Template:Burghelea&Lashof1974

D. Burghelea and R. Lashof, The homotopy type of the space of diffeomorphisms. I, II, Trans. Amer. Math. Soc. 196 (1974), 1–36; ibid. 196 (1974), 37–50. MR0356103 (50 #8574) Zbl 0296.58003

Correction

There is a small mistake in Corollary 5.5 of part II of these papers. This corollary states in particular that the stabilization homomorphism

$\displaystyle \pi_i(PL_n) \to \pi_i(PL)$ $$\displaystyle \pi_i(PL_n) \to \pi_i(PL)$$

is onto if $n \geq 5$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}n \geq 5$ and $i \leq n+1$$i \leq n+1$. However the stabilization homomorphism

$\displaystyle \pi_7(PL_6) \to \pi_7(PL)$ $$\displaystyle \pi_7(PL_6) \to \pi_7(PL)$$

is not onto. This follows from the fact that the stabilization homomorphism $\pi_7(O_6) \to \pi_7(O)$$\pi_7(O_6) \to \pi_7(O)$ is not onto (it is isomorphic to $\times 4 \colon \Z \to \Z$$\times 4 \colon \Z \to \Z$). The proof of Corollary 5.5 given by Burghelea and Lashof proves the following slightly weaker statement:

Corollary 5.5': The stabilization homomorphism $\pi_i(PL_n) \to \pi_i(PL)$$\pi_i(PL_n) \to \pi_i(PL)$ is onto if $n \geq 5, i \leq n+1$$n \geq 5, i \leq n+1$ and the stabilization homomorphism $\pi_i(O_n) \to \pi_i(O)$$\pi_i(O_n) \to \pi_i(O)$ is onto.

The analogous modification is needed for the stabilization homomorphism $\pi_i(Top_n) \to \pi_i(Top)$$\pi_i(Top_n) \to \pi_i(Top)$.