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@@ -394,247 +394,4 @@ a filtration $0\subseteq F^0_n\subseteq\cdots\subseteq F^n_n=H_n(X;G)$ of ...
 \end{enumerate}
 \end{thm}
 
-\part{K-theory}
-
-\section{Vector bundles}
-
-\subsection{Basic spaces}
-\begin{defn}
-The \define{$n$-sphere} $\Sn^n$ is the subspace of $\R^{n+1}$ consisting of unit vectors.
-The \define{real projective $n$-space} $\R P^n$  is the space of lines in
-$\R^{n+1}$ through the origin. Equivalently, we may regard $\R P^n$ as the quotient
-space of $\Sn^n$ in which the antipodal pairs of points are identified. Notice
-that $\R P^1\approx \Sn^1$. 
-\end{defn}
-
-\begin{defn}
-For each $n$, we may include the $n$-sphere $\Sn^n$ into $\Sn^{n+1}$ by mapping
-it into the equator. These inclusions induce inclusions $\R P^n\to \R P^{n+1}$.
-We define $\R P^\infty$ to be the sequential colimit of $\R P^n$. 
-\end{defn}
-
-\subsection{Definition and basic properties}
-\begin{defn}
-An \define{$n$-dimensional vector bundle} is a map $p:E\to B$ together with a
-real vector space structure on $p^{-1}(b)$ for each $b\in B$, satisfying the
-\define{local triviality condition}, which says that there is an open cover 
-$\mathcal{C}$ of $B$, with homeomorphisms $h_U:p^{-1}(U)\to U\times\mathbb{R}^n$
-for each $U\in\mathcal{C}$, which maps $p^{-1}(b)$ to $\{b\}\times\mathbb{R}^n$
-for each $b\in U$. 
-
-The functions $h_U$ are also called \define{local trivializations}. Given a
-vector bundle $p:E\to B$, the space $B$ is called the \define{base space}, the
-space $E$ is called the \define{total space}, and the spaces $p^{-1}(b)$ are
-called the \define{fibers}. A $1$-dimensional
-vector bundle is also called a \define{line bundle}.
-\end{defn}
-
-\begin{defn}
-An \define{isomorphism of vector bundles} from $p:E\to B$ to $p':E'\to B$
-consists of a map $h:E\to E'$ satisfying $p'\circ h=p$, 
-which induces a linear isomorphism
-$p^{-1}(b)\to p'^{-1}(b)$ between each of the fibers.
-\end{defn}
-
-\begin{lem}
-If $h:E\to E'$ is an isomorphism of vector bundles, then the underlying map
-of type $E\to E'$ is a homeomorphism. 
-\end{lem}
-
-\begin{proof}
-Suppose $h:E\to E'$ induces isomorphisms $p^{-1}(b)\to p'^{-1}(b)$ for each
-$b\in B$. Then, for each $x\in E'$ we have an isomorphism from
-$p^{-1}(p'(x))$ to $p'^{-1}(p'(x))$. Since $x\in p'^{-1}(p'(x))$, we find
-an element $y\in p^{-1}(p'(x))\subseteq E$. Thus, $h$ is surjective. Now suppose
-that $x,x'\in E$ are two elements for which $h(x)=h(x')$. Since $p'\circ h=p$,
-it follows that $x'\in p^{-1}(x)$. Now, the fact that $h$ induces an isomorphism
-between fibers implies that $x=x'$. 
-
-Thus, $h$ has an inverse function $k:E'\to E$, and we need to show that this
-function is continuous. It suffices to show that $k|_U$ is continuous for each
-$U$ on which $p'$ is trivial. Let $x\in B$, and compose the map
-$h_U:p^{-1}(U)\to p'^{-1}(U)$ with its local trivializations. Thus, we obtain
-a map $g_U:U\times\R^n\to U\times \R^n$, mapping $(x,y)$ to $(x,A(y))$, where
-$A$ is a linear isomorphism. 
-\end{proof}
-
-In the following definition, we give a vector bundle by a gluing construction.
-
-\begin{defn}
-Consider a space $B$, and an open cover $\mathcal{C}$ which is closed under
-finite intersections. Then $\mathcal{C}$ may be considered a poset ordered by
-inclusion.
-
-A \define{collection of gluing functions} consists of a continuous choice of linear 
-isomorphisms $g_{U,V}:U\cap V\to GL_n(\R)$ satisfying the \define{cocycle
-condition} 
-\begin{equation*}
-g_{V,W}\circ g_{U,V}=g_{U,W}
-\end{equation*}
-on $U\cap V\cap W$, for every 
-$U,V,W\in\mathcal{C}$. Such a collection of gluing functions determines a functor
-$\mathcal{C}\to\mathbf{Top}$, which is given on points by $U\mapsto U\times\R^n$,
-and on morphisms by $(x,v)\mapsto(x,A(v))$, for each $U\subseteq V$ determining
-a linear isomorphism $A$. 
