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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Introduction to Stable Homotopy Theory} \vspace{.5em} \hrule \vspace{.5em} $\,$ This entry is a detailed introduction to the [[stable homotopy category]] and to its key computational tool, the [[Adams spectral sequence]]. To that end we introduce the modern tools, such as [[model categories]] and [[highly structured ring spectra]]. In the accompanying [[Introduction to Stable homotopy theory -- S|seminar]] we consider applications to [[cobordism theory]] and [[complex oriented cohomology]] such as to converge in the end to a glimpse of the modern picture of [[chromatic homotopy theory]]. $\,$ \vspace{.5em} \hrule \vspace{.5em} Lecture notes. $\;\;\;\;\;\;\;$ (web version requires Firefox browser -- \href{https://www.mozilla.org/en-US/firefox/new/}{free download}) \emph{[[Introduction to Stable homotopy theory -- P|Prelude -- Classical homotopy theory]]} ([[IntroductionToStableHomotopyTheory-P-170509.pdf:file]] 111 pages) \emph{[[Introduction to Stable homotopy theory -- 1|Part 1 -- Stable homotopy theory]]} $\,\,$ \emph{[[Introduction to stable homotopy theory -- 1-1|Part 1.1 -- Sequential Spectra]]} ([[IntroductionToStableHomotopyTheory-1-1-170509.pdf:file]], 79 pages) $\,\,$ \emph{[[Introduction to Stable homotopy theory -- 1-2|Part 1.2 -- Structured Spectra]]} ([[IntroductionToStableHomotopyTheory-1-2-170509.pdf:file]], 75 pages) \emph{[[Introduction to Stable homotopy theory -- I|Interlude -- Spectral sequences]]} ([[IntroductionToStableHomotopyTheory-I-170509.pdf:file]], 15 pages) \emph{[[Introduction to Stable homotopy theory -- 2|Part 2 -- Adams spectral sequences]]} ([[IntroductionToStableHomotopyTheory-2-170509.pdf:file]], 53 pages) \emph{[[Introduction to Cobordism and Complex Oriented Cohomology|Examples and Applications -- Cobordism and Complex Oriented Cohomology]]} ([[IntroductionToStableHomotopyTheory-S-161227.pdf:file]], 76 pages) \textbf{total file} ([[IntroductionToStableHomotopyTheory-170509.pdf:file]], 418 pages) Background reading: \emph{[[schreiber:Introduction to Homological algebra|Background -- Introduction to Homological algebra]]} ([[IntroductionToHomologicalAlgebra-170509.pdf:file]], 83 pages) \emph{[[Introduction to Topology|Background -- Introduction to Topology]]} (\href{https://ncatlab.org/nlab/files/IntroductionToTopologyI-170509.pdf}{pdf}, 122 pages) \vspace{.5em} \hrule \vspace{.5em} $\,$ $\,$ \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Introduction}{Survey}\dotfill \pageref*{Introduction} \linebreak \noindent\hyperlink{StableHomotopyTheorySurvey}{1) Stable homotopy theory}\dotfill \pageref*{StableHomotopyTheorySurvey} \linebreak \noindent\hyperlink{ANSS}{2) Adams spectral sequences}\dotfill \pageref*{ANSS} \linebreak \noindent\hyperlink{s_complex_oriented_cohomology}{S) Complex oriented cohomology}\dotfill \pageref*{s_complex_oriented_cohomology} \linebreak \noindent\hyperlink{ClassicalHomotopyTheory}{\textbf{Prelude) Classical homotopy theory}}\dotfill \pageref*{ClassicalHomotopyTheory} \linebreak \noindent\hyperlink{StableHomotopyTheory}{\textbf{Part 1) Stable homotopy theory}}\dotfill \pageref*{StableHomotopyTheory} \linebreak \noindent\hyperlink{SpectralSequences}{\textbf{Interlude) Spectral sequences}}\dotfill \pageref*{SpectralSequences} \linebreak \noindent\hyperlink{part_2_adams_spectral_sequences}{\textbf{Part 2) Adams spectral sequences}}\dotfill \pageref*{part_2_adams_spectral_sequences} \linebreak \noindent\hyperlink{ComplexOrientedCohomology}{\textbf{Seminar) Complex oriented cohomology}}\dotfill \pageref*{ComplexOrientedCohomology} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \noindent\hyperlink{basic_reading}{Basic reading}\dotfill \pageref*{basic_reading} \linebreak \noindent\hyperlink{further_reading}{Further reading}\dotfill \pageref*{further_reading} \linebreak $\,$ \vspace{.5em} \hrule \vspace{.5em} $\,$ \begin{quote}% My initial inclination was to call this book [[The Music of the Spheres]], but I was dissuaded from doing so by my diligent publisher, who is ever mindful of the sensibilities of librarians. (\hyperlink{Ravenel86}{Ravenel 