\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\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*{Complex cobordism and stable homotopy groups of spheres}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

This entry collected pointers related to the book

\begin{itemize}%
\item [[Doug Ravenel]]

\emph{Complex cobordism and stable homotopy groups of spheres}

1986/2003

(\href{http://www.math.rochester.edu/people/faculty/doug/mu.html}{web})



\end{itemize}
on [[stable homotopy theory]] in general and in particular the computation of the [[homotopy groups of spheres]] via the [[Adams-Novikov spectral sequence]] and its use of [[complex cobordism cohomology theory]].

\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. (preface to the first edition)


\end{quote}
See also

\begin{itemize}%
\item [[Frank Adams]], \emph{[[Stable homotopy and generalised homology]]}, 1974


\item [[Mike Hopkins]], \emph{[[Complex oriented cohomology theories and the language of stacks]]}, 1999


\item [[Jacob Lurie]], \emph{[[Chromatic Homotopy Theory]]}, 2010



\end{itemize}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{chapter_1_an_introduction_to_the_homotopy_groups_of_spheres}{Chapter 1. An introduction to the Homotopy Groups of Spheres}\dotfill \pageref*{chapter_1_an_introduction_to_the_homotopy_groups_of_spheres} \linebreak
\noindent\hyperlink{1_classical_theorems_old_and_new}{1. Classical theorems Old and New}\dotfill \pageref*{1_classical_theorems_old_and_new} \linebreak
\noindent\hyperlink{2_methods_of_computing_}{2. Methods of computing $\pi_\ast(S^n)$}\dotfill \pageref*{2_methods_of_computing_} \linebreak
\noindent\hyperlink{3_the_adamsnovikov_term_formal_group_laws_and_the_greek_letter_construction}{3. The Adams-Novikov $E_2$-term, Formal Group Laws, and the Greek Letter Construction}\dotfill \pageref*{3_the_adamsnovikov_term_formal_group_laws_and_the_greek_letter_construction} \linebreak
\noindent\hyperlink{4_more_formal_group_law_theory_moravas_point_of_view_and_the_chromatic_spectral_sequence}{4. More formal group law theory, Morava's point of view, and the Chromatic Spectral Sequence}\dotfill \pageref*{4_more_formal_group_law_theory_moravas_point_of_view_and_the_chromatic_spectral_sequence} \linebreak
\noindent\hyperlink{5_unstable_homotopy_groups_and_the_ehp_spectral_sequence}{5. Unstable homotopy groups and the EHP spectral sequence}\dotfill \pageref*{5_unstable_homotopy_groups_and_the_ehp_spectral_sequence} \linebreak
\noindent\hyperlink{chapter_2_setting_up_the_adams_spectral_sequence}{Chapter 2. Setting up the Adams Spectral sequence}\dotfill \pageref*{chapter_2_setting_up_the_adams_spectral_sequence} \linebreak
\noindent\hyperlink{1_the_classical_adams_spectral_sequence}{1. The classical Adams spectral sequence}\dotfill \pageref*{1_the_classical_adams_spectral_sequence} \linebreak
\noindent\hyperlink{2_the_adams_spectral_sequence_based_on_a_generalized_homology_theory}{2. The Adams spectral sequence based on a generalized homology theory}\dotfill \pageref*{2_the_adams_spectral_sequence_based_on_a_generalized_homology_theory} \linebreak
\noindent\hyperlink{3_the_smash_product_pairing_and_the_generalized_connecting_homomorphism}{3. The smash product pairing and the Generalized connecting homomorphism}\dotfill \pageref*{3_the_smash_product_pairing_and_the_generalized_connecting_homomorphism} \linebreak
\noindent\hyperlink{chapter_3_the_classical_adams_spectral_sequence}{Chapter 3. The Classical Adams Spectral Sequence}\dotfill \pageref*{chapter_3_the_classical_adams_spectral_sequence} \linebreak
\noindent\hyperlink{1_the_steenrod_algebra_and_some_easy_calculuation}{1. The Steenrod algebra and some easy calculuation}\dotfill \pageref*{1_the_steenrod_algebra_and_some_easy_calculuation} \linebreak
\noindent\hyperlink{2_the_may_spectral_sequence}{2. The May spectral sequence}\dotfill \pageref*{2_the_may_spectral_sequence} \linebreak