-
-The colimit of this functor is the total space of a vector bundle.
-\end{defn}
-
-\begin{eg}
-There are lots of examples of vector bundles:
-\begin{enumerate}
-\item The \define{$n$-dimensional trivial bundle} over $B$ is defined to be 
-$\proj1:B\times\mathbb{R}^n\to B$. So the trivial bundle is the one which is
-\emph{globally} trivial. We will write the $n$-dimensional trivial bundle over
-$B$ as $\epsilon^n\to B$. 
-\item The circle may be regarded as the quotient of $[0,1]$ modulo the end points.
-The \define{Mobius bundle} is the line bundle over $\Sn^1$ to have total space
-$E\defeq [0,1]\times\R$, with the identifications $(0,t)\sim(1,-t)$. 
-\item The \define{tangent bundle} of the unit sphere $\Sn^n$, viewed as a subspace of
-$\R^{n+1}$, is defined to be the subspace $E\defeq\{(x,v)\in\Sn^n\times\R^{n+1}
-\mid x\perp v\}$ of $\R^{2n+2}$, which projects onto $\Sn^n$. 
-
-The $n$-sphere is covered by $2n+2$ open hemispheres, centering at $\pm e_i$,
-where $e_i\in\R^{n+1}$ is a basis vector. 
-\item The \define{normal bundle} of the unit sphere $\Sn^n$ is the line bundle
-with $E$ consisting of pairs $(x,v)\in\Sn^n\times\R^{n+1}$ such that $v=tx$ for
-some $t\in\R$. \emph{The normal bundle on $\Sn^n$ is isomorphic to the trivial line 
-bundle $\Sn^n\times\R\to\Sn^n$.}
-\item The \define{canonical line bundle} $p:E\to \R P^n$ has as its total space
-the subspace $E\subseteq \R P^{n+1}\times\R ^{n+1}$ consisting of pairs
-$(l,v)$ with $v\in l$. \emph{The M\"obius line bundle is isomorphic to the
-canonical line bundle on $\Sn^1$.}
-\item The inclusions $\R P^n\subseteq \R P^{n+1}$ induce inclusions of the
-canonical line bundles. The sequential colimit of the canonical line bundles
-produces the canonical line bundle on $\R P^\infty$. 
-\end{enumerate}
-\end{eg}
-
-\begin{defn}
-Given two vector bundles $p:E\to B$ and $p':E'\to B$ over the same base space
-$B$, we obtain a vector bundle $p\oplus p': E\oplus E'\to B$, fitting in the
-pullback square
-\begin{equation*}
-\begin{tikzcd}
-E\oplus E' \arrow[r] \arrow[d] \arrow[dr,"{p\oplus p'}" description ] & E' \arrow[d,"{p'}"] \\
-E \arrow[r,swap,"p"] & B
-\end{tikzcd}
-\end{equation*}
-\end{defn}
-
-\begin{eg}
-\begin{enumerate}
-\item The direct sum of the tangent and normal bundles on $\Sn^n$ is the trivial
-bundle $\Sn^n\times\R^{n+1}$. 
-\end{enumerate}
-\end{eg}
-
-\begin{defn}
-Let $p:E\to B$ and $p':E'\to B$ be two vector bundles over the same space $B$,
-and choose an open cover $\mathcal{C}$ such that both $E$ and $E'$ are locally
-trivial with respect to $\mathcal{C}$. We define $E\otimes E'$ by gluing. 
-
-Then we can define, for each $U,V\in\mathcal{C}$ satisfying $U\subseteq V$, we
-have linear isomorphisms $g_{U,V}(x):\R^n\to\R^n$ and $g'_{U,V}(x):\R^m\to\R^m$,
-induced by the local trivializations of $E$ and $E'$ respectively. These give
-gluing functions $g_{U,V}(x)\otimes g'_{U,V}(x):\R^n\otimes\R^m\to\R^n\otimes\R^m$
-for each $x\in U$, and these gluing functions satisfy the cocycle condition.
-
-Thus, we obtain a vector bundle $E\otimes E'$ from these gluing functions.
-\end{defn}
-
-\begin{lem}
-The tensor product of vector bundles over a fixed base space is commutative,
-associative, it has an identity element (the trivial bundle), and it is
-distributive with respect to direct sum.
-\end{lem}
-
-Change of base $f:B'\to B$ turns a vector bundle $E$ over $B$ to a vector
-bundle $f^\ast(E)$ over $B'$.
