Fiber Sequences and Suspension Methods

Opening obstruction

A single encode–decode computation does not scale to higher homotopy groups. Fiber sequences and connectivity organize calculations across dimensions.

Development contract

For a pointed fiber sequence $F\to E\to B$, construct the connecting maps and prove exactness of ${\pi }_{k+1}(B)\to {\pi }_k(F)\to {\pi }_k(E)\to {\pi }_k(B)$ in the justified range. If $A$ and $B$ are respectively $m$- and $n$-connected, prove that $A\ast B$ is $(m+n+2)$-connected; deduce suspension connectivity and the $2n$-connected Freudenthal map $X\to \mathrm{\Omega }\mathrm{\Sigma }X$ for $n$-connected $X$.