I’m interested in understanding the importance of the local coefficients in the definition of the obstruction cocycle for a lift of a map $f\colon X \to B$ along a fibration $p \colon E \to B$. I’m following the explanation given at page $189$ in Davis & Kirk’s Lecture Notes in Algebraic Topology. Aim of this […]

Consider the Hopf bundles $$S^1\rightarrow S^{2n+1}\rightarrow \mathbb{C}P^n$$ and $$S^3\rightarrow S^{4n+3}\rightarrow \mathbb{H}P^n.$$ In this question (and also here), it is shown that for any continuous map $f:X\rightarrow \mathbb{C}P^n$, there is a lift $\tilde{f}:X\rightarrow S^{2n+1}$ iff $f^\ast:H^2(\mathbb{C}P^2)\rightarrow H^2(X)$ is the $0$ map. I’d like to know a similar characterization for the quaternionic Hopf bundles. Suppose $f:X\rightarrow \mathbb{H}P^n$ […]

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