-
-\begin{lem}
-For any two vector bundles $E$ and $E'$ over $B$, and any $f:B'\to B$, we have
-natural isomorphisms $f^\ast(E\oplus E')\approx f^\ast(E)\oplus f^\ast(E')$, and
-$f^\ast(E\otimes E')\approx f^\ast(E)\otimes f^\ast(E')$. Moreover, if $f$
-is homotopic to $g$, then $f^\ast=g^\ast$. 
-\end{lem}
-
-\subsection{K-theory}
-
-\begin{defn}
-Two vector bundles $E\to B$ and $E'\to B$ are callec \define{stably isomorphic},
-if there is an $n$ for which $E\oplus\epsilon^n\approx E'\oplus\epsilon^n$, and
-we write $E\approx_s E'$ if $E$ and $E'$ are stably isomorphic. Also,
-we will define the relation $E\sim E'$ if there are $m$ and $n$ such that
-$E\oplus\epsilon^m\approx E'\oplus^n$. 
-\end{defn}
-
-\begin{lem}
-The direct sum preserves both $\approx_s$ and $\sim$. Moreover, if $B$ is compact,
-then the set of ${\sim}$-equivalence classes of vector bundles forms an abelian
-group, called $\tilde{K}(B)$. If $B$ is pointed, then the tensor product turns 
-$\tilde{K}(B)$ into a ring.
-\end{lem}
-
-\begin{lem}
-The direct sum satisfies the cancellation property with respect to $\approx_s$,
-i.e.~we have that $E\oplus E'\approx_s E\oplus E''$ implies $E'\oplus E''$. 
-Thus, if we define two pairs $(E,F)$ and $(E',F')$ to be equivalent to each
-other whenever $E\oplus F'=E'\oplus F$, we obtain an abelian group $K(B)$ for
-any compact space $B$. The tensor product turns $K(B)$ into a ring.
-\end{lem}
-
-\begin{lem}
-We have a ring isomorphism
-\begin{equation*}
-K(B)\approx \tilde{K}(B)\oplus\Z.
-\end{equation*}
-\end{lem}
-
-Both $K$ and $\tilde{K}$ are contravariant functors.
-
-\begin{lem}
-If $X$ is compact Hausdorff and $A\subseteq X$ is a closed subspace, then the
-inclusion and quotient maps $A\stackrel{i}{\to}X\stackrel{q}{\to}X/A$ induces
-an sequence
-\begin{equation*}
-\begin{tikzcd}
-\tilde{K}(X/A) \arrow[r,"q^\ast"] & \tilde{K}(X) \arrow[r,"i^\ast"] & \tilde{K}(A)
-\end{tikzcd}
-\end{equation*}
-which is exact at $\tilde{K}(X)$.
-\end{lem}
-
-\begin{lem}
-If $A$ is contractible, the quotient map $q:X\to X/A$ induces a bijection
-$q^\ast:\mathrm{Vect}^n(X/A)\to\mathrm{Vect}^n(X)$.
-\end{lem}
-
-Apparently, this gives a long exact sequence of $\tilde{K}$-groups:
-\begin{equation*}
-\begin{tikzcd}[column sep=small]
-\cdots\arrow[r] & \tilde{K}(\Sn(X)) \arrow[r] & \tilde{K}(\Sn(A)) \arrow[r]
-& \tilde{K}(X/A) \arrow[r] & \tilde{K}(X) \arrow[r] & \tilde{K}(A).
-\end{tikzcd}
-\end{equation*}
-Still considering pointed spaces, we may consider the long exact sequence of the pair $(X\times Y,
-X\vee Y)$. Recall that $(X\times Y)/(X\vee Y)$ is the smash product
-$X\wedge Y$, i.e.~the smash product is the pushout of $\unit\leftarrow
-X\vee Y\rightarrow X\times Y$. The long exact sequence of the pair
-$(X\times Y,X\vee Y)$ looks as follows:
-\begin{equation*}
-\begin{tikzcd}[column sep=.8em]
-\cdots\arrow[r] & \tilde{K}(\Sn(X\times Y)) \arrow[r] & \tilde{K}(\Sn(X\vee Y)) \arrow[r]
-& \tilde{K}(X\wedge Y) \arrow[r] & \tilde{K}(X\times Y) \arrow[r] & \tilde{K}(X\vee Y).
-\end{tikzcd}
-\end{equation*}  
-
-\subsection{Bott periodicity}
-
-\begin{defn}
-We define an \define{external product} $\mu:K(X)\otimes K(Y)\to K(X\times Y)$,
-by $\mu(a\otimes b)\defeq \proj1^\ast(a)\cdot\proj2^\ast(b)$. 
-\end{defn}
-
-
 \end{document}