\noindent\hyperlink{3_the_lambda_algebra}{3. The Lambda Algebra}\dotfill \pageref*{3_the_lambda_algebra} \linebreak
\noindent\hyperlink{4_some_general_properties_of_}{4. Some general properties of $Ext$}\dotfill \pageref*{4_some_general_properties_of_} \linebreak
\noindent\hyperlink{5_survey_and_further_reading}{5. Survey and further reading}\dotfill \pageref*{5_survey_and_further_reading} \linebreak
\noindent\hyperlink{chapter_4_theory_and_the_adamsnovikov_spectral_sequence}{Chapter 4. $B P$-Theory and the Adams-Novikov Spectral Sequence}\dotfill \pageref*{chapter_4_theory_and_the_adamsnovikov_spectral_sequence} \linebreak
\noindent\hyperlink{1_quillens_theorem_and_the_structure_of_}{1. Quillen's theorem and the structure of $BP_\bullet(BP)$}\dotfill \pageref*{1_quillens_theorem_and_the_structure_of_} \linebreak
\noindent\hyperlink{2_a_survey_of_theory}{2. A survey of $B P$-theory}\dotfill \pageref*{2_a_survey_of_theory} \linebreak
\noindent\hyperlink{3_some_calculations_in_}{3. Some calculations in $B P_\bullet(B P)$}\dotfill \pageref*{3_some_calculations_in_} \linebreak
\noindent\hyperlink{4_beginning_calculations_with_the_adamsnovikov_spectral_sequence}{4. Beginning calculations with the Adams-Novikov Spectral Sequence}\dotfill \pageref*{4_beginning_calculations_with_the_adamsnovikov_spectral_sequence} \linebreak
\noindent\hyperlink{chapter_5_the_chromatic_spectral_sequence}{Chapter 5. The Chromatic Spectral Sequence}\dotfill \pageref*{chapter_5_the_chromatic_spectral_sequence} \linebreak
\noindent\hyperlink{1_the_algebraic_construction}{1. The algebraic construction}\dotfill \pageref*{1_the_algebraic_construction} \linebreak
\noindent\hyperlink{2__and_hopf_invariant_one}{2. $Ext^1(B P_\bullet/I_n)$ and Hopf Invariant One}\dotfill \pageref*{2__and_hopf_invariant_one} \linebreak
\noindent\hyperlink{3__and_the_homomorphism}{3. $Ext(M^1)$ and the $J$-Homomorphism}\dotfill \pageref*{3__and_the_homomorphism} \linebreak
\noindent\hyperlink{4__and_the_thom_reduction}{4. $Ext^2$ and the Thom Reduction}\dotfill \pageref*{4__and_the_thom_reduction} \linebreak
\noindent\hyperlink{5_periodic_families_in_}{5. Periodic families in $Ext^2$}\dotfill \pageref*{5_periodic_families_in_} \linebreak
\noindent\hyperlink{6_elements_in__and_beyond}{6. Elements in $Ext^3$ and Beyond}\dotfill \pageref*{6_elements_in__and_beyond} \linebreak
\noindent\hyperlink{chapter_6_morava_stabilizer_algebras}{Chapter 6. Morava Stabilizer Algebras}\dotfill \pageref*{chapter_6_morava_stabilizer_algebras} \linebreak
\noindent\hyperlink{chapter_7_computing_stable_homotopy_groups_with_the_adamsnovikov_spectral_sequence}{Chapter 7. Computing Stable Homotopy Groups with the Adams-Novikov Spectral Sequence}\dotfill \pageref*{chapter_7_computing_stable_homotopy_groups_with_the_adamsnovikov_spectral_sequence} \linebreak
\noindent\hyperlink{appendix_1_hopf_algebras_and_hopf_algebroids}{Appendix 1. Hopf Algebras and Hopf Algebroids}\dotfill \pageref*{appendix_1_hopf_algebras_and_hopf_algebroids} \linebreak
\noindent\hyperlink{1_basic_definitions}{1. Basic definitions}\dotfill \pageref*{1_basic_definitions} \linebreak
\noindent\hyperlink{2_homological_algebra}{2. Homological algebra}\dotfill \pageref*{2_homological_algebra} \linebreak
\noindent\hyperlink{3_some_spectral_sequences}{3. Some spectral sequences}\dotfill \pageref*{3_some_spectral_sequences} \linebreak
\noindent\hyperlink{4_massey_products}{4. Massey products}\dotfill \pageref*{4_massey_products} \linebreak
\noindent\hyperlink{5_algebraic_steenrod_operations}{5. Algebraic Steenrod operations}\dotfill \pageref*{5_algebraic_steenrod_operations} \linebreak
\noindent\hyperlink{appendix_2_formal_group_laws}{Appendix 2. Formal Group Laws}\dotfill \pageref*{appendix_2_formal_group_laws} \linebreak
\noindent\hyperlink{appendix_3_table_of_homotopy_groups_of_spheres}{Appendix 3. Table of homotopy groups of spheres}\dotfill \pageref*{appendix_3_table_of_homotopy_groups_of_spheres} \linebreak


