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\section*{Chromatic Homotopy Theory}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

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

[[!include stable homotopy theory - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

This page collects links related to the lecture notes

\begin{itemize}%
\item [[Jacob Lurie]],

\emph{Chromatic Homotopy Theory}, Lecture series 2010

(\href{http://www.math.harvard.edu/~lurie/252x.html}{web})



\end{itemize}
on [[complex oriented cohomology]], the [[Adams spectral sequence]] and [[chromatic homotopy theory]] from the modern point of view of [[E-infinity geometry]].

Based on the program of

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

\end{itemize}
as nicely laid out in more detail in

\begin{itemize}%
\item [[Stanley Kochmann]], \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996

\end{itemize}
First indications of the big picture developed here are due to

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

\end{itemize}
Via the chromatic stratification of the [[moduli stack of formal groups]] one recovers as, roughly, the second chromatic stage the [[moduli stack of elliptic curves]]. For the story in that case see

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[A Survey of Elliptic Cohomology]]}.

\end{itemize}
See also

\begin{itemize}%
\item [[Doug Ravenel]], \emph{[[Complex cobordism and stable homotopy groups of spheres]]}, 1986/2003

\end{itemize}
\textbf{Lectures}

\begin{itemize}%
\item Lecture 1 \emph{Introduction} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture1.pdf}{pdf})


\item Lecture 2 \emph{[[Lazard's theorem]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture2.pdf}{pdf})


\item Lecture 3 \emph{Lazard's theorem (continued)} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture3.pdf}{pdf})


\item Lecture 4 \emph{[[complex oriented cohomology theory|Complex-oriented cohomology theories]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf}{pdf})


\item Lecture 5 \emph{[[complex bordism|Complex bordism]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture5.pdf}{pdf})


\item Lecture 6 \emph{[[MU and complex orientations]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture6.pdf}{pdf})


\item Lecture 7 \emph{[[homology of MU|The homology of MU]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture7.pdf}{pdf})


\item Lecture 8 \emph{The [[Adams spectral sequence]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture8.pdf}{pdf})


\item Lecture 9 \emph{The Adams spectral sequence for MU} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture9.pdf}{pdf})


\item Lecture 10 \emph{The proof of [[Quillen's theorem]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture10.pdf}{pdf})


\item Lecture 11 \emph{[[formal groups|Formal groups]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf}{pdf})


\item Lecture 12 \emph{[[height of a formal group|Heights and formal groups]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture12.pdf}{pdf})


\item Lecture 13 \emph{The stratification of $\mathcal{M}_{FG}$} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture13.pdf}{pdf})


\item Lecture 14 \emph{Classification of formal groups} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture14.pdf}{pdf})


\item Lecture 15 \emph{Flat modules over $\mathcal{M}_{FG}$} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture15.pdf}{pdf})


\item Lecture 16 \emph{The [[Landweber exact functor theorem]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture16.pdf}{pdf})


\item Lecture 17 \emph{[[phantom maps|Phantom maps]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture17.pdf}{pdf})


\item Lecture 18 \emph{Even periodic cohomology theories} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture18.pdf}{pdf})


\item Lecture 19 \emph{[[Morava stabilizer groups]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture19.pdf}{pdf})


\item Lecture 20 \emph{[[Bousfield localization of spectra|Bousfield localization]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture20.pdf}{pdf})


\item Lecture 21 \emph{Lubin-Tate theory} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture21.pdf}{pdf})


\item Lecture 22 \emph{Morava E-theory and Morava K-theory} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf}{pdf})


\item Lecture 23 \emph{The Bousfield Classes of $E(n)$ and $K(n)$} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture23.pdf}{pdf})


\item Lecture 24 \emph{Uniqueness of Morava K-theory} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture24.pdf}{pdf})


\item Lecture 25 \emph{The Nilpotence lemma} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture25.pdf}{pdf})


\item Lecture 26 \emph{Thick subcategories} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture26.pdf}{pdf})


\item Lecture 27 \emph{The periodicity theorem} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture27.pdf}{pdf})


\item Lecture 28 \emph{Telescopic localization} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture28.pdf}{pdf})


\item Lecture 29 \emph{Telescopic vs $E_n$-localization} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture29.pdf}{pdf})


\item Lecture 30 \emph{Localizations and the Adams-Novikov spectral sequence} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture30.pdf}{pdf})


