Talk:Connected sum of topological manifolds (Ex)

Part 1

Let $M^n$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M^n$ be an (oriented) topological $n$$n$-manifold. Choose any two locally flat embeddings $f,g:S^{n-1}\to M^n$$f,g:S^{n-1}\to M^n$ so that $f(S^{n-1})$$f(S^{n-1})$ and $g(S^{n-1})$$g(S^{n-1})$ bound topological balls in $M^n$$M^n$. We will show that $f(S^{n-1})$$f(S^{n-1})$ is topologically isotopic to $g(S^{n-1})$$g(S^{n-1})$. This fact implies that connected sum of topological manifolds is well-defined, as the connected sum of $M^n$$M^n$ and $N^n$$N^n$ can then be defined by deleting a topological ball from each and then gluing the boundary, respecting the orientations on $M^n$$M^n$ and $N^n$$N^n$ (the gluing map is determined up to homotopy by the orientations, as an orientation-preserving automorphism of $S^{n-1}$$S^{n-1}$. We will show this in Part 2).

So now let $x$$x$ and $y$$y$ be the center points of the balls bounded by $F:=f(S^{n-1})$$F:=f(S^{n-1})$ and $G:=g(S^{n-1})$$G:=g(S^{n-1})$. Let $B_F$$B_F$ and $B_G$$B_G$ be balls bounded by $F/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_jnhban$$F$ and $G$$G$ respectively. Since $F$$F$ and $G$$G$ are locally flat, they have collar neighborhoods. Then by extending $B_F$$B_F$ and $B_G$$B_G$ along these collars, we find open balls $U_F$$U_F$ and $U_G$$U_G$ containing $F$$F$ and $G$$G$ (respectively).

Our current goal is to ambiently isotope $B_G$$B_G$ so that $x$$x$ and $y$$y$ agree. Since $M^n$$M^n$ is a manifold, it is path-connected. Then there exists some continuous map $\gamma:[0,1]\to M^n$$\gamma:[0,1]\to M^n$ so that $\gamma(0)=y$$\gamma(0)=y$ and $\gamma(1)=x$$\gamma(1)=x$. Pick charts $\{(O_\alpha,f_\alpha)\}_{\alpha\in\Lambda}$$\{(O_\alpha,f_\alpha)\}_{\alpha\in\Lambda}$ covering $M$$M$, where $f_\alpha:O_\alpha\to\mathbb{R}^n$$f_\alpha:O_\alpha\to\mathbb{R}^n$ is a homeomorphism. For each $p\in [0,1]$$p\in [0,1]$, Let $I_p$$I_p$ be an open (in $[0,1]$$[0,1]$) subinterval of $[0,1]$$[0,1]$ containing $p$$p$ so that $\gamma(I_p)$$\gamma(I_p)$ is contained in a single $O_{\alpha}$$O_{\alpha}$ (call this $O_p$$O_p$). Then $\{I_p\}_{p\in[0,1]}$$\{I_p\}_{p\in[0,1]}$ is an open cover of $[0,1]$$[0,1]$, so from compactness there exists a finite collection $p_1<\ldots<p_k\in[0,1]$$p_1<\ldots<p_k\in[0,1]$ so that $[0,1]=\cup_{i=1}^k I_{p_i}$$[0,1]=\cup_{i=1}^k I_{p_i}$. If $I_{p_i}\subset I_{p_j}$$I_{p_i}\subset I_{p_j}$ for $i\neq j$$i\neq j$, delete $p_i$$p_i$ from the collection $\{p_1,\ldots, p_k\}$$\{p_1,\ldots, p_k\}$ and relabel. Let $p_{k+1}:=1$$p_{k+1}:=1$. In $O_{p_1}\cong\mathbb{R}^4$$O_{p_1}\cong\mathbb{R}^4$, choose a path $\eta_{p_1}$$\eta_{p_1}$ from $x$$x$ to $\gamma(p_1)$$\gamma(p_1)$ which has a tubular neighborhood $\eta_{p_1}\times \mathring{D}^{n-1}$$\eta_{p_1}\times \mathring{D}^{n-1}$. Ambiently isotope $B_G$$B_G$ along $\eta_{p_1}$$\eta_{p_1}$ so that $\gamma(p_1)\in B_G$$\gamma(p_1)\in B_G$. Then ambiently isotope the interior of $B_G$$B_G$ to take $y$$y$ along $\eta_{p_1}$$\eta_{p_1}$ so that $y$$y$ agrees with $\gamma(p_1)$$\gamma(p_1)$. Repeat for each $p_i$$p_i$ for $i=2,\ldots, k$$i=2,\ldots, k$ until $y$$y$ agrees with $\gamma(p_{k+1})=\gamma(1)=x$$\gamma(p_{k+1})=\gamma(1)=x$.