\hypertarget{chapter_1_an_introduction_to_the_homotopy_groups_of_spheres}{}\subsection*{{Chapter 1. An introduction to the Homotopy Groups of Spheres}}\label{chapter_1_an_introduction_to_the_homotopy_groups_of_spheres}

\hypertarget{1_classical_theorems_old_and_new}{}\subsubsection*{{1. Classical theorems Old and New}}\label{1_classical_theorems_old_and_new}

\begin{itemize}%
\item [[homotopy group]]


\item [[Hurewicz theorem]]


\item [[suspension]], [[reduced suspension]]


\item [[Freudenthal suspension theorem]]


\item [[stable homotopy groups of spheres]]


\item [[Hopf fibration]]


\item [[Serre finiteness theorem]]


\item [[Nishida nilpotence theorem]]


\item [[Bott periodicity theorem]]


\item [[J-homomorphism]]


\item [[Adams conjecture]]



\end{itemize}
\hypertarget{2_methods_of_computing_}{}\subsubsection*{{2. Methods of computing $\pi_\ast(S^n)$}}\label{2_methods_of_computing_}

\begin{itemize}%
\item [[Postnikov tower]]


\item [[Serre spectral sequence]]


\item [[Adams spectral sequence]]


\item [[Steenrod algebra]]


\item [[parallelizable sphere]]


\item [[Hopf invariant one theorem]]


\item [[complex cobordism cohomology]]


\item [[cobordism ring]]


\item [[Adams-Novikov spectral sequence]]



\end{itemize}
\hypertarget{3_the_adamsnovikov_term_formal_group_laws_and_the_greek_letter_construction}{}\subsubsection*{{3. The Adams-Novikov $E_2$-term, Formal Group Laws, and the Greek Letter Construction}}\label{3_the_adamsnovikov_term_formal_group_laws_and_the_greek_letter_construction}

\begin{itemize}%
\item [[formal group law]]


\item [[Lazard ring]]


\item [[Greek letter construction]]



\end{itemize}
\hypertarget{4_more_formal_group_law_theory_moravas_point_of_view_and_the_chromatic_spectral_sequence}{}\subsubsection*{{4. More formal group law theory, Morava's point of view, and the Chromatic Spectral Sequence}}\label{4_more_formal_group_law_theory_moravas_point_of_view_and_the_chromatic_spectral_sequence}

\begin{itemize}%
\item [[Brown-Peterson cohomology]]


\item [[p-typical formal group law]]


\item [[height of a formal group]]


\item [[chromatic spectral sequence]]



\end{itemize}
\hypertarget{5_unstable_homotopy_groups_and_the_ehp_spectral_sequence}{}\subsubsection*{{5. Unstable homotopy groups and the EHP spectral sequence}}\label{5_unstable_homotopy_groups_and_the_ehp_spectral_sequence}

\begin{itemize}%
\item [[EHP spectral sequence]]

\end{itemize}
\hypertarget{chapter_2_setting_up_the_adams_spectral_sequence}{}\subsection*{{Chapter 2. Setting up the Adams Spectral sequence}}\label{chapter_2_setting_up_the_adams_spectral_sequence}

\hypertarget{1_the_classical_adams_spectral_sequence}{}\subsubsection*{{1. The classical Adams spectral sequence}}\label{1_the_classical_adams_spectral_sequence}

\begin{itemize}%
\item [[exact couple]]


\item [[spectral sequence]]


\item [[Adams resolution]]


\item [[Adams spectral sequence]]


\item [[Eilenberg-MacLane spectrum]], [[ordinary cohomology]]