86, preface}) \end{quote} \hypertarget{Introduction}{}\subsection*{{Survey}}\label{Introduction} We are concerned with the theory of \emph{[[spectra]]} in the sense of [[algebraic topology]]: the proper generalization of [[abelian groups]] to [[homotopy theory]]. \hypertarget{StableHomotopyTheorySurvey}{}\subsubsection*{{1) Stable homotopy theory}}\label{StableHomotopyTheorySurvey} A [[group]] in homotopy theory is equivalently a [[loop space]] under concatenation of loops (``[[∞-group]]''). A double loop space is a group with some commutativity structure (``[[Eckmann-Hilton argument]]''), a triple loop space has more commutativity structure, and so forth. A \emph{spectrum} is where this progression of [[looping and delooping]] \emph{stabilizes} (an ``$\infty$-abelian group''). Therefore one speaks of \emph{[[stable homotopy theory]]}: \begin{displaymath} Spaces \;\; \underoverset{(linearization)}{stabilization}{\mapsto} \;\; Spectra \,. \end{displaymath} Most of [[linear algebra]] and [[algebraic geometry]] passes along as [[abelian groups]] are generalized to [[spectra]] and turns into something remarkably rich, called \emph{[[brave new algebra]]}, \emph{[[higher algebra]]} and \emph{[[E-∞ geometry|spectral geometry]]}. In particular the analog of the theory of ([[commutative ring|commutative]]) [[rings]] and their [[modules]] exists, given by ([[commutative ring spectrum|commutative]]) [[ring spectra]] ([[E-∞ rings]], [[A-∞ rings]]) and [[module spectra]] ([[∞-modules]]). \hypertarget{ANSS}{}\subsubsection*{{2) Adams spectral sequences}}\label{ANSS} Since spectra are considerably richer than abelian groups, stable homotopy is much concerned with ``[[fracture theorem|fracturing]]'' stable homotopy types into more tractable components: To that end, notice that from the point of view of [[arithmetic geometry]], an [[abelian group]] $A$ is equivalently a [[quasicoherent sheaf]] over [[Spec(Z)]]. \begin{displaymath} AbelianGroups \simeq QCoh(Spec(\mathbb{Z})) \,. \end{displaymath} This point of view generalizes to homotopy theory and turns out to be very fruitful there. The analog of the [[integers]] $\mathbb{Z}$ is the [[sphere spectrum]] $\mathbb{S}$, and this is naturally the [[initial object in an (infinity,1)-category|initial]] [[commutative ring spectrum]] (``[[E-∞ ring]]''), just as $\mathbb{Z}$ is the [[initial object|initial]] [[commutative ring]]. The [[formal dual]] [[Spec(S)]] of $\mathbb{S}$ is hence the [[terminal object in an (infinity,1)-category|terminal]] [[space]] in [[E-∞ arithmetic geometry]] (``[[spectral geometry]]'') and [[spectra]] are equivalently the [[quasicoherent ∞-stacks]] over $Spec(\mathbb{S})$ \begin{displaymath} Spectra \simeq QCoh(Spec(\mathbb{S})) \,. \end{displaymath} Therefore the study of spectra ``[[fracture theorem|fractures]]'' into the various [[localizations]] and [[formal completions]] of $Spec(\mathbb{S})$. Since this is like the white light of $Spec(\mathbb{S})$ decomposing into various wavelengths, one speaks of \emph{[[chromatic homotopy theory]]}. In particular, an [[E-∞ ring]] $E$ is [[formal dual|dually]] a morphism of $E_\infty$-algebraic spaces $Spec(E) \longrightarrow Spec(\mathbb{S})$ and under good conditions the [[1-image]] of this map is the formal dual of the [[Bousfield localization of spectra|localization]] $L_E \mathbb{S}$ at $E$: \begin{displaymath} Spec(E) \stackrel{epi_1}{\longrightarrow} Spec(L_E \mathbb{S}) \stackrel{mono_1}{\longrightarrow} Spec(\mathbb{S}) \,. \end{displaymath} This means that $Spec(E) \longrightarrow Spec(L_E \mathbb{S})$ is a [[cover]] and that hence $E$-local spectra are equivalently [[quasicoherent ∞-stacks]] on $Spec(E)$ equipped with [[descent data]]: [[formal dual|dually]] they are [[∞-modules]] over $E$ equipped with [[comodule]] structure over the [[Hopf algebroid]] ([[Sweedler coring]]) $E \otimes_{\mathbb{S}} E$. The computation of [[homotopy groups]] of spectra that make use of their decomposition this way into $E$-[[∞-modules]] equipped with [[descent]] data is the \emph{$E$-[[Adams spectral sequence]]}, a central tool of the theory. \hypertarget{s_complex_oriented_cohomology}{}\subsubsection*{{S) Complex oriented