Timeline for Classification of oriented vector bundles of rank 5 over closed oriented 5-manifolds

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Dec 21, 2018 at 10:19	comment	added	Panagiotis Konstantis		Geometrically, on a vector bundle $V$ of rank $n$ over an $n$-manifold $M$, you always can "add" a vector bundle $E\to S^n$ to $V$ in the following sense: Pinch of a small neighboorhood of a point in $M$ and you obtain a map $M \to M\vee S^n$. You can postcompone this map with $\phi\vee \psi$ where $\phi$ is the classifying map of $V$ and $\psi$ that of $E$. For $n=5$ there is only the tangent bundle of the sphere to add.
Dec 21, 2018 at 10:12	comment	added	Panagiotis Konstantis		Moreover, you are right with your last comment, there are at most two isomorphism classes if one fixes a stable class of a vector bundle. In my preprint I show that it depends if $w_4$ if there the stable class contains only one or two isomorphism classes.
Dec 21, 2018 at 10:08	comment	added	Panagiotis Konstantis		Thank you for your detailed comments! What I did in my preprint, if I understand this right, is almost the same as you said, however I considered the infinite quaternionic projective spaces instead the Grassmanian. The short exact sequence, which I am using is described in your fifth observation. What you asking, namely to determine the image of $\pi_0[M^4,\tilde G_5] \to \pi_0[S^5,\tilde G_5]$ is probably "my" $\kappa$-invariant in case one substitutes $\tilde G_5$ with $\mathbb H P^\infty$.
Dec 20, 2018 at 17:57	history	answered	Ian Agol	CC BY-SA 4.0