\item [[inverse limit]]


\item [[p-completion]]



\end{itemize}
\hypertarget{2_the_adams_spectral_sequence_based_on_a_generalized_homology_theory}{}\subsubsection*{{2. The Adams spectral sequence based on a generalized homology theory}}\label{2_the_adams_spectral_sequence_based_on_a_generalized_homology_theory}

\begin{itemize}%
\item [[commutative Hopf algebroid]] structure on [[Steenrod algebra]]

\end{itemize}
\hypertarget{3_the_smash_product_pairing_and_the_generalized_connecting_homomorphism}{}\subsubsection*{{3. The smash product pairing and the Generalized connecting homomorphism}}\label{3_the_smash_product_pairing_and_the_generalized_connecting_homomorphism}

\begin{itemize}%
\item [[connecting homomorphism]]

\end{itemize}
\hypertarget{chapter_3_the_classical_adams_spectral_sequence}{}\subsection*{{Chapter 3. The Classical Adams Spectral Sequence}}\label{chapter_3_the_classical_adams_spectral_sequence}

\begin{itemize}%
\item [[classical Adams spectral sequence]]

\end{itemize}
\hypertarget{1_the_steenrod_algebra_and_some_easy_calculuation}{}\subsubsection*{{1. The Steenrod algebra and some easy calculuation}}\label{1_the_steenrod_algebra_and_some_easy_calculuation}

\begin{itemize}%
\item [[Hopf algebra]] structure on [[Steenrod algebra]]


\item [[cobar complex]], [[cosimplicial object]]



\end{itemize}
\hypertarget{2_the_may_spectral_sequence}{}\subsubsection*{{2. The May spectral sequence}}\label{2_the_may_spectral_sequence}

\begin{itemize}%
\item [[May spectral sequence]]


\item [[Massey product]]



\end{itemize}
\hypertarget{3_the_lambda_algebra}{}\subsubsection*{{3. The Lambda Algebra}}\label{3_the_lambda_algebra}

\begin{itemize}%
\item [[Lambda algebra]]


\item [[EHP spectral sequence]]



\end{itemize}
\hypertarget{4_some_general_properties_of_}{}\subsubsection*{{4. Some general properties of $Ext$}}\label{4_some_general_properties_of_}

\begin{itemize}%
\item [[periodicity theorem]]

\end{itemize}
\hypertarget{5_survey_and_further_reading}{}\subsubsection*{{5. Survey and further reading}}\label{5_survey_and_further_reading}

\hypertarget{chapter_4_theory_and_the_adamsnovikov_spectral_sequence}{}\subsection*{{Chapter 4. $B P$-Theory and the Adams-Novikov Spectral Sequence}}\label{chapter_4_theory_and_the_adamsnovikov_spectral_sequence}

\begin{itemize}%
\item [[Adams-Novikov spectral sequence]]

\end{itemize}
\hypertarget{1_quillens_theorem_and_the_structure_of_}{}\subsubsection*{{1. Quillen's theorem and the structure of $BP_\bullet(BP)$}}\label{1_quillens_theorem_and_the_structure_of_}

\begin{itemize}%
\item [[Quillen's theorem on MU]]


\item [[Landweber-Novikov theorem]]


\item [[Brown-Peterson spectrum]]


\item [[p-typical formal group]]


\item \href{cobordism+cohomology+theory#UniversalComplexOrientation}{universal complex orientation on MU}, [[complex orientation and MU]]


\item [[Adams-Quillen theorem]]



\end{itemize}
\hypertarget{2_a_survey_of_theory}{}\subsubsection*{{2. A survey of $B P$-theory}}\label{2_a_survey_of_theory}

\begin{itemize}%
\item [[cobordism group]]


\item [[Thom spectrum]]



\end{itemize}
\hypertarget{3_some_calculations_in_}{}\subsubsection*{{3. Some calculations in $B P_\bullet(B P)$}}\label{3_some_calculations_in_}

\hypertarget{4_beginning_calculations_with_the_adamsnovikov_spectral_sequence}{}\subsubsection*{{4. Beginning calculations with the Adams-Novikov Spectral Sequence}}\label{4_beginning_calculations_with_the_adamsnovikov_spectral_sequence}

\begin{itemize}%
\item [[BP]]-[[Adams-Novikov spectral sequence]]

