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