Shrink $U_G$$U_G$ so that $B_G\subset U_G\subset B_F$$B_G\subset U_G\subset B_F$. So in $U_F\cong\mathbb{R}^n$$U_F\cong\mathbb{R}^n$, $F$$F$ and $G$$G$ are nested disjoint $(n-1)$$(n-1)$-spheres. By the annulus conjecture, $F$$F$ and $G$$G$ cobound an $S^{n-1}\times I$$S^{n-1}\times I$ in $U_F$$U_F$. Therefore, we may isotope $F$$F$ in $U_F$$U_F$ along the $I$$I$ direction of this product until $F$$F$ and $G$$G$ agree.

Part 2

Finally, we must show that there is a unique orientation-preserving automorphism of $S^n$$S^n$ up to topological isotopy (for each $n$$n$). First we remark that if $f_1:D^n\to D^n$$f_1:D^n\to D^n$ is a continuous map fixing $\partial D^n$$\partial D^n$ pointwise, then $f_1$$f_1$ is isotopic rel boundary to the identity map $f_0:D^n\to D^n$$f_0:D^n\to D^n$. (This is called Alexander's trick.) This can be shown constructively; consider the isotopy $f_t(x)=\begin{cases}tf_1(x/t)&|x|\le t\\x&|x|>t,\end{cases}$$f_t(x)=\begin{cases}tf_1(x/t)&|x|\le t\\x&|x|>t,\end{cases}$ where $D^n$$D^n$ has radius $1$$1$.

Now let $h:S^n\to S^n$$h:S^n\to S^n$ be an orientation-preserving automorphism. We will proceed by induction; as a base case consider $n=0$$n=0$. There are two automorphisms of $S^0$$S^0$, one of which is orientation-preserving. Then the claim holds trivially.

Now let $n$$n$ be arbitrary. Let $C$$C$ be a locally flat copy of $S^{n-1}$$S^{n-1}$ embedded in $S^n$$S^n$ which splits $S^n$$S^n$ into two $n$$n$-balls. Then $h(C)$$h(C)$ is another such copy of $S^{n-1}$$S^{n-1}$. By the argument we used above to isotope $F$$F$ and $G$$G$, we may isotope $h$$h$ (and hence $h(C)$$h(C)$) so that $C$$C$ and $h(C)$$h(C)$ are disjoint and contained in an $n$$n$-ball. By the annular embedding theorem, $C$$C$ and $h(C)$$h(C)$ cobound an annulus, so we may isotope $h$$h$ to fix $C$$C$ setwise (isotoping $h(C)$$h(C)$ through the $I$$I$ direction of the annulus). Call the closures of the $n$$n$-ball components of $S^n\setminus C$$S^n\setminus C$ $A$$A$ and $B$$B$. If $h$$h$ reverses the orientation of $C$$C$, isotope $h$$h$ to isotope one hemisphere of $h(C)$$h(C)$ through $A$$A$ and the other through $B$$B$ so that $h$$h$ fixes $C$$C$ setwise and preserves the orientation of $C$$C$.

By induction hypothesis, isotope $h$$h$ to fix $C$$C$ pointwise. Since $h$$h$ is orientation preserving, $h(A)=A$$h(A)=A$ and $h(B)=B$$h(B)=B$. Then by Alexander's trick, $h$$h$ can be isotoped rel boundary to be the identity within $A$$A$ and $B$$B$. Thus, $h$$h$ is isotopic to the identity.