\item Lecture 31 \emph{The smash product theorem} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture31.pdf}{pdf})


\item Lecture 32 \emph{The chromatic convergence theorem} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture32.pdf}{pdf})


\item Lecture 33 \emph{Complex bordism and $E(n)$-localization} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture33.pdf}{pdf})


\item Lecture 34 \emph{Monochromatic layers} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture34.pdf}{pdf})


\item Lecture 35 \emph{The image of $J$} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture35.pdf}{pdf})



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

\noindent\hyperlink{lecture_2_lazards_theorem}{Lecture 2 \emph{Lazard's theorem}}\dotfill \pageref*{lecture_2_lazards_theorem} \linebreak
\noindent\hyperlink{lecture_3_lazards_theorem_continued}{Lecture 3 \emph{Lazard's theorem (continued)}}\dotfill \pageref*{lecture_3_lazards_theorem_continued} \linebreak
\noindent\hyperlink{lecture_4_complexoriented_cohomology_theories}{Lecture 4 \emph{Complex-oriented cohomology theories}}\dotfill \pageref*{lecture_4_complexoriented_cohomology_theories} \linebreak
\noindent\hyperlink{lecture_5_complex_bordism}{Lecture 5 \emph{Complex bordism}}\dotfill \pageref*{lecture_5_complex_bordism} \linebreak
\noindent\hyperlink{lecture_6_mu_and_complex_orientations}{Lecture 6 \emph{MU and complex orientations}}\dotfill \pageref*{lecture_6_mu_and_complex_orientations} \linebreak
\noindent\hyperlink{lecture_7_the_homology_of_mu}{Lecture 7 \emph{The homology of MU}}\dotfill \pageref*{lecture_7_the_homology_of_mu} \linebreak
\noindent\hyperlink{lecture_8_the_adams_spectral_sequence}{Lecture 8 \emph{The Adams spectral sequence}}\dotfill \pageref*{lecture_8_the_adams_spectral_sequence} \linebreak
\noindent\hyperlink{lecture_9_the_adams_spectral_sequence_for_mu}{Lecture 9 \emph{The Adams spectral sequence for MU}}\dotfill \pageref*{lecture_9_the_adams_spectral_sequence_for_mu} \linebreak
\noindent\hyperlink{lecture_10_the_proof_of_quillens_theorem}{Lecture 10 \emph{The proof of Quillen's theorem}}\dotfill \pageref*{lecture_10_the_proof_of_quillens_theorem} \linebreak
\noindent\hyperlink{lecture_11_formal_groups}{Lecture 11 \emph{Formal groups}}\dotfill \pageref*{lecture_11_formal_groups} \linebreak
\noindent\hyperlink{lecture_12_heights_and_formal_groups}{Lecture 12 \emph{Heights and formal groups}}\dotfill \pageref*{lecture_12_heights_and_formal_groups} \linebreak
\noindent\hyperlink{lecture_13_the_stratification_of_}{Lecture 13 \emph{The stratification of $\mathcal{M}_{FG}$}}\dotfill \pageref*{lecture_13_the_stratification_of_} \linebreak
\noindent\hyperlink{lecture_14_classification_of_formal_groups}{Lecture 14 \emph{Classification of formal groups}}\dotfill \pageref*{lecture_14_classification_of_formal_groups} \linebreak
\noindent\hyperlink{lecture_15_flat_modules_over_}{Lecture 15 \emph{Flat modules over $\mathcal{M}_{FG}$}}\dotfill \pageref*{lecture_15_flat_modules_over_} \linebreak
\noindent\hyperlink{lecture_16_the_landweber_exact_functor_theorem}{Lecture 16 \emph{The Landweber exact functor theorem}}\dotfill \pageref*{lecture_16_the_landweber_exact_functor_theorem} \linebreak
