Talk:Euler characteristic as surgery obstruction (Ex)

If $n$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}n$ is odd, then $\chi(X)=\chi(M’)=\chi(M)=0$$\chi(X)=\chi(M’)=\chi(M)=0$ and there is nothing to check. (This does not matter in the ensuing proof, but might be worth mentioning.)

We will check the effect surgery has on Euler characteristic.

Let $S^k\subset M^n$$S^k\subset M^n$ be an embedding of a $k$$k$-sphere into $n$$n$-manifold $M$$M$, where the embedded sphere has trivial normal bundle. Say $N$$N$ is obtained from $M$$M$ by surgery along $S^k$$S^k$. That is, $N\cong M\setminus (S^k\times D^{n-k})\cup (D^{k+1}\times S^{n-k-1})$$N\cong M\setminus (S^k\times D^{n-k})\cup (D^{k+1}\times S^{n-k-1})$, so $\chi(N)=\chi(M)-\chi(S^k)+\chi(S^{n-k-1})\equiv\chi(M)\pmod{2}$$\chi(N)=\chi(M)-\chi(S^k)+\chi(S^{n-k-1})\equiv\chi(M)\pmod{2}$. Then inductively, since $M’$$M’$ is obtained from $M$$M$ by a finite sequence of surgeries, $\chi(X)=\chi(M’)\equiv\chi(M)\pmod{2}$$\chi(X)=\chi(M’)\equiv\chi(M)\pmod{2}$.