Talk:Borel conjecture in dimensions 1 and 2 (Ex)

Let us do the case of dimension $1$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}1$ first. Here it follows from the classification that every connected closed $1$$1$-manifold is homeomorphic to a circle $S^1$$S^1$. Moreover we know that up to homotopy every self map of $S^1$$S^1$ is of the from $z \mapsto z^k$$z \mapsto z^k$ for some $k\in \Z$$k\in \Z$. Moreover homotopy equivalences have to fulfill $k \in \{-1,1\}$$k \in \{-1,1\}$ and those maps are homeomorphisms, so every homotopy equivalence is homotopic to a homeomorphism.

For the case of dimension $2$$2$, first we note that by handle body decomposition it follows that two orientable surfaces are homotopy equivalent if and only if they are homeomorphic. For any space $X$$X$ let us denote by $\mathrm{Homeo}(X)$$\mathrm{Homeo}(X)$ the space of homeomorphisms of $X$$X$ and by $\mathrm{HoEq}(X)$$\mathrm{HoEq}(X)$ the space of homotopy equivalences of $X$$X$, each equipped with the compact open topology. We now claim that for $M$$M$ an orientable surface, the natural inclusion

$\displaystyle \xymatrix{\mathrm{Homeo}(M) \ar[r] & \mathrm{HoEq}(M)}$

induces a surjection on path components which follows from a theorem that this is even a bijection on path components.