cohomology}}\label{s_complex_oriented_cohomology} For this reason special importance is carried by those [[E-∞ rings]] such that $Spec(E) \to Spec(\mathbb{S})$ is already a [[covering]], in a suitable sense, for these the $E$-[[∞-modules]] equipped with descent data give an equivalent, but in general more tractable, incarnation of the stable homotopy theory of spectra. Curiously, this way a good bit of [[differential topology]] -- [[cobordism theory]] -- arises within stable homotopy theory: the archetypical $Spec(E)$ which covers $Spec(\mathbb{S})$ in a suitable sense is $E =$ [[MU]], the [[Thom spectrum]] representing [[complex cobordism cohomology]]. An [[commutative ring spectrum]] $E$ over $MU$, hence a $Spec(E)\to Spec(MU)$ is now a [[multiplicative cohomology theory|multiplicative]] ``[[complex oriented cohomology theory]]''. $\,$ \vspace{.5em} \hrule \vspace{.5em} $\,$ \hypertarget{ClassicalHomotopyTheory}{}\subsection*{{\textbf{Prelude) Classical homotopy theory}}}\label{ClassicalHomotopyTheory} $\,$ This section is at: \emph{[[Introduction to Stable homotopy theory -- P]]} $\,$ $\,$ \hypertarget{StableHomotopyTheory}{}\subsection*{{\textbf{Part 1) Stable homotopy theory}}}\label{StableHomotopyTheory} $\,$ This section is at \emph{[[Introduction to Stable homotopy theory -- 1]]} $\,$ $\,$ \hypertarget{SpectralSequences}{}\subsection*{{\textbf{Interlude) Spectral sequences}}}\label{SpectralSequences} $\,$ This section is at \emph{[[Introduction to Stable homotopy theory -- I]]} $\,$ $\,$ \hypertarget{part_2_adams_spectral_sequences}{}\subsection*{{\textbf{Part 2) Adams spectral sequences}}}\label{part_2_adams_spectral_sequences} $\,$ This section is at \emph{[[Introduction to Stable homotopy theory -- 2]]} $\,$ $\,$ \hypertarget{ComplexOrientedCohomology}{}\subsection*{{\textbf{Seminar) Complex oriented cohomology}}}\label{ComplexOrientedCohomology} $\,$ This section is at \emph{[[Introduction to Stable homotopy theory -- S]]} $\,$ $\,$ \vspace{.5em} \hrule \vspace{.5em} $\,$ \hypertarget{References}{}\subsection*{{References}}\label{References} \hypertarget{basic_reading}{}\subsubsection*{{Basic reading}}\label{basic_reading} For \textbf{Prelude) Classical homotopy theory} a concise and self-contained re-write of the proof (\hyperlink{Quillen67}{Quillen 67}) of the [[classical model structure on topological spaces]] is in \begin{itemize}% \item [[Philip Hirschhorn]], \emph{The Quillen model category of topological spaces} (\href{http://arxiv.org/abs/1508.01942}{arXiv:1508.01942}). \end{itemize} For general [[model category]] theory a decent concise account is in \begin{itemize}% \item [[William Dwyer]], J. Spalinski, \emph{[[Homotopy theories and model categories]]} (\href{http://folk.uio.no/paularne/SUPh05/DS.pdf}{pdf}) in [[Ioan Mackenzie James]] (ed.), \emph{[[Handbook of Algebraic Topology]]} 1995 \end{itemize} For the restriction to the [[convenient category of topological spaces|convenient category]] of [[compactly generated topological spaces]] good sources are \begin{itemize}% \item [[Gaunce Lewis]], \emph{Compactly generated spaces} (\href{http://www.math.uchicago.edu/~may/MISC/GaunceApp.pdf}{pdf}), appendix A of \emph{The Stable Category and Generalized Thom Spectra} PhD thesis Chicago, 1978 \item [[Neil Strickland]], \emph{The category of CGWH spaces}, 2009 (\href{http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf}{pdf}) \end{itemize} For section \textbf{1) Stable homotopy theory} we follow the modern picture of the stable homotopy category for which an enjoyable survey may be found in \begin{itemize}% \item [[Cary Malkiewich]], \emph{The stable homotopy category}, 2014 (\href{http://math.uiuc.edu/~cmalkiew/stable.pdf}{pdf}). \end{itemize} The classical account in (\hyperlink{Adams74}{Adams 74, part III sections 2, 4-7}) is still a good read, but ignore the ``[[Adams category]]''-construction of the [[stable homotopy category]] in sections III.2 and III.3. What we actually do follows \begin{itemize}% \item [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], \emph{[[Model categories of diagram spectra]]}, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (\href{http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf}{pdf}) \end{itemize} For