\noindent\hyperlink{lecture_17_phantom_maps}{Lecture 17 \emph{Phantom maps}}\dotfill \pageref*{lecture_17_phantom_maps} \linebreak
\noindent\hyperlink{lecture_18_even_periodic_cohomology_theories}{Lecture 18 \emph{Even periodic cohomology theories}}\dotfill \pageref*{lecture_18_even_periodic_cohomology_theories} \linebreak
\noindent\hyperlink{lecture_19_morava_stabilizer_groups}{Lecture 19 \emph{Morava stabilizer groups}}\dotfill \pageref*{lecture_19_morava_stabilizer_groups} \linebreak
\noindent\hyperlink{lecture_20_bousfield_localization}{Lecture 20 \emph{Bousfield localization}}\dotfill \pageref*{lecture_20_bousfield_localization} \linebreak
\noindent\hyperlink{lecture_21_lubintate_theory}{Lecture 21 \emph{Lubin-Tate theory}}\dotfill \pageref*{lecture_21_lubintate_theory} \linebreak
\noindent\hyperlink{lecture_22_morava_etheory_and_morava_ktheory}{Lecture 22 \emph{Morava E-theory and Morava K-theory}}\dotfill \pageref*{lecture_22_morava_etheory_and_morava_ktheory} \linebreak
\noindent\hyperlink{lecture_23_the_bousfield_classes_of__and_}{Lecture 23 \emph{The Bousfield Classes of $E(n)$ and $K(n)$}}\dotfill \pageref*{lecture_23_the_bousfield_classes_of__and_} \linebreak
\noindent\hyperlink{lecture_24_uniqueness_of_morava_ktheory}{Lecture 24 \emph{Uniqueness of Morava K-theory}}\dotfill \pageref*{lecture_24_uniqueness_of_morava_ktheory} \linebreak
\noindent\hyperlink{lecture_25_the_nilpotence_lemma}{Lecture 25 \emph{The Nilpotence lemma}}\dotfill \pageref*{lecture_25_the_nilpotence_lemma} \linebreak
\noindent\hyperlink{lecture_26_thick_subcategories}{Lecture 26 \emph{Thick subcategories}}\dotfill \pageref*{lecture_26_thick_subcategories} \linebreak
\noindent\hyperlink{lecture_27_the_periodicity_theorem}{Lecture 27 \emph{The periodicity theorem}}\dotfill \pageref*{lecture_27_the_periodicity_theorem} \linebreak
\noindent\hyperlink{lecture_28_telescopic_localization}{Lecture 28 \emph{Telescopic localization}}\dotfill \pageref*{lecture_28_telescopic_localization} \linebreak
\noindent\hyperlink{lecture_29_telescopic_vs_localization}{Lecture 29 \emph{Telescopic vs $E_n$-localization}}\dotfill \pageref*{lecture_29_telescopic_vs_localization} \linebreak
\noindent\hyperlink{lecture_30_localizations_and_the_adamsnovikov_spectral_sequence}{Lecture 30 \emph{Localizations and the Adams-Novikov spectral sequence}}\dotfill \pageref*{lecture_30_localizations_and_the_adamsnovikov_spectral_sequence} \linebreak
\noindent\hyperlink{lecture_31_the_smash_product_theorem}{Lecture 31 \emph{The smash product theorem}}\dotfill \pageref*{lecture_31_the_smash_product_theorem} \linebreak
\noindent\hyperlink{lecture_32_the_chromatic_convergence_theorem}{Lecture 32 \emph{The chromatic convergence theorem}}\dotfill \pageref*{lecture_32_the_chromatic_convergence_theorem} \linebreak
\noindent\hyperlink{lecture_33_complex_bordism_and_localization}{Lecture 33 \emph{Complex bordism and $E(n)$-localization}}\dotfill \pageref*{lecture_33_complex_bordism_and_localization} \linebreak
\noindent\hyperlink{lecture_34_monochromatic_layers}{Lecture 34 \emph{Monochromatic layers}}\dotfill \pageref*{lecture_34_monochromatic_layers} \linebreak
\noindent\hyperlink{lecture_35_the_image_of_}{Lecture 35 \emph{The image of $J$}}\dotfill \pageref*{lecture_35_the_image_of_} \linebreak


