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\documentclass[11pt]{memoir}
\usepackage[margin=1.375in]{geometry}
\input{header.tex}
%\usepackage[urw-garamond]{mathdesign}
\begin{document}
\title{Algebraic Topology}
\author{Lectures by Haynes Miller\\
Notes based on live{\TeX}ed record made by Sanath Devalapurkar\\
Images created by John Ni
}
%\date{Fall 2016 -- Spring 2017}
\frontmatter
\maketitle
\input{preface.tex}
\newpage
\tableofcontents
\newpage
\mainmatter
%The current division of labor is: John'll "work on the first part (like, most
%of 905)", and skd will start off by editing/rewriting the whole of 906.
\part{18.905: an introduction to algebraic topology}\label{905}
\chapter{Homology and CW-complexes}
\input{905-hrm-edited/18-905/905/lec-01-intro.tex}
\input{905-hrm-edited/18-905/905/lec-02-simplices.tex}
\input{905-hrm-edited/18-905/905/lec-03-categories.tex}\label{lec:3-categories}
\input{905-hrm-edited/18-905/905/lec-04-more-on-categories.tex}
\input{905-hrm-edited/18-905/905/lec-05-homotopy.tex}
\input{905-hrm-edited/18-905/905/lec-06-homotopy-invariance.tex}\label{lec:6-homotopy-invariance-of-homology}
\input{905-hrm-edited/18-905/905/lec-07-eilenberg-zilber.tex}
\input{905-hrm-edited/18-905/905/lec-08-relative-homology.tex}
\input{905-hrm-edited/18-905/905/lec-09-long-exact.tex}
\input{905-hrm-edited/18-905/905/lec-10-excision.tex}
\input{905-hrm-edited/18-905/905/lec-11-eilenberg-steenrod.tex}
\input{905-hrm-edited/18-905/905/lec-12-subdivision.tex}
\input{905-hrm-edited/18-905/905/lec-13-locality.tex}
\input{905-hrm-edited/18-905/905/lec-14-cw-complexes.tex}
\input{905-hrm-edited/18-905/905/lec-15-cw-complexes-2.tex}
\input{905-hrm-edited/18-905/905/lec-16-homology-cw-complexes.tex}
\input{905-hrm-edited/18-905/905/lec-17-RPn.tex}
\input{905-hrm-edited/18-905/905/lec-18-euler-char.tex}
\input{905-hrm-edited/18-905/905/lec-19-coefficients.tex}
\input{905-hrm-edited/18-905/905/lec-20-tensor-products.tex}
\input{905-hrm-edited/18-905/905/lec-21-tensor-and-tor.tex}
\input{905-hrm-edited/18-905/905/lec-22-more-on-tor.tex}
\input{905-hrm-edited/18-905/905/lec-23-direct-limits.tex}
\input{905-hrm-edited/18-905/905/lec-24-uct.tex}
\input{905-hrm-edited/18-905/905/lec-25-kunneth-eilenberg-zilber.tex}
\chapter{Cohomology and duality}
\input{905-hrm-edited/18-905/905/lec-26-cohomology.tex}
\input{905-hrm-edited/18-905/905/lec-27-ext-cup-product.tex}
\input{905-hrm-edited/18-905/905/lec-28-uct-products-in-cohomology.tex}
\input{905-hrm-edited/18-905/905/lec-29-cup-products-contd.tex}
\input{905-hrm-edited/18-905/905/lec-30-surfaces-bilinear-forms.tex}
\input{905-hrm-edited/18-905/905/lec-31-plethora-of-products.tex}
\input{905-hrm-edited/18-905/905/lec-32-cap-product-cech-cohomology.tex}
\input{905-hrm-edited/18-905/905/lec-33-fully-relative-cap-product.tex}
\input{905-hrm-edited/18-905/905/lec-34-cech-cohomology.tex}
\input{905-hrm-edited/18-905/905/lec-35-topological-manifolds.tex}
\input{905-hrm-edited/18-905/905/lec-36-fundamental-classes.tex}
\input{905-hrm-edited/18-905/905/lec-37-covering-spaces-poincare.tex}
\input{905-hrm-edited/18-905/905/lec-38-the-end-applications.tex}
\part{18.906 -- homotopy theory}\label{906}
\section*{Introduction}
Here is an overview of this part of the book.