the discussion of [[ring spectra]] we pass to [[symmetric spectra]] and [[orthogonal spectra]]. A compendium on the former is in \begin{itemize}% \item [[Stefan Schwede]], \emph{[[Symmetric spectra]]}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf}) \end{itemize} For \textbf{Interlude: Spectral sequences} a discussion streamlined for our purposes is in (\hyperlink{Rognes12}{Rognes 12, section 2}). In \textbf{2) Adams spectral sequence} for the general theory we follow \begin{itemize}% \item [[Frank Adams]], \emph{[[Stable homotopy and generalized homology]]}, Chicago Lectures in mathematics, 1974 \item [[Aldridge Bousfield]], sections 5 and 6 of \emph{The localization of spectra with respect to homology}, Topology 18 (1979), no. 4, 257--281. (\href{http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf}{pdf}) \end{itemize} For the special case of the [[classical Adams spectral sequence]] we follow (\hyperlink{Kochman96}{Kochman 96, chapter V}). For the \textbf{Seminar on Complex oriented cohomology} an excellent textbook to hold on to is \begin{itemize}% \item [[Stanley Kochman]], \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996 \end{itemize} Specifically for \textbf{S.1) Generalized cohomology} a neat account is in: \begin{itemize}% \item Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 12 of \emph{Algebraic topology from a homotopical viewpoint}, Springer (2002) (\href{http://tocs.ulb.tu-darmstadt.de/106999419.pdf}{toc pdf}) \end{itemize} For \textbf{S.2) Cobordism theory} an efficient collection of the highlights is in \begin{itemize}% \item [[Cary Malkiewich]], \emph{Unoriented cobordism and $M O$}, 2011 (\href{http://math.uiuc.edu/~cmalkiew/cobordism.pdf}{pdf}) \end{itemize} except that it omits proof of the [[Leray-Hirsch theorem]]/[[Serre spectral sequence]] and that of the [[Thom isomorphism]], but see the references there and see (\hyperlink{Kochman96}{Kochman 96}, \hyperlink{AguilarGitlerPrieto02}{Aguilar-Gitler-Prieto 02, section 11.7}) for details. For \textbf{S.3) Complex oriented cohomology} besides (\hyperlink{Kochman96}{Kochman 96, chapter 4}) have a look at \hyperlink{Adams74}{Adams 74, part II} and \begin{itemize}% \item [[Jacob Lurie]], lectures 1-10 of \emph{[[Chromatic Homotopy Theory]]}, 2010 \end{itemize} (These overlap, pick the one that seems more inviting on first reading.) \hypertarget{further_reading}{}\subsubsection*{{Further reading}}\label{further_reading} The two originals \begin{itemize}% \item [[Daniel Quillen]], \emph{Axiomatic homotopy theory} in \emph{Homotopical algebra}, Lecture Notes in Mathematics, No. 43 43, Berlin (1967) \item [[Kenneth Brown]], \emph{[[Abstract Homotopy Theory and Generalized Sheaf Cohomology]]}, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458 (\href{http://www.jstor.org/stable/1996573}{JSTOR}) \end{itemize} are still an excellent source. For further reading on homotopy theory and stable homotopy theory a useful collection is \begin{itemize}% \item [[Ioan Mackenzie James]], \emph{[[Handbook of Algebraic Topology]]} 1995 \end{itemize} The modern chromatic picture originates around \begin{itemize}% \item [[Mike Hopkins]], \emph{[[Complex oriented cohomology theories and the language of stacks]]}, 1999 \end{itemize} a useful survey is in \begin{itemize}% \item [[Dylan Wilson]] section 1.2 of \emph{Spectral Sequences from Sequences of Spectra: Towards the Spectrum of the Category of Spectra} lecture at \emph{\href{http://math.harvard.edu/~hirolee/pretalbot2013/}{2013 Pre-Talbot Seminar}}, March 2013 ([[DylanWilsonOnANSS.pdf:file]]) \end{itemize} a wealth of details is in \begin{itemize}% \item [[Doug Ravenel]], \emph{[[Complex cobordism and stable homotopy groups of spheres]]}, 1987/2003 (\href{http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenelA1.pdf}{pdf}) \end{itemize} and new foundations have been laid in \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Higher Algebra]]} \end{itemize} [[!redirects Introduction to Stable homotopy theory]] [[!redirects geometry of physics -- stable homotopy types]] [[!redirects geometry of physics -- stable homotopy theory]] \end{document}