Template:Kreck1979

M. Kreck, Isotopy classes of diffeomorphisms of $(k-1)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(k-1)$-connected almost-parallelizable $2k$$2k$-manifolds, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Springer (1979), 643–663. MR561244 (81i:57029) Zbl 0421.57009

Correction

The final two sentences of part (c) of Theorem 3 are incorrect. In fact, if $M = \Sigma^{10} \in \Theta_{10} \cong \Z_6$$M = \Sigma^{10} \in \Theta_{10} \cong \Z_6$ is a homotopy sphere with non-trivial $\alpha$$\alpha$-invariant then $\Sigma_M$$\Sigma_M$ is the non-trivial element of order two in $bP_{12} \cong \Z_{992}$$bP_{12} \cong \Z_{992}$. This, in fact, can be seen from Corollary 1 since there is a homotopy 10-sphere as above representing $\eta \circ \mu + \beta_1$$\eta \circ \mu + \beta_1$ in the stable $10$$10$-stem, $\pi_{10}^S$$\pi_{10}^S$ but $\eta \circ(\mu + \beta) = 0 \in \pi_{11}^S$$\eta \circ(\mu + \beta) = 0 \in \pi_{11}^S$. Hence $\Sigma_M$$\Sigma_M$ must be the non-trivial element of order two in $bP_{12}$$bP_{12}$. It also follows from the proof of Corollary 3 which is given just prior to the statement of Corollary 3.

The second to last sentece of part (c) of Theorem 3 should be reformulated as follows to add the assumption that $M$$M$ bounds over a spin manifold:

Especially it follows for $k$$k$ odd and $[M] = 0 \in \Omega_{2k}^{Spin}$$[M] = 0 \in \Omega_{2k}^{Spin}$ that $\Sigma_M = 0 \Leftrightarrow \Sigma_M \in bP_{2k+2}$$\Sigma_M = 0 \Leftrightarrow \Sigma_M \in bP_{2k+2}$.

The proof of this statement uses the results of Kreck in combination with Proposition 8 of [Levine1970].