# Even dimensional surgery obstruction (Ex)

This question is intended to illustrate the importance of the normal map in defining the even-dimensional surgery obstruction.

1. Show that any closed, orientable $m$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}m$-manifold $M^m$$M^m$, possesses a degree 1 map $f:M^m\to S^m$$f:M^m\to S^m$.
2. For $M=\Sigma_g$$M=\Sigma_g$, find all degree 1 normal maps $(\bar{f},f)$$(\bar{f},f)$ that cover $f:\Sigma_g\to S^2$$f:\Sigma_g\to S^2$.
3. For each of the $(\bar{f},f)$$(\bar{f},f)$, calculate the surgery obstruction $\sigma_*(\bar{f},f)\in L_2(\mathbb{Z})$$\sigma_*(\bar{f},f)\in L_2(\mathbb{Z})$. If this vanishes, write down an explicit surgery on $(\bar{f},f)$$(\bar{f},f)$ that describes a cobordism between $\Sigma_g$$\Sigma_g$ and $S^2$$S^2$.

Note if there is surgery obstruction that even though we know $\Sigma_g$$\Sigma_g$ is cobordant to $S^2$$S^2$, the wrong normal map in the surgery programme will not find this cobordism.