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At least in the case when $\pi_1(B)=0$, $\mathfrak{o}(f)$ is just the first non-trivial differential, $d_k$ in disguise (let's work over some field, for simplicity; then $d_k\in\operatorname{Hom}(H^k(F),H^{k+1}(B))\cong H^{k+1}(B)\otimes H_k(F)\ni \mathfrak o(f)$).

Reference (well, kind of: it doesn't even give precise statement, let alone proof): Mosher, Tangora. Cohomology operations and applications in homotopy theory (pp. 103, 109).

I'm afraid I can't say anything about non-simple case, though (not even sure what is the correct statement in this case).
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At least in the case when $\pi_1(B)=0$, $\mathfrak{o}(f)$ is just the first non-trivial differential, $d_k$ in disguise (let's work over some field, for simplicity; then $d_k\in\operatorname{Hom}(H^k(F),H^{k+1}(B))\cong H^{k+1}(B)\otimes H_k(F)\ni \mathfrak o(f)$).
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$\mathfrak{o}(f)$ is just the first non-trivial differential, $d_k$ in disguise (let's work over some field, for simplicity; then $d_k\in\operatorname{Hom}(H^k(F),H^{k+1}(B))\cong H^{k+1}(B)\otimes H_k(F)\ni \mathfrak o(f)$).