\begin{enumerate}
\item \textbf{General homotopy theory.} This includes category theory;
because it started as a part of algebraic topology, we'll speak freely
about it here. We'll also cover the general theory of homotopy groups,
long exact sequences, and obstruction theory.
\item \textbf{Bundles.} One of the major themes of this part of the book is
the use of bundles to understand spaces. This will include the theory
of classifying spaces; later, we will touch upon connections with
cohomology.
\item \textbf{Spectral sequences.} It is impossible to describe everything
about spectral sequences in the duration of a single course, so we will
focus on a special (and important) example: the Serre spectral
sequence. As a consequence, we will derive some homotopy-theoretic
applications. For instance, we will relate homotopy and homology (via
the Hurewicz theorem, Whitehead's theorem, and ``local'' versions like
Serre's mod C theory).
\item \textbf{Characteristic classes.} This relates the geometric theory of
bundles to algebraic constructions like cohomology described earlier in
the book. We will discuss many examples of characteristic classes,
including the Thom, Euler, Chern, and Stiefel-Whitney classes. This
will allow us to apply a lot of the theory we built up to geometry.
%\item Time permitting, there's a beautiful story that comes out (on
%cobordism, etc).
\end{enumerate}
\chapter{Homotopy groups}
\todo[inline]{Insert an outline of the content of each lecture.}
\input{906/lec-39-limits-colimits-adjunctions.tex}
\input{906/lec-40-compactly-generated.tex}
\input{906/lec-41-basepoints.tex}
\input{906/lec-42-fiber-bundles.tex}
\input{906/lec-43-fibrations-cofibrations.tex}
\input{906/lec-44-homotopy-fibers.tex}
\input{906/lec-45-barratt-puppe.tex}
\input{906/lec-46-relative-homotopy.tex}
\input{906/lec-47-pi_1-action-hurewicz.tex}
\input{906/lec-48-examples-cw-complexes.tex}
\input{906/lec-49-relative-hurewicz.tex}
\input{906/lec-50-cellular-approximation.tex}
\input{906/lec-51-and-all-the-rest.tex}
\chapter{Vector bundles}
\input{906/lec-52-vector-bundles.tex}
\input{906/lec-53-principal-bundles.tex}
\input{906/lec-54-I-invariance.tex}
\input{906/lec-55-grassmann-model-classifying-spaces.tex}
\input{906/lec-56-ssets.tex}
\input{906/lec-57-recall.tex}
\input{906/lec-58-bundles-classifying-spaces.tex}
\input{906/lec-59-classifying-spaces.tex}
\chapter{Spectral sequences}
\input{906/lec-60-why-sseqs.tex}
\input{906/lec-61-serre-sseq.tex}
\input{906/lec-62-exact-couples.tex}
\input{906/lec-63-sseq-applications.tex}
\input{906/lec-64-edge-homomorphisms.tex}
\input{906/lec-65-serre-classes.tex}
\input{906/lec-66-mod-C-hurewicz.tex}
\input{906/lec-67-dress-sseq.tex}
\input{906/lec-68-leray-hirsch.tex}
\input{906/lec-69-integration.tex}
\chapter{Characteristic classes}
\input{906/lec-70-grothendieck-chern-classes.tex}
\input{906/lec-71-homology-of-BUn.tex}
\input{906/lec-72-chern-roots.tex}
\input{906/lec-73-steifel-whitney-classes.tex}
\input{906/lec-74-oriented-bundles.tex}
\backmatter
\nocite{*}
\bibliographystyle{alpha}
\bibliography{main}
\end{document}