# Talk:5-manifolds: 1-connected/1st edition

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##  Conjecture about mapping class groups of 1-connected 5-manifolds

Let $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_lj9Weu$$\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 \DeclareMathOrd\GL{GL} % general linear group \DeclareMathOrd\SL{SL} % special linear group \DeclareMathOrd\SO{SO} % special orthogonal group \DeclareMathOrd\SU{SU} % special unitary group \DeclareMathOrd\Spin{Spin} % Spin group \DeclareMathOrd\RP{\Rr\mathrm P} % real projective space \DeclareMathOrd\CP{\Cc\mathrm P} % complex projective space \DeclareMathOrd\HP{\Hh\mathrm P} % quaternionic projective space \DeclareMathOrd\Top{\mathrm{Top}} % topological category \DeclareMathOrd\PL{\mathrm{PL}} % piecewise linear category \DeclareMathOrd\Cat{\mathrm{Cat}} % any category \DeclareMathOrd\KS{KS} % Kirby-Siebenmann class \DeclareMathOrd\Hud{Hud} % Hudson torusM$ be a closed, smooth, 1-connected 5-manifold, Matthias Kreck and Diarmuid Crowley conjecture that there is an isomorphism of abelian groups $\displaystyle \pi_0(\SDiff(M)) \cong \Omega_6(B_2(M))$

where $B_2(M)$$B_2(M)$ is the normal $2$$2$-type of $M$$M$ as defined in [Kreck1999]. For example, if $M$$M$ is Spinable with $H_2(M) \cong H$$H_2(M) \cong H$ then $\displaystyle \Omega_6(B_2(M)) \cong \Omega_6^{Spin}(K(H, 2)).$

At present we are checking the details of the proof of this conjecture using the methods of [Kreck1999].

Diarmuid Crowley 10:02, 29 September 2009 (UTC)