxGRl9wIl0sWzgsNywiXFxTaWdtYV57NHBebn0iXSxbMCwzLCJ2X25eMiJdLFswLDUsInZfbiJdLFs0LDVdLFs1LDNdLFs3LDgsIlFfbiJdLFs2LDcsIlFfbiJdLFsyLDZdLFszLDJdLFswLDNdLFsxLDJdXQ==}{Link to Diagram} \end{quote} \begin{itemize} \item 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. \item 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. \end{itemize} 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 \emph{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. \end{remark} \begin{remark} How to build \(k(n)\) as an \(E_1\) ring instead of a spectrum: \begin{itemize} \tightlist \item Write down the object parameterizing two-stage Postnikov towers in the category of \(E_1\) rings. This is well-known to be the \textbf{\(E_1\dash\) 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. \end{itemize} \end{remark} \begin{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 \begin{align*} S^0[a_{2\ell}] \to \THC(\FF_p) .\end{align*} \end{remark} \end{document}