Talk:Borel Conjecture for the 2-torus (Ex)

The selfhomotopy equivalences of an Eilenberg-Mac Lane space of type $(G,n)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(G,n)$ generally have $\pi_0 = Out(G)$$\pi_0 = Out(G)$, detected by the action of fundamental groups, and $\pi_n = C(G)$$\pi_n = C(G)$, with all other homotopy groups vanishing. In particular, $\pi_0(hAut(T^2)) = Gl_2(\mathbb Z)$$\pi_0(hAut(T^2)) = Gl_2(\mathbb Z)$ is detected by the action on $H_1(T^2)$$H_1(T^2)$ and all we have to do is produce a homeomorphism that acts by a given matrix $A \in Gl_2(\mathbb Z)$$A \in Gl_2(\mathbb Z)$. But via the identification $T^2 = \mathbb R^2/\mathbb Z^2$$T^2 = \mathbb R^2/\mathbb Z^2$ the action of $A$$A$ on $\mathbb R^2$$\mathbb R^2$ produces such an element.

The same argument shows that the inclusion $Diff(T^n) \rightarrow hAut(T^n)$$Diff(T^n) \rightarrow hAut(T^n)$ splits on $pi_0$$pi_0$ In arbitrary dimensions.

Note: as described by a (now erased) suggestion, there is a more elementary method. You can treat the map as a periodic map $\mathbb R^2 \to \mathbb R^2$$\mathbb R^2 \to \mathbb R^2$ sending every lattice point to itself. This is because the image of the origin's generators also generate the target's homology. This map of $\mathbb R^2$$\mathbb R^2$ can be homotoped to the identity map linearly, producing a diffeomorphism. This also works for any $T^n$$T^n$, as the above proof does.