\end{itemize}
\hypertarget{chapter_5_the_chromatic_spectral_sequence}{}\subsection*{{Chapter 5. The Chromatic Spectral Sequence}}\label{chapter_5_the_chromatic_spectral_sequence}

\hypertarget{1_the_algebraic_construction}{}\subsubsection*{{1. The algebraic construction}}\label{1_the_algebraic_construction}

\begin{itemize}%
\item [[chromatic spectral sequence]]

\end{itemize}
\hypertarget{2__and_hopf_invariant_one}{}\subsubsection*{{2. $Ext^1(B P_\bullet/I_n)$ and Hopf Invariant One}}\label{2__and_hopf_invariant_one}

\begin{itemize}%
\item [[Hopf invariant one]]

\end{itemize}
\hypertarget{3__and_the_homomorphism}{}\subsubsection*{{3. $Ext(M^1)$ and the $J$-Homomorphism}}\label{3__and_the_homomorphism}

\begin{itemize}%
\item [[J-homomorphism]]

\end{itemize}
\hypertarget{4__and_the_thom_reduction}{}\subsubsection*{{4. $Ext^2$ and the Thom Reduction}}\label{4__and_the_thom_reduction}

\hypertarget{5_periodic_families_in_}{}\subsubsection*{{5. Periodic families in $Ext^2$}}\label{5_periodic_families_in_}

\hypertarget{6_elements_in__and_beyond}{}\subsubsection*{{6. Elements in $Ext^3$ and Beyond}}\label{6_elements_in__and_beyond}

\hypertarget{chapter_6_morava_stabilizer_algebras}{}\subsection*{{Chapter 6. Morava Stabilizer Algebras}}\label{chapter_6_morava_stabilizer_algebras}

\begin{itemize}%
\item [[Morava stabilizer group]]

\end{itemize}
\hypertarget{chapter_7_computing_stable_homotopy_groups_with_the_adamsnovikov_spectral_sequence}{}\subsection*{{Chapter 7. Computing Stable Homotopy Groups with the Adams-Novikov Spectral Sequence}}\label{chapter_7_computing_stable_homotopy_groups_with_the_adamsnovikov_spectral_sequence}

\hypertarget{appendix_1_hopf_algebras_and_hopf_algebroids}{}\subsection*{{Appendix 1. Hopf Algebras and Hopf Algebroids}}\label{appendix_1_hopf_algebras_and_hopf_algebroids}

\hypertarget{1_basic_definitions}{}\subsubsection*{{1. Basic definitions}}\label{1_basic_definitions}

\begin{itemize}%
\item [[Hopf algebra]]


\item [[commutative Hopf algebroid]]


\item [[module]], [[comodule]]


\item [[cotensor product]]


\item [[change of rings theorem]]



\end{itemize}
\hypertarget{2_homological_algebra}{}\subsubsection*{{2. Homological algebra}}\label{2_homological_algebra}

\begin{itemize}%
\item [[derived functor]]


\item [[Ext]] and [[Cotor]] for [[comodules]]



\end{itemize}
\hypertarget{3_some_spectral_sequences}{}\subsubsection*{{3. Some spectral sequences}}\label{3_some_spectral_sequences}

\begin{itemize}%
\item [[spectral sequence]]


\item [[Cartan-Eilenberg spectral sequence]]



\end{itemize}
\hypertarget{4_massey_products}{}\subsubsection*{{4. Massey products}}\label{4_massey_products}

\begin{itemize}%
\item [[Massey product]]

\end{itemize}
\hypertarget{5_algebraic_steenrod_operations}{}\subsubsection*{{5. Algebraic Steenrod operations}}\label{5_algebraic_steenrod_operations}

\begin{itemize}%
\item [[Steenrod algebra]] on [[Cotor]]-groups of [[comodules]] over [[commutative Hopf algebroids]]

\end{itemize}
\hypertarget{appendix_2_formal_group_laws}{}\subsection*{{Appendix 2. Formal Group Laws}}\label{appendix_2_formal_group_laws}

\begin{itemize}%
\item [[formal group law]]

\end{itemize}
\hypertarget{appendix_3_table_of_homotopy_groups_of_spheres}{}\subsection*{{Appendix 3. Table of homotopy groups of spheres}}\label{appendix_3_table_of_homotopy_groups_of_spheres}

category: reference



\end{document}