The Unapologetic Mathematician

Mathematics for the interested outsider

A Family of Nontrivial Homology Classes (part 2)

We continue investigating the differential forms we defined last time. Recall that we started with the position vector field P(p)=\mathcal{I}_pp and use the interior product to produce the n-form \hat{\omega}=\iota_P(du^0\wedge\dots\wedge du^n) on the punctured n+1-dimensional space \mathbb{R}^{n+1}\setminus\{0\}. We restrict this form to the n-dimensional sphere S^n and then pull back along the retraction mapping \pi:p\mapsto\frac{p}{\lvert p\rvert} to get the form \omega=\pi^*\hat\omega.

I’ve asserted that \omega=\frac{1}{\lvert p\rvert^n}\hat{\omega}, and now we will prove it; let \{v_1,\dots,v_n\} be n tangent vectors at p and calculate

\displaystyle\begin{aligned}{}[\omega(p)]&(v_1,\dots,v_n)\\&=[[\pi^*\hat{\omega}](p)](v_1,\dots,v_n)\\&=[\hat{\omega}(\pi(p))](\pi_{*p}v_1,\dots,\pi_{*p}v_n)\\&=\left[\hat{\omega}\left(\frac{p}{\lvert p\rvert}\right)\right]\left(\frac{1}{\lvert p\rvert}\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_1,\dots,\frac{1}{\lvert p\rvert}\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_n\right)\\&=\frac{1}{\lvert p\rvert^n}\left[\hat{\omega}\left(\frac{p}{\lvert p\rvert}\right)\right]\left(\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_1,\dots,\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_n\right)\\&=\frac{1}{\lvert p\rvert^n}\left[[du^0\wedge\dots\wedge du^n]\left(\frac{p}{\lvert p\rvert}\right)\right]\left(P\left(\frac{p}{\lvert p\rvert}\right),\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_1,\dots,\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_n\right)\\&=\frac{1}{\lvert p\rvert^n}\left[[du^0\wedge\dots\wedge du^n]\left(\frac{p}{\lvert p\rvert}\right)\right]\left(\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}P(p),\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_1,\dots,\mathcal{I}_{\frac{p}{\lvert p\rvert}}\mathcal{I}_p^{-1}v_n\right)\\&=\frac{1}{\lvert p\rvert^n}[du^0\wedge\dots\wedge du^n]\left(\mathcal{I}_p^{-1}P(p),\mathcal{I}_p^{-1}v_1,\dots,\mathcal{I}_p^{-1}v_n\right)\\&=\frac{1}{\lvert p\rvert^n}\left[[du^0\wedge\dots\wedge du^n](p)\right](P(p),v_1,\dots,v_n)\\&=\left[\left[\frac{1}{\lvert p\rvert^n}\hat{\omega}\right](p)\right](v_1,\dots,v_n)\end{aligned}

as asserted. Along the way we’ve used two things that might not be immediately apparent. First: the derivative \pi_* works by transferring a vector from \mathcal{T}_p\mathbb{R}^{n+1} to \mathcal{T}_{\frac{p}{\lvert p\rvert}}\mathbb{R}^{n+1} and scaling down by a factor of \frac{1}{\lvert p\rvert}, which is a consequence of the linear action of \pi_* and the usual canonical identifications. Second: the volume form on \mathcal{T}_p\mathbb{R}^{n+1} can be transferred to essentially the same form on \mathbb{R}^{n+1} itself by using the canonical identification \mathcal{I}_p.

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December 20, 2011 - Posted by | Differential Topology, Topology

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