\hypertarget{lecture_2_lazards_theorem}{}\subsection*{{Lecture 2 \emph{Lazard's theorem}}}\label{lecture_2_lazards_theorem}

\begin{itemize}%
\item [[Lazard ring]]


\item [[Lazard's theorem]]



\end{itemize}
\hypertarget{lecture_3_lazards_theorem_continued}{}\subsection*{{Lecture 3 \emph{Lazard's theorem (continued)}}}\label{lecture_3_lazards_theorem_continued}

\begin{itemize}%
\item [[Lazard's theorem]]

\end{itemize}
\hypertarget{lecture_4_complexoriented_cohomology_theories}{}\subsection*{{Lecture 4 \emph{Complex-oriented cohomology theories}}}\label{lecture_4_complexoriented_cohomology_theories}

\begin{itemize}%
\item [[complex oriented cohomology theory]]


\item [[splitting principle]]


\item [[generalized Chern classes]]



\end{itemize}
\hypertarget{lecture_5_complex_bordism}{}\subsection*{{Lecture 5 \emph{Complex bordism}}}\label{lecture_5_complex_bordism}

\begin{itemize}%
\item [[complex cobordism cohomology theory]], [[MU]]

\end{itemize}
\hypertarget{lecture_6_mu_and_complex_orientations}{}\subsection*{{Lecture 6 \emph{MU and complex orientations}}}\label{lecture_6_mu_and_complex_orientations}

\begin{itemize}%
\item [[universal complex orientation in MU]]

\end{itemize}
\hypertarget{lecture_7_the_homology_of_mu}{}\subsection*{{Lecture 7 \emph{The homology of MU}}}\label{lecture_7_the_homology_of_mu}

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


\item [[homology of MU]]



\end{itemize}
\hypertarget{lecture_8_the_adams_spectral_sequence}{}\subsection*{{Lecture 8 \emph{The Adams spectral sequence}}}\label{lecture_8_the_adams_spectral_sequence}

\begin{itemize}%
\item [[Adams resolution]]


\item [[Adams spectral sequence]]



\end{itemize}
\hypertarget{lecture_9_the_adams_spectral_sequence_for_mu}{}\subsection*{{Lecture 9 \emph{The Adams spectral sequence for MU}}}\label{lecture_9_the_adams_spectral_sequence_for_mu}

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

\end{itemize}
\hypertarget{lecture_10_the_proof_of_quillens_theorem}{}\subsection*{{Lecture 10 \emph{The proof of Quillen's theorem}}}\label{lecture_10_the_proof_of_quillens_theorem}

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

\end{itemize}
\hypertarget{lecture_11_formal_groups}{}\subsection*{{Lecture 11 \emph{Formal groups}}}\label{lecture_11_formal_groups}

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

\end{itemize}
\hypertarget{lecture_12_heights_and_formal_groups}{}\subsection*{{Lecture 12 \emph{Heights and formal groups}}}\label{lecture_12_heights_and_formal_groups}

\begin{itemize}%
\item [[height of a formal group]]

\end{itemize}
\hypertarget{lecture_13_the_stratification_of_}{}\subsection*{{Lecture 13 \emph{The stratification of $\mathcal{M}_{FG}$}}}\label{lecture_13_the_stratification_of_}

\begin{itemize}%
\item [[chromatic filtration]]

\end{itemize}
\hypertarget{lecture_14_classification_of_formal_groups}{}\subsection*{{Lecture 14 \emph{Classification of formal groups}}}\label{lecture_14_classification_of_formal_groups}

\begin{itemize}%
\item [[moduli stack of formal groups]]

\end{itemize}
\hypertarget{lecture_15_flat_modules_over_}{}\subsection*{{Lecture 15 \emph{Flat modules over $\mathcal{M}_{FG}$}}}\label{lecture_15_flat_modules_over_}

\begin{itemize}%
\item [[quasicoherent sheaf]], [[flat module]], [[exact functor]]


\item [[Landweber exactness]]



\end{itemize}
\hypertarget{lecture_16_the_landweber_exact_functor_theorem}{}\subsection*{{Lecture 16 \emph{The Landweber exact functor theorem}}}\label{lecture_16_the_landweber_exact_functor_theorem}

\begin{itemize}%
\item [[Landweber exact functor theorem]]

\end{itemize}
\hypertarget{lecture_17_phantom_maps}{}\subsection*{{Lecture 17 \emph{Phantom maps}}}\label{lecture_17_phantom_maps}

\begin{itemize}%
\item [[Brown representability theorem]], [[Landweber exact spectrum]]


\item [[phantom map]]



\end{itemize}
\hypertarget{lecture_18_even_periodic_cohomology_theories}{}\subsection*{{Lecture 18 \emph{Even periodic cohomology theories}}}\label{lecture_18_even_periodic_cohomology_theories}

\begin{itemize}%
\item [[periodic cohomology theory]]


\item [[periodic ring spectrum]]



\end{itemize}
\hypertarget{lecture_19_morava_stabilizer_groups}{}\subsection*{{Lecture 19 \emph{Morava stabilizer groups}}}\label{lecture_19_morava_stabilizer_groups}

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

\end{itemize}
\hypertarget{lecture_20_bousfield_localization}{}\subsection*{{Lecture 20 \emph{Bousfield localization}}}\label{lecture_20_bousfield_localization}

\begin{itemize}%
\item [[Bousfield localization]]


\item [[Bousfield localization of spectra]]


\item [[p-localization]]



\end{itemize}
\hypertarget{lecture_21_lubintate_theory}{}\subsection*{{Lecture 21 \emph{Lubin-Tate theory}}}\label{lecture_21_lubintate_theory}

\begin{itemize}%
\item [[Lubin-Tate formal group]]


\item [[Lubin-Tate theory]]



\end{itemize}
\hypertarget{lecture_22_morava_etheory_and_morava_ktheory}{}\subsection*{{Lecture 22 \emph{Morava E-theory and Morava K-theory}}}\label{lecture_22_morava_etheory_and_morava_ktheory}

\begin{itemize}%
\item [[Morava K-theory]]


\item [[Morava E-theory]]



\end{itemize}
\hypertarget{lecture_23_the_bousfield_classes_of__and_}{}\subsection*{{Lecture 23 \emph{The Bousfield Classes of $E(n)$ and $K(n)$}}}\label{lecture_23_the_bousfield_classes_of__and_}

\begin{itemize}%
\item [[Bousfield equivalence]]


\item [[smash product theorem]]



\end{itemize}
\hypertarget{lecture_24_uniqueness_of_morava_ktheory}{}\subsection*{{Lecture 24 \emph{Uniqueness of Morava K-theory}}}\label{lecture_24_uniqueness_of_morava_ktheory}

\begin{itemize}%
\item [[∞-field]]

\end{itemize}
\hypertarget{lecture_25_the_nilpotence_lemma}{}\subsection*{{Lecture 25 \emph{The Nilpotence lemma}}}\label{lecture_25_the_nilpotence_lemma}

\begin{itemize}%
\item [[nilpotence lemma]]

\end{itemize}
\hypertarget{lecture_26_thick_subcategories}{}\subsection*{{Lecture 26 \emph{Thick subcategories}}}\label{lecture_26_thick_subcategories}

\begin{itemize}%
\item [[thick subcategory]]

\end{itemize}
\hypertarget{lecture_27_the_periodicity_theorem}{}\subsection*{{Lecture 27 \emph{The periodicity theorem}}}\label{lecture_27_the_periodicity_theorem}

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

\end{itemize}
\hypertarget{lecture_28_telescopic_localization}{}\subsection*{{Lecture 28 \emph{Telescopic localization}}}\label{lecture_28_telescopic_localization}

\begin{itemize}%
\item [[telescopic localization]]

\end{itemize}
\hypertarget{lecture_29_telescopic_vs_localization}{}\subsection*{{Lecture 29 \emph{Telescopic vs $E_n$-localization}}}\label{lecture_29_telescopic_vs_localization}

\begin{itemize}%
\item [[smashing localization]]


\item [[chromatic tower]]



\end{itemize}
\hypertarget{lecture_30_localizations_and_the_adamsnovikov_spectral_sequence}{}\subsection*{{Lecture 30 \emph{Localizations and the Adams-Novikov spectral sequence}}}\label{lecture_30_localizations_and_the_adamsnovikov_spectral_sequence}

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

\end{itemize}
\hypertarget{lecture_31_the_smash_product_theorem}{}\subsection*{{Lecture 31 \emph{The smash product theorem}}}\label{lecture_31_the_smash_product_theorem}

\begin{itemize}%
\item [[smash product theorem]]

\end{itemize}
\hypertarget{lecture_32_the_chromatic_convergence_theorem}{}\subsection*{{Lecture 32 \emph{The chromatic convergence theorem}}}\label{lecture_32_the_chromatic_convergence_theorem}

\begin{itemize}%
\item [[chromatic convergence theorem]]

\end{itemize}
\hypertarget{lecture_33_complex_bordism_and_localization}{}\subsection*{{Lecture 33 \emph{Complex bordism and $E(n)$-localization}}}\label{lecture_33_complex_bordism_and_localization}

\hypertarget{lecture_34_monochromatic_layers}{}\subsection*{{Lecture 34 \emph{Monochromatic layers}}}\label{lecture_34_monochromatic_layers}

\begin{itemize}%
\item [[monochromatic layer]]

\end{itemize}
\hypertarget{lecture_35_the_image_of_}{}\subsection*{{Lecture 35 \emph{The image of $J$}}}\label{lecture_35_the_image_of_}

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


\item [[Adams operation]]



\end{itemize}
category: reference



\end{document}