Real spin bundles over $\mathbb{C}\mathrm{P}^3$ and a new Euclidean embedding of $\mathbb{R}\mathrm{P}^7$
Dominik Gdesz
TL;DR
The paper addresses the problem of classifying real Spin-bundles over $CP^3$ and uses a new $\rho$-invariant, built from Spin(n) representations and Clifford algebra actions, to achieve a complete (for $n\le6$) classification via $K$-theory groups ($KO$, $K$, $KSP$). In the stable range ($n\ge7$) the invariants reduce to the image in $\widetilde{K}(X)$, linking spin structures with characteristic classes like $p_1$ and $sp_1$. Leveraging these classifications, the authors construct a smooth embedding of $RP^7$ into $RP^{11}$ by aligning the relevant real vector bundles on $CP^3$ so that the sphere bundle of $\eta$ embeds in $\mathbb{R}^{11}$, ultimately yielding the desired Euclidean embedding. The work combines explicit representation-theoretic computations (via exceptional isomorphisms of Spin groups) with $K$-theoretic techniques to derive concrete embedding results, representing an improvement on previously known codimension bounds for $RP^7$. The methods have potential implications for studying spin structures on complex projective spaces and related embedding problems.
Abstract
We generalize the $α$-invariant introduced by Atiyah and Rees to an invariant of real spin bundles and use it to classify real bundles over $\mathbb{C}\mathrm{P}^3$ admitting spin structure. We apply this result to show that $\mathbb{R}\mathrm{P}^7$ can be smoothly embedded in $\mathbb{R}\mathrm{P}^{11}$.