Talk:Self-maps of simply connected manifolds

1 Details for Arkowitz and Lupton's paper

Diarmuid Crowley and Clara Löh are working on showing that the algebras $\mathcal{M}_1$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\mathcal{M}_1$ and $\mathcal{M}_2$$\mathcal{M}_2$ of [Arkowitz&Lupton2000, Examples 5.1 & 5.2] satisfy the theorem of Barge-Sullivan. In particular, we hope to give a detailed proof of the following:

Conjecture 1.1. The rational intersection forms of $\mathcal{M}_1$$\mathcal{M}_1$ and $\mathcal{M}_2$$\mathcal{M}_2$ represent zero in the Witt group of $\Qq$$\Qq$.

Diarmuid Crowley 14:13, 9 June 2010 (UTC), Clara Löh

Remark 1.2. The conjecture above was proven in [Crowley&Löh2015].

2 References

• [Arkowitz&Lupton2000] M. Arkowitz and G. Lupton, Rational obstruction theory and rational homotopy sets, Math. Z. 235 (2000), no.3, 525–539. MR1800210 (2001h:55012) Zbl 0968.55005
• [Crowley&Löh2015] D. Crowley and C. Löh, Functorial seminorms on singular homology and (in)flexible manifolds, Algebr. Geom. Topol. 15 (2015), no.3, 1453–1499. MR3361142 Zbl 1391.57008