= Freudenthal suspension from two cones
{c}
Write the suspension of an $(n-2)$-connected based <CW complex> as two cones meeting in the original space. Each cone pair is $(n-1)$-connected because its relative <groups> shift the original <homotopy groups> by one. <Homotopy excision theorem> compares the cone pair with the suspension relative to the other cone, giving the suspension <homomorphism> as an <isomorphism> for $i<2n-2$ and a surjection at $i=2n-2$ in the indexing $\pi_{i-1}X\to\pi_i\Sigma X$.
Back to article page
