what is the image of $\partial( 1_{S^n})$ for the exact sequence for the fibration $X \to E \to S^n$

Question

what is the image of $\partial 1_{S^n}$ where $\cdots \pi_n(S^n)\rightarrow \pi_{n-1}(X) \rightarrow \pi_{n-1}(B)\rightarrow\cdots$

Answer 1

Any fibration $X \to E \to S^n$ up to fiber homotopy equivalence is given by the "clutching construction" applied to a map $f: S^{n-1} \to G(X)$, where $G(X)$ denotes the topological monoid of self homotopy equivalences of $X$. Let $e: G(X) \to X$ be the evaluation map (evaluate an equivalence at the basepoint of $X$). The answer to your question is given by taking the homotopy class of the composition $$ S^{n-1} \overset{f}\to G(X) \overset{e} \to X . $$ Note on the clutching construction: Given $f$ as above, form $E = (D^n_- \times X) \cup (D^n_+ \times X)$ where the gluing is done along the equivalence $S^{n-1} \times X \to S^{n-1}\times X$ given by $(x,v) \mapsto (x,g(x)(v))$. The first factor projection $E \to S^n$ gives your fibration up to fiberwise equivalence. Small technical point: clutching doesn't give a fibration necessarily (it is what one calls a quasifibration), so we have to convert the map to a fibration in the usual way. However, if we had started with a map $S^{n-1} \to \text{homeo}(X)$ it would clutch to a fiber bundle.