# Normal maps - (non)-examples (Ex)

1a) Give an example of a degree one map of closed $n$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}n$-manifolds $f \colon M \to X$$f \colon M \to X$ which cannot be covered by a map $\overline{f} \colon \nu_M \to \nu_X$$\overline{f} \colon \nu_M \to \nu_X$ of normal bundles.

1b) Give an example of a degree one map of closed $n$$n$-manifolds $f \colon M \to X$$f \colon M \to X$ which cannot be covered by a map $\overline{f} \colon \nu_M \to \xi$$\overline{f} \colon \nu_M \to \xi$ of bundles, for any stable bundle $\xi$$\xi$.

2) For every integer $d$$d$, give an example of a degree $d$$d$ map $f_d \colon M \to X$$f_d \colon M \to X$ of closed $n$$n$-manifolds which can be covered by a map $\overline{f_d} \colon \nu_M \to \nu_X$$\overline{f_d} \colon \nu_M \to \nu_X$ of normal bundles.

3) Let $F_g$$F_g$ denote the oriented surface of genus $g$$g$. Determine the values of $(g, g')$$(g, g')$ for which there is a degree one normal map $(f, \overline{f}) \colon F_g \to F_{g'}$$(f, \overline{f}) \colon F_g